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Noisy intermediate-scale quantum computers
Bin Cheng1,2,3, Xiu-Hao Deng1,2,3, Xiu Gu1,2,3, Yu He1,2,3, Guangchong Hu1,2,3, Peihao Huang1,2,3,
Jun Li1,2,3, Ben-Chuan Lin1,2,3, Dawei Lu1,2,3,4, Yao Lu1,2,3, Chudan Qiu1,2,3,4, Hui Wang5,6,7,
Tao Xin1,2,3, Shi Yu1,2,3, Man-Hong Yung1,2,3, Junkai Zeng1,2,3, Song Zhang1,2,3,
Youpeng Zhong1,2,3, Xinhua Peng6,7,8, Franco Nori9,10, Dapeng Yu1,2,3,4,†
1Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology,
Shenzhen 518055, China
2International Quantum Academy, Shenzhen 518048, China
3Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and
Technology, Shenzhen 518055, China
4Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
5Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei 230026, China
6CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei 230026, China
7Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
8CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China
9Quantum Computing Center and Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan
10Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA
Corresponding author. E-mail: †yudp@sustech.edu.cn
Received December 18, 2022; accepted January 2, 2023
©The Authors 2023
ABSTRACT
Quantum computers have made extraordinary progress over the past
decade, and significant milestones have been achieved along the path
of pursuing universal fault-tolerant quantum computers. Quantum
advantage, the tipping point heralding the quantum era, has been
accomplished along with several waves of breakthroughs. Quantum
hardware has become more integrated and architectural compared to
its toddler days. The controlling precision of various physical systems
is pushed beyond the fault-tolerant threshold. Meanwhile, quantum
computation research has established a new norm by embracing industrialization and commercialization. The joint power
of governments, private investors, and tech companies has significantly shaped a new vibrant environment that accelerates
the development of this field, now at the beginning of the noisy intermediate-scale quantum era. Here, we first discuss the
progress achieved in the field of quantum computation by reviewing the most important algorithms and advances in the
most promising technical routes, and then summarizing the next-stage challenges. Furthermore, we illustrate our confidence
that solid foundations have been built for the fault-tolerant quantum computer and our optimism that the emergence of
quantum killer applications essential for human society shall happen in the future.
Keywordsquantum computer, quantum algorithm, quantum chip
*Special Topic: Embracing the Quantum Era: Celebrating the 5th Anniversary of Shenzhen Institute for Quantum Science and Engineering
(Eds.: Dapeng Yu, Dawei Lu & Zhimin Liao).
REVIEW ARTICLE
Volume 18 / Issue 2 / 21308 / 2023
https://doi.org/10.1007/s11467-022-1249-z
Contents
1 Introduction 2
2 Quantum algorithms 4
3 Superconducting qubits 6
4 Trapped-ion qubits 11
5 Semiconductor spin qubits 15
6 NV centers 20
7 NMR system 22
8 Neutral atom arrays 25
9 Photonic quantum computing 28
10Outlook and conclusion 31
Acknowledgements 32
Author contributions 32
References 32
1Introduction
Quantum computing exploits phenomena of quantum
nature, such as superposition, interference, and entangle-
ment, to provide beyond-classical computational
resources. Its ultimate goal is to build a quantum
computer that can be significantly more powerful than
classical computers in solving certain tasks. Historically,
quantum computing dates back to the early 1980s, when
Benioff developed a quantum-mechanical model of the
Turing machine [1], and Feynman [2] and Manin [3]
proposed the idea of harnessing the laws of quantum
mechanics to simulate phenomena that a classical
computer could not feasibly do. In 1994, Shor devised an
efficient quantum algorithm for finding the prime factors
of an integer, a very concrete and important problem for
which no efficient classical algorithm is known [4]. Shor’s
algorithm, along with a number of other quantum algo-
rithms [5], strengthened the foundations of quantum
computing, inspired the community of quantum physicists
and stimulated research in finding actual realizations of
quantum computing. The first implementation scheme
came in 1995, when Cirac and Zoller made a proposal
for quantum logic gates with trapped ions [6]. In the
following years, other physical routes to realize quantum
computing, such as nuclear magnetic resonance (NMR)
[7–9], spin qubits [10, 11] and superconducting qubits
[12], were proposed, and there has been substantial
experimental progress in the area since then. Several
hardware platforms, including cavity quantum electrody-
namics systems, ion traps, and NMR, have successfully
realized more than one qubit in experiments since the
start of this century. In the following decade, various
platforms have achieved quantum information processing
on small-scale quantum systems composed of several
qubits. In recent years, the field has advanced to the
point where research groups have been able to demonstrate
quantum devices at a scale around or even beyond forty
qubits, particularly in trapped ions and superconducting
circuits.
Remarkable progress has been achieved toward fault-
tolerant quantum computing. In the beginning, as Fig. 1
shows, universal quantum gates and precise readout
have been realized in various physical qubit systems and
demonstrated the fulfillment of DiVincenzo criteria.
Hardware-level developments and the progress in fabri-
cation further enable the integration of qubits. These
achievements enable excessive trials of prototype demon-
Fig.1Quantum computing development levels. The left panel illustrates the three development stages of quantum
computing with some iconic progress classified as the physical and logical levels. The right panel lists some potential applications
according to different stages. Detailed discussion about this diagram could be referred to Section 1.
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stration of quantum computing, including analog/digital
quantum simulation, quantum error correction (QEC),
fault-tolerant quantum operations, quantum algorithms,
etc. Google achieved quantum supremacy using random-
ized circuit sampling on their 53-qubit sycamore processor
[13]. Afterward, several “quantum advantage” experi-
ments, including superconducting systems [14, 15] and
photons [16–18] have been realized, and the gap between
the computational power of quantum computers and
their classical counterparts was greatly widened.
Another milestone is that the realization of quantum
annealing in commercialized quantum machines triggers
the industrialization of quantum computing. Efforts
have been made to develop specialized quantum computers
for certain tasks, such as the D-wave annealing machines
[19] and the aforementioned photonic boson sampling
circuits [16–18]. Moreover, practical QEC has been
explored in various physical systems, such as supercon-
ducting circuits [20–24], ion traps, semiconductors [25,
26], and nitrogen–vacancy (NV) centers in diamond
[27–29]. Considering its speeding-up strides, the break-
through toward universal fault-tolerant quantum compu-
tation is closely tangible. Another noteworthy achievement
is the construction of functional quantum simulators
[30, 31], digitally or analogously, towards practical prob-
lems in quantum chemistry [32] and condensed matter
physics [33].
99%
50
100
99.4%
With ever-increasing abilities to precisely manipulate
quantum-mechanical systems, the quantum computing
community has been shifting the focus from laboratory
curiosities to technical realities, from investigating the
underlying physics to solving the engineering problems
in building a scalable system, from searching for a well-
behaved qubit to seriously addressing the question of
how to make our near-term quantum hardware practically
useful. During the first decade of the 21st century,
superconducting qubits, the leading candidate for building
scalable quantum computers, were used to demonstrate
prototype algorithms with no more than two
programmable qubits in most cases. Many efforts have
been spent on proof-of-principle tests of various hardware
modules. In 2014, two-qubit gate fidelities, an overall
performance metric that evaluates the degree of control
of a quantum processor, greater than were achieved
for the first time in a multi-qubit superconducting
circuit, surpassing the error-correction threshold [34].
Since then, the community has seen a trend of growing
system size, with – qubits integrated into state-of-
the-art processors. It is remarkable that the average
fidelities across these processors are also advancing; In
Google’s 53-qubit processor, an average of two-
qubit gate fidelity was achieved with simultaneous oper-
ations across the chip [13]; such enhanced reproducibility
indicates immense engineering efforts in all aspects of
the experiment, including design, fabrication, wiring,
electronics, and software.
Along this grand trend, we have already entered the
second stage of quantum computing—noisy intermediate-
scale quantum (NISQ) [121] and cloud service of quantum
computers, as shown in the left panel of Fig. 1. NISQ for
quantum computing is analogous to the early stage of
classical quantum computers, when analog/digital
signals are hybrid as the exploration of the limit of
information processing and the applications of the
computing tasks are limited to several areas. During this
stage, logic qubits and operation might reach the break-
even point by encoding a limited but enough number of
noisy qubits in medium-sized integrated systems. As a
result, demonstrations of quantum algorithms can be
performed using a small amount of logic qubits. And
further quantum advantage utilizing quantum computa-
tional algorithms or quantum simulation would also be
demonstrated with applications on quantum chemistry,
variational quantum computers, quantum machine
learning, or quantum optimization. Eventually, it is
generally believed that fault-tolerant universal quantum
computers would be realized in large-scale and integrated
quantum systems.
In addition to the advances in hardware, commercially
valuable algorithms and applications are beginning to
burgeon [35]. A typical example is the variational quantum
eigensolver (VQE) algorithm [36], which is shown to be
worthwhile eventually from a two-atom molecule calcu-
lation to bigger quantum systems. Algorithms for
general purposes, in a similar spirit to the forerunner
textbook algorithms —Shor's algorithm and Grover’s
algorithm [5], were developed recently [37], such as the
Harrow–Hassidim–Lloyd (HHL) algorithm [38] and the
quantum singular value transformation (QSVT) algorithm
[39].
The paradigm of quantum computing research has
been revolutionized over the years from solely academic
research. Nowadays, the great impetus comes not only
from its intrinsic scientific interest, but also from
companies and societies [40]. With the aforementioned
tremendous progress, the approach to full-stack quantum
computing [40–42] approach is encouraging. As commer-
cialization is becoming a trend, many large companies
and start-ups are contributing to this field jointly with
the scientific community. We shall see further contributions
and incentives to the development of this field coming
from commercialization, as has already happened for
classical computers, genetic technology, and artificial
intelligence. From a broader society scope, cloud quantum
computing, such as IBM’s quantum network, makes it
possible for global users around the world to explore new
quantum algorithms without their own hardware devices.
As quantum computing lessons and experiences in
schools and universities are becoming a routine for the
next generation, more well-educated engineers and scien-
tists are being enrolled in the field, equipped with
insights and knowledge of quantum science. Thus, the
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-3
positive feedback from society is creating a new norm for
quantum computing research compared to its primitive
days.
In this review, we will focus on hardware platforms
that have the potential to realize the ultimate large-scale
quantum computers, including superconducting circuits,
trapped ions, semiconductors, neutral atoms, NMR,
photonics, and NV centers. In particular, we will focus
more on the important advances in these platforms over
the past decade. By following the guidelines of DiVin-
cenzo’s criteria [43], we will introduce how to implement
the key elements of quantum computing in each physical
system and their typical features and advantages. The
scalability of each platform and critical challenges in
recent developments will also be discussed here. More-
over, recent progress on quantum algorithms will also be
mentioned in this review. By combining fast-developing
hardware platforms and potential applications, we hope
to shed light on the innovations that quantum computing
can bring in the foreseeable future.
Topological quantum computation provides another
approach to tackling quantum errors by keeping the
computational states to the desired pure quantum states
without erroneous results. A typical type of topological
qubits is made of Majorana zero modes, which are
immune to environmental noise and thus overcome the
inevitable decoherence at the hardware level through the
Majorana non-locality and braiding operations. However,
the non-topological in-gap states or trivial zero-energy
states can also mimic the typical Majorana behavior,
making the detection and other operations of Majorana
zero modes difficult. So far, the non-locality and braiding
operations to demonstrate the non-Abelian statistics
have yet to be proved before the realization of topological
qubits. A complete discussion of topological quantum
computation is beyond the scope of this review. The
interested reader is referred to the literature for further
details [44–46].
2Quantumalgorithms
f:{0,1}n→ {0,1}
2n
Introduction.— It is anticipated that quantum computers
utilizing the exotic quantum features can solve computa-
tional problems more efficiently than their classical
counterparts. For example, in the query model, given an
oracle access to a function , a classical
computer can only query it once at a time, whereas a
quantum computer can query the oracle once and obtain
all values simultaneously, a phenomenon known as
quantum parallelism. Formally,
x|x⟩|0⟩ 7→
x|x⟩|f(x)⟩(1)
can be achieved on a quantum computer. However,
quantum parallelism alone is not useful because when
one performs a measurement, the quantum state
collapses, and only one bit of information can be
obtained. To design quantum algorithms, quantum
parallelism needs to be combined with other features
such as interference and entanglement.
f:{0,1} → {0,1}
f(0) = f(1)
In 1985, Deutsch [47] combined quantum parallelism
with interference to design the first quantum algorithm
that can solve a black-box problem with fewer queries
than a classical computer. Specifically, in Deutsch’s
problem, one is given a function and
asked whether the function is constant, that is,
, or not. Classically, we would need two
queries to solve this problem; but with quantum
computers, only one query is needed. Later, it was
generalized to a multi-qubit version called the
Deutsch–Jozsa algorithm, which can achieve an expo-
nential speedup over any classical deterministic algorithms
[48]. However, the quantum speed-up vanishes in the
presence of a small error probability. In 1993, Berstein
and Vazirani proposed another problem and designed a
quantum algorithm for it that can achieve polynomial
speedup even over classical randomized algorithms [49].
After one year, Simon strengthened their result by
designing Simon’s problem and a quantum algorithm for
it that yields an exponential speedup [50].
These early-stage explorations focused mostly on the
search for problems that quantum computers can solve
more efficiently than classical computers, instead of
focusing on real-world applications. But interestingly, as
it turned out later, Simon’s algorithm inspired Shor to
design quantum algorithms to solve discrete logarithmic
and integer factoring problems [4], which are widely
used in cryptography.
Quantum Fourier transform.—In the next stage of the
development of quantum algorithms, several quantum
algorithmic primitives emerged and appeared to be
extensively used in designing new quantum algorithms.
One such primitive is the quantum Fourier transform
(QFT), which implements the Fourier transform matrix
(FN)jk := ωj k
N/√N(2)
N:= 2n
ωN:= e2πi/N
N
N= 2
ωN=−1
FN
ZN
N
{0,1,·· · , N −1}
Zn
2
with a polynomial-sized quantum circuit on a quantum
computer, where and is the -th
root of unity. Simon’s algorithm uses a special instance
of QFT, namely the Hadamard transform, which corre-
sponds to the case and . From a group-
theoretic point of view, is the Fourier transform over
, the additive group of integers modulo , consisting
of elements ; Simon’s Hadamard transform
is the Fourier transform over [51].
ZN
There are two steps in Shor’s factoring algorithm, a
classical polynomial-time reduction from integer factoring
to period finding, followed by an efficient quantum algo-
rithm for solving the period finding problem [4], which
uses QFT over . Combining these two steps, Shor
obtained a polynomial-time quantum algorithm for solving
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ϕ
U|ψ⟩=ei2πϕ |ψ⟩
integer factorization, which has super-polynomial
speedup over the best classical algorithm. Kitaev gave a
generalized QFT over an arbitrary finite Abelian group,
with which he designed a polynomial-time quantum
algorithm for finding the stabilizer of an Abelian group;
the Abelian stabilizer problem includes integer factoring
and discrete logarithm as special instances [52]. It is
worth mentioning that Kitaev [52] also gave the phase
estimation algorithm in the same paper, which estimates
the phase in and can be used to solve
the period finding problem. In a coherent picture, all
these problems belong to the hidden subgroup problems
category [51, 53].
f:{0,1}n→ {0,1}
x0
f(x0) = 1
x0
Ω(2n)
x0
O(√2n)
Quantum search.—The second primitive starts from
Grover’s search algorithm [54, 55], which concerns
searching over an unsorted database for a target.
Formally, given a function and the
promise that there is exactly one such that ,
the search problem is to find the target . Since there is
no structure in this problem, a classical algorithm will
need times of queries to find the target with
sizable probability. Grover’s algorithm allows a quantum
computer to find the target with queries to the
database, which achieves a quadratic speedup over classical
computation. Grover’s algorithm repeatedly applies the
Grover iterate
G= (2|u⟩⟨u| − I) (I−2|x0⟩⟨x0|),(3)
|u⟩:= 1
√Ny|y⟩
I−2|x0⟩⟨x0|
Ω(√2n)
A
|0n⟩
1/p
O(1/√p)
which is the product of two reflections; here,
is the uniform superposition and
is the quantum query operator. Grover’s
algorithm is optimal in the sense that any quantum
algorithm that solves this problem will take at least
queries [56]. Grover’s algorithm can be extended
to amplitude amplification, which can handle the case of
multiple numbers of targets [57–59]. More precisely,
given a quantum (or classical) algorithm applied to
that can output a correct target when measured
with probability p, it would require running the algorithm
times to obtain the targeted result. But amplitude
amplification can obtain the target in time ,
which is also a quadratic speedup. The fixed-point
version of Grover’s algorithm or amplitude amplification
can even handle the case when the number of targets is
unknown [60–62]. Grover’s algorithm has inspired more
applications than Shor’s algorithm, as it can be used to
speed up the search subroutine for solving many opti-
mization problems [63–67].
One may consider the search problem in an alternative
paradigm, namely, the Markov chains or random walks.
The quantum version of random walks includes the
continuous-time quantum walk [68–70] and the discrete-
time quantum walk; we focus on the latter here. The
framework of the discrete-time quantum walk was incre-
mentally developed in several works [71–77]. Later, this
framework was applied to obtain a different formulation
of Grover’s search algorithm [78]. In a breakthrough
work, Ambainis designed a quantum walk algorithm for
element distinctness [79] that achieves better query
complexity than a direct application of Grover’s algorithm
and matches the theoretical lower bound [80]. Ambainis’
result was generalized subsequently [81, 82], and, in
particular, Szegedy gave a general framework for quan-
tizing classical Markov chains [82], which was further
improved in [83]. This quantum walk-based search algo-
rithm finds many applications [84, 85], including triangle
finding [86], testing group commutativity [87], etc.
e−iHt
H
H=H1+H2
H1
H2
Hamiltonian simulation.—The third primitive that
will be discussed in this review is Hamiltonian simula-
tion, which approximates the time evolution operator
of a Hamiltonian on a quantum computer. In
fact, Hamiltonian simulation is one of the initial motiva-
tions for developing quantum computing [2]. The first
quantum algorithm for implementing the time evolution
operator is given by Lloyd [88], which is based on the
Lie-Trotter formula. For example, suppose that we have
a local Hamiltonian such that the time
evolution of the local terms and can be efficiently
implemented on a quantum computer. The Lie–Trotter
formula gives
e−iHt = (e−iH1t/se−iH2t/s)s+O(t2/s),(4)
e−iHt
H1
H2
t/s
H
t
which means that can be implemented by alternating
and over an incremental time . One can also
use the higher-order formula [89, 90] to approximate the
time evolution of . The general scheme is called the
product-formula approach, or Trotterization, which was
later applied to simulate a sparse Hamiltonian [91–93].
Later, a Hamiltonian simulation method based on quantum
walk [94, 95] was proposed, which achieved linear gate
and query complexities in the evolution time , matching
the lower bound imposed by no fast-forwarding theorem
[92].
Another important approach to simulate Hamiltonian
dynamics is using a linear combination of unitaries
(LCU) [96], and it is shown to have the optimal dependence
on the simulation precision [97, 98]. The LCU approach
is combined with the quantum walk approach to give an
algorithm that has optimal dependence on all parameters
of interest, such as precision, sparsity of the Hamiltonian,
and the evolution time [99]. Moreover, there is a subroutine
used in the LCU approach that was later named block
encoding, which turns out to provide a versatile toolkit
in designing quantum algorithms. In a series of works,
Low et al. [100–103] gave improved Hamiltonian simulation
algorithms based on block encoding and quantum signal
processing. The idea is to encode the Hamiltonian as a
block of a unitary, and then apply the polynomial trans-
formation to the Hamiltonian using the quantum signal
processing technique [100]. This method was further
generalized to a framework called QSVT [104], which
covers most existing quantum algorithms as special cases,
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achieving a grand unification of quantum algorithms
[105]. However recently, an in-depth analysis of the
Trotter error showed that the product-formula approach
can achieve a competitive scaling of gate complexity
compared to other approaches [106].
A
b
Ax=b
x
|b⟩
b
|x⟩=A−1|b⟩
x
|b⟩
k
Quantum linear algebra and quantum machine learn-
ing.— The previous primitives can be combined to
design new quantum algorithms. Here, we discuss quantum
algorithms for linear algebra and machine learning.
Quantum linear algebra starts with the HHL algorithm,
named after Harrow, Hassidim, and Lloyd [38]. The
problem they considered is to solve linear systems of
equations; that is, given a matrix and a vector ,
solve for . Given a quantum state that
encodes the vector in its amplitudes, HHL uses Hamil-
tonian evolution and phase estimation to approximately
prepare the state . Provided that the whole
description of the solution is not required and that the
state can be prepared, the HHL algorithm can achieve
exponential speedup over any classical algorithm [38].
HHL was applied to many quantum machine learning
algorithms to obtain exponential quantum speedup,
including quantum -means clustering [107], quantum
principal component analysis [108], quantum support
vector machine [109], quantum data fitting [110], etc.;
see Ref. [111] for a review of these algorithms.
However, it is not clear whether such an exponential
quantum speedup is artificial or not. Specifically, these
quantum machine learning algorithms typically made
strong input assumptions, such as quantum random
access memory (QRAM) with access to the classical
data [112]. It might be possible to derive efficient classical
algorithms in an analogous setting. In 2018, the break-
through work by Tang [113] gave a classical algorithm
that dequantizes the quantum algorithm for recommen-
dation systems [114], which was previously believed to
have an exponential speedup, with only a polynomial
slowdown. Tang’s result stimulated a series of subsequent
work on dequantizing various quantum machine learning
algorithms, such as those for principal component analysis
[115], solving low-rank linear systems [116, 117], and
solving low-rank semidefinite programming [118]. The
sample and query access model [115] to the input data is
assumed in those works, which is a classical analogue to
the input assumptions in many quantum machine learning
algorithms. Since the QSVT provides a primitive for
unifying quantum algorithms, especially quantum linear
algebra, these dequantization results were later extended
to a unifying framework by dequantizing the QSVT [39].
Therefore, whether exponential quantum speedup can be
achieved in machine learning is still under debate.
Variational quantum algorithms.—Apart from quan-
tizing machine learning algorithms with HHL, another
exploration is inspired by neural networks, which are
variational quantum algorithms. Variational quantum
algorithms are hybrid quantum algorithms that prepare
parameterized quantum states on a quantum computer
and use classical computers to optimize the parameters.
The first variational quantum algorithm is the VQE [36],
designed to tackle quantum chemistry problems. Its goal
is to find the ground state and ground energy of local
Hamiltonians. Before VQE, a common approach was to
use quantum phase estimation [119, 120]. However, such
an approach, just like other quantum algorithms,
imposes a stringent coherence requirement on the quantum
devices, which is challenging in the current NISQ era
[121].
Since VQE, more works have been done in this direc-
tion. Inspired by quantum adiabatic algorithm [122],
Farhi et al. [123] proposed the quantum approximate
optimization algorithm (QAOA) for solving combinatorial
optimization problems such as the max-cut problem.
The third family of variational quantum algorithms is
the quantum neural networks, which aims to solve
machine learning problems such as classification [124,
125] and generative modeling [126, 127].
In the current NISQ era, although we have demonstrated
quantum computational supremacy in various models
[13, 15, 128, 129], these problems are not designed to be
of practical relevance. Variational quantum algorithms
are regarded as promising approaches for demonstrating
“killer applications” on quantum computers. Such appli-
cations might appear in various areas including quantum
chemistry, material science and biological science. For
example, in quantum chemistry, VQE can be used to
compute the low-energy eigenstates of electronic Hamil-
tonians, which helps understand chemical reactions and
design new catalysts [130]. As for biological science,
optimization is often involved in many fields like
sequence analysis and functional genomics [131]. This
opens opportunities for potential quantum speedup by
using quantum neural networks, and quantum variational
auto-encoders [132], etc. However, there is a long way to
go along this path and continuous efforts should be put
into the study of variational quantum algorithms. More-
over, to make it practical for the neat-term quantum
devices, perhaps error mitigation techniques are also
required [133–137].
3Superconductingqubits
10
kBT/h∼0.2
Introduction.—Superconducting qubits are nonlinear
superconducting circuits based on Josephson junctions,
with quantized electromagnetic fields in the microwave
frequency domain (typically 0.1–12 GHz). They operate
at cryogenic temperatures (~ mK; equivalent to
GHz) provided by dilution refrigerators in
order to suppress thermal fluctuations. Superconducting
qubits recently emerge as a leading platform for scalable
quantum information processing. Some recent milestones
include the demonstration of quantum supremacy using
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a 53-qubit superconducting quantum processor [13],
which is further strengthened with a 66-qubit processor
[128]. Offering scalable high-fidelity control and config-
urable interactions, superconducting circuits have
become a versatile playground for quantum computational
tasks [125, 128, 138–141], quantum simulation [142–150],
quantum annealing [19, 151], quantum chemistry
[152–155], exotic many-body physics [156–161], new
regimes for light–matter interaction [162–165], quantum
sensing [166, 167] and studying biological processes
[168].
Some facts about superconducting qubits are summa-
rized in Fig. 2(a), and a list of excellent reviews on
superconducting qubits can be found in Refs. [169,
173–190]. The charge carriers in superconductors, known
Ic
CJ
LJ= Φ0/(2π)Iccos ϕ
Φ0=h/(2e)
ϕ
as Cooper pairs, can flow without dissipation, a desirable
feature for preserving quantum coherence of a macroscopic
system. More importantly, non-trivial quantum properties
emerge from the integration of a special superconducting
circuit element, the Josephson junction, which is usually
in the form of a sandwich structure consisting of two
superconducting electrodes separated by a nanometer-
thick insulating layer [Fig. 3(a)]; Cooper pairs can
tunnel through the insulating barrier with a supercurrent
no larger than the critical current of the junction
which depends on the material, thickness, and size [191,
192]. From a circuit point of view, a Josephson junction
can be modeled as a native capacitor in parallel
with a nonlinear inductor , where
is the superconducting flux quantum and is
F1
F2
Fig.2Schematic summary of different types of quantum bits (top half) and their corresponding pros and cons. (bottom
half). ( ) is the one-qubit (two-qubit) gate fidelity.
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-7
EJ= Φ0Ic/(2π)
EC=e2/(2CJ)
the superconducting phase difference across the junction.
Two characteristic parameters of a Josephson junction
are its Josephson energy and the charging
energy .
EC=
e2/2(CS+CJ)
EJ/EC>50
Qubit construction.—There have been numerous
explorations of how to construct a superconducting
qubit using Josephson junctions. Traditionally, super-
conducting qubit designs are categorized into charge
[193], flux [194, 195] and phase qubits [196]; all are
successful in many early demonstrations [12, 197–208].
In recent years, the transmon qubit [209] and a modified
version for implementing scalability, i.e., the Xmon
qubit [171] have become popular. These modified designs
of the charge qubit shunt a Josephson junction with a
large capacitor CS to strongly suppress their sensitivity
to the charge fluctuations [210]. Typically, this shunt
capacitor lowers the effective charging energy
to the regime of [Fig. 3(b)]. The
fact that the transmon design has the simplest possible
circuit geometry makes it more tolerant of fabrication
variations and excellent at reproducibility. The Hamilto-
nian of the transmon qubit is the same one for the
charge qubit and can be expressed as
H= 4ECn2−EJcos ϕ, (5)
n
n
ϕ
[ϕ, n] = i
where is the number of Cooper pairs that traverse the
junction; and satisfy the commutation relation
. Note that the Hamiltonian is identical to the
CS≫CJ
ϕ
one that describes a quantum particle in a one-dimensional
potential [Fig. 3(c)]. In the limit, the low-
energy eigenstates are, to a good approximation, localized
states in the potential well, and the superconducting
phase is small. We can therefore expand the potential
term into a power series:
−EJcos ϕ=1
2EJϕ2−1
24EJϕ4+O(ϕ6).(6)
ℏω10
ℏω21
|1⟩
|2⟩
|0⟩
|1⟩
|0⟩
|1⟩
The first quadratic potential term leads to a quantum
harmonic oscillator with equidistant energy levels ,
whereas the second quartic potential arising from the
Josephson nonlinearity introduces anharmonicity to the
level structure, allowing the transition energy
between first-excited state and second excited state
to be different from that between the ground state
and first-excited state . This nonlinearity allows one to
define a qubit in the computational subspace consisting
of the lowest two energy levels and only. The
design may be further modified by replacing a single
junction with a pair of junctions so that the effective
Josephson energy, and consequently the qubit frequency,
can be tuned by adjusting the magnetic flux threading
the two-junction loop.
The transmon design can be implemented with a
qubit circuit embedded in a three dimensional cavity
[Fig. 3(d)] and in the form of lithographically defined
circuits based on superconducting materials such as
LJ
CJ
CS
CS≫CJ
Fig.3(a) Schematic of a Josephson junction composed of two superconductors separated by a thin insulating layer
through which Cooper pairs can tunnel. Reproduced from Ref. [169], Springer Nature Limited. (b) Circuit diagram of a
transmon qubit consisting of a Josephson junction (Josephson inductance , self-capacitance ) and a shunt capacitor
( ). (c) Potential profile and level diagram of the transmon qubit, a quantum anharmonic oscillator. (d) Image of a
transmon qubit embedded in a three dimensional cavity. Reproduced from Ref. [170]. (e) Image of a planar transmon qubit.
Reproduced from Ref. [171]. (f) Photograph of the Sycamore quantum processor. Reproduced from Ref. [13], Springer Nature
Limited. (g) Device schematic of the Zuchongzhi quantum processor. Reproduced from Ref. [128]. (h) Photograph of a
modular quantum processor consisting of two nodes. Reproduced from Ref. [172], Springer Nature Limited.
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21308-8 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
π
aluminum and niobium [Fig. 3(e)] and many other variants
[170, 171, 211–213]. Qubit designs with alternative
topology such as capacitively shunted flux qubit [210,
218–221], fluxonium [214–217], and 0– qubit [222] have
also been under active development and shown promising
progresses. By engineering the energy-level spectra and
the coupling matrix elements, some of these designs
have a better-defined two-level system and intrinsic
protection against external perturbations at the cost of
increased circuit complexity. The remarkable flexibility
in configuring the Hamiltonian offers a rich parameter
space to search for desired qubit properties and therefore
gives superconducting qubits the name “artificial
atoms”.
Readout and initialization.—Having a well-defined
two-level system is not enough for quantum computing;
the ability to faithfully measure and initialize the qubit
is also indispensable. The prevailing technique for
discerning the qubit state is the dispersive readout
scheme. Utilizing the cavity or circuit quantum electro-
dynamics (cQED) architecture, a qubit is strongly
coupled to but sufficiently detuned from a readout
resonator [223, 224]; the qubit induces a state-dependent
shift in the resonator frequency from which the qubit
state can be inferred by interrogating the resonator. The
cQED scheme has been successful in achieving fast, high-
fidelity, non-demolition readout, assisted by a list of
technologies that have been invented around this
approach. To avoid extra decoherence introduced by the
readout resonator, a Purcell filter can be placed between
the resonator and the external circuitry to reshape the
environmental mode density seen by the qubit and the
resonator [225–228]; in this way, one may enhance the
readout speed while inhibiting qubit relaxation. In addi-
tion, the use of Josephson parametric amplifiers (JPA)
[229–237] at the beginning stage of the readout signal
amplification can also bring an immediate improvement
to the measurement fidelity. It is noteworthy that other
techniques including multiplexed readout [238], multilevel
encoding [239], and photon counting method [240] also
help improve the measurement efficiency and scalability.
Between consecutive measurements, superconducting
qubits are typically initialized by simply waiting for the
qubit to relax to its ground state. Various conditional
and unconditional reset techniques have been developed
for superconducting qubits to accelerate this process
[241–245].
Gates.—Controlling superconducting qubits is chal-
lenging because performing a quantum logic or unitary
operation is fundamentally an analog process governed
by the Schrodinger equation and the realistic Hamiltonian
is often far from ideal. A single qubit XY operation,
rotation around an axis in the XY plane in the Bloch
sphere, is commonly implemented by driving the qubit
with a resonant microwave pulse. For weakly anharmonic
qubits such as the transmon, the resonant drive can
induce unwanted leakage to higher excited states and
additional phase errors; the derivative-removal-by-adia-
batic-gate (DRAG) scheme, which adds additional
quadrature components to the pulse, has become a
routine in pulse calibration to combat these coherent
errors at no additional hardware cost [246, 247]. The
single-qubit phase gate or Z gate, rotation around the Z
axis, can be realized either by combining XY rotations
or by adding a physical Z pulse provided that the qubit
frequency is adjustable or by performing the more efficient
virtual Z gate through shifting the phases of XY rotations
[248]. Heat dissipation is another important concern for
cryogenic experiments when scaling to a large number of
qubits; a more energy-efficient approach for single-qubit
operations using single-flux-quantum (SFQ) circuits has
been demonstrated recently [249].
√iSWAP
g
Entangling operations are currently the performance
bottleneck for existing quantum processors. Among the
numerous entangling gate schemes, most of them are
between two qubits and they generally belong to two
families. One general approach is to frequency-tune the
relevant energy levels into resonance to initiate interac-
tions; related demonstrations include the implementation
of iSWAP or gate [250, 251] and the controlled-
Z gate [252–257]. The other method is to apply
microwave pulses at certain non-local transitions; for
example, the cross-resonance gate [258, 259], resonator-
induced phase gate [260] and parametrically driven gate
[261, 262]. The gist for obtaining high-fidelity two-qubit
gates is to engineer an effective interaction strength such
that it is strong during gate operation for achieving
short gate time while as weak as possible outside of the
gate window for avoiding unwanted interactions, in
another word, a high on/off ratio. In a fixed-coupling
architecture where the qubit–qubit coupling strength is
almost constant, a straightforward way to turn the
interaction on or off is to tune the qubit frequencies into
or away from resonance. However, as the qubit–qubit
Table1Typical performance reported in superconducting qubits.
No. of qubits
T1
(μs)
T∗
2
(μs)
t1q
(ns)
Err1q (10−3)
t2q
(ns)
Err2q (10−3)
tr
(μs)
Errr
10−2
( ) Fridge
Size
53,66
[13, 128]
16−30.6
∼5.3
∼25
∼1.4
∼12
∼5
∼1
[281]
∼3.1
[281]
∼20 mK
∼1
mm2
<10 [273–280]
15−76
12−105
∼30
∼1
10−200
∼1.5
t1q
t2q
tr
Errr
( ) is the duration of one-qubit (two-qubit) gate.
( ) stands for the duration (error rate) of the readout.
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-9
g
99.9%
connectivity increases, each qubit sees more transitions
in its surroundings; it becomes increasingly difficult to
manipulate the whole system in a clean fashion. This is
known as the frequency-crowding problem, one of the
main challenges in scaling up quantum processors. The
problem also exists for alternative coupling schemes such
as the all-to-all connection via a bus resonator [263]. It
may be resolved by the tunable-coupling schemes in
which the coupling strength can be independently
controlled over a large dynamic range [264–271]. In
recent years, a tunable-coupling architecture based on
native capacitive coupling and interference effect [272]
has become a trending solution; many research groups
have made tremendous progress in gate fidelities, including
some results approaching the mark [273–280].
Typical performance of superconducting processors is
summarized in Table 1.
20
Decoherence.—Quantum information can be quickly
destroyed by decoherence. The bad news is that super-
conducting qubits are extremely susceptible to external
fluctuations due to their macroscopic nature. One imme-
diate solution is, of course, to make the qubit lifetime
longer. Ever since the first observation of quantum
coherence in superconducting qubits [12], the lifetime of
the qubit has been improved by six orders of magnitude
from nanoseconds to milliseconds [282–284] in about
years. This remarkable progress is attributed to a combi-
nation of advances in design, material, fabrication qual-
ity, and testing environment. The current common belief
is that spurious two-level systems (TLS) that reside in
the vicinity of the qubits are a major source of decoherence
[285] and unpredictable fluctuations in coherence and
qubit frequencies, which can be troublesome in large-
scale implementations [286–290].
1/f
Besides coherence improvement on the hardware side,
another way to combat noise and decoherence is through
quantum control methods. A particularly useful technique
is dynamical decoupling (DD) [291] which uses tailored
pulse sequences to correct for coherent noise, in particu-
lar, the notorious noise [292] that is ubiquitous in
these solid-state devices. Designing an optimal sequence
requires detailed knowledge of the noise such as its spectral
properties which may be extracted using various techniques
at different frequency ranges [251, 293–298].
Given the limited coherence, the performance of a
quantum processor may also be enhanced through opti-
mized quantum compiling, i.e., to translate high-level
operations into a shorter sequence of native gates [299].
Compiling on a superconducting quantum processor can
be exceptionally challenging due to the planar geometry
and limited connectivity; often, the final sequence to
execute is too time-consuming. An effective strategy is
to fully explore the hardware capabilities and diversify
the available gate alphabets to optimize compilation.
Recent progress on continuous gate set, multi-qubit
gates, and qudit operations have shown considerable
potential in this respect [275, 300–304].
10−3
10−12
10−15
Quantum error correction.—Since the state-of-the-art
gate error rate ( ) is many orders of magnitude
higher than that a logical qubit would require
( – ), QEC is necessary for building a universal
quantum computer. Surface codes [305, 306], which
encode logical qubits into a square lattice of physical
qubits, are appealing for planar architectures. Recently,
we have observed a surge of exciting experimental devel-
opments in this respect [21, 22, 24, 307–309]. In some of
these experiments the performance is getting close to or
partially exceeds the QEC threshold. Still, it is challenging
to achieve substantial error-correction gain and most
importantly, to have the performance reproducible at an
even larger scale. In the near future, a logical qubit
made of a few hundreds to a thousand physical qubits is
highly anticipated; in the next five to ten years, we may
have an idea about whether a fault-tolerant quantum
computer is feasible and how powerful it can be. A relevant
issue with increasing attention is the presence of cosmic
rays which can cause chip-wide failure and is catastrophic
to surface codes [310–312].
Another promising route to QEC is bosonic codes,
where logical qubits are encoded in microwave photon
states of three-dimensional superconducting cavities.
Depending on how a logical qubit is encoded into
harmonic oscillator states, there are different kinds of
bosonic codes [313–315]. The cat codes and associated
variants utilize a superposition of two photonic cats of
the same parity as logical qubit [316–320]; the binomial
codes instead use definite photon number parity as code
words [321, 322]; the Gottesman–Kitaev–Preskill (GKP)
codes implement a coherent state lattice in phase space
[323–327] with the advantage that errors, measurements,
and gates are simple displacements of the oscillators. To
date, only bosonic codes have reached the break-even
point in QEC experiments, which means that the error-
corrected qubit has longer lifetime than the otherwise.
This is due to the fact that microwave photons have
fewer error syndromes and three-dimensional cavities
usually have higher quality.
Scalability.—Lastly, we would like to touch upon the
most concerning question: how to make the supercon-
ducting quantum processor more scalable. With the
continuous improvement in planar circuit design and
fabrication and the development of flip-chip packaging,
dozens of superconducting qubits have been integrated
on a single processor so far, allowing for the demonstration
of quantum supremacy [13] [Fig. 3(f)] or quantum
computational advantage [128] [Fig. 3(g)]. It is worth
emphasizing that simply printing thousands of qubits is
straightforward, but the real challenge is to achieve high-
fidelity operations for all qubits, simultaneously. For
this purpose, many existing architectures may need to
be reinvented. First of all, hosting more qubits in a
limited space requires reducing the qubit footprint.
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21308-10 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
100
L×L
∝L
∝L2
Recent developments show that the shunt capacitor can
be miniaturized by folds or more using two-dimensional
materials while maintaining coherence [328–330]. Even if
the qubits can be densely packed up on a chip (size:
), it is nevertheless extremely difficult to route all
the control wires from the perimeter (length ) to
individual qubits (density ) due to the unmatched
scaling law; let alone to avoid crosstalk between wires.
In recent years, there have been substantial efforts to
exploit the third dimension to relieve this pain with
various technologies borrowed from the semiconductor
chip packaging, such as flip-chip bonding and through-
silicon vias [331–338]. Aside from expanding the space
for wiring, a different approach is to reuse the wire for
multiple targets. Signal multiplexing and control line
sharing schemes can alleviate the problem for future
large-scale devices [304, 339, 340]. They also help reduce
the cable density inside and out of the dilution refrigera-
tor.
∼80
∼5×104
∼90
In the future, we may end up with insurmountable
engineering challenges, including available wafer size,
device yield, and crosstalk, all constraining the scalability
of monolithic quantum processor designs. This suggests
the desirability of developing alternative modular
approaches, where smaller-scale quantum modules are
individually constructed and calibrated, then assembled
into a larger architecture using high-quality quantum
coherent interconnects [341–345]. Several recent experi-
ments have demonstrated deterministic quantum state
transfers (QSTs) between two superconducting quantum
modules, with interconnects provided by commercial
niobium-titanium (NbTi) superconducting coaxial cables
[346–348], copper coaxial cables [349], and aluminum
waveguides [350], showing fidelities up to %, primarily
limited by lossy components including connectors, circu-
lators, and printed circuit board traces. More recent
efforts using wirebond [172] or clamped [351] connections
between the quantum modules and the superconducting
cables have eliminated the need for normal-metal
components, improving cable quality factors to
and inter-module QST fidelities to % [Fig. 3(h)].
Flip-chip modular approaches have also been pursued
[352], where the qubits on separate chips are closely
spaced and directly coupled, achieving high fidelities
while retaining many of the benefits of a modular archi-
tecture.
In addition to the architectural design of quantum
processors, supporting technologies are also crucial for
scaled implementations. As a result of non-ideal fabrication
conditions, the critical current of a Josephson junction
usually varies by a few percent, equivalent to a few
hundred megahertz in terms of qubit frequency; such
unpredictable variation severely affects the quality of
qubit operations. Techniques for improving junction
uniformity during and after fabrication may open up
new possibilities in hardware and software design
ZZ
[353–356]. Moreover, the capacity of a dilution refrigerator
will ultimately be limited by its cooling power. Promising
solutions include a careful wiring plan [357] and energy-
efficient cryogenic electronics [358–360]. Simultaneous
high-fidelity operations require low crosstalk. Crosstalk
mitigation is a must-do. Besides optimization through
design and packaging [220, 272, 361–365], various
control techniques have been developed to reduce different
kinds of crosstalk such as microwave signal crosstalk
[279, 366], spectator effect [367–369], and residual
interactions [370, 371]. In the future, by integrating
together various technologies in a large system, more
powerful quantum processors based on superconducting
qubits can be anticipated.
4Trapped-ionqubits
Introduction.—Trapped-ion systems are the leading
physical platform in pursuit of fault-tolerant quantum
computing. Laser-cooled atomic ions confined in an ultra-
high vacuum environment are well isolated from noises,
being able to encode high-quality qubits into a stable
pair of electronic energy levels in each ion, as shown in
Fig. 2(b). Ion qubits hold the longest coherence time
beyond any other systems [372–374], and can be initialized
and measured with extremely high fidelity [372, 375,
376]. Quantum logical operations are typically performed
through tailored laser or microwave-ion interactions, and
gate fidelities achieved experimentally on a few ion
qubits have gone well beyond the fault-tolerant threshold
[372, 377–379]. Ion-based quantum processors can
already primely manipulate dozens of qubits [380, 381],
and quantum algorithms such as Shor’s algorithm and
Grover’s search algorithm have been exemplified in
small-scale systems [382, 383]. Meanwhile, quantum
simulators with up to 53 qubits have been demonstrated
to study various novel features of complex many-body
systems represented by quantum spin models [384–392].
The above progress illustrates the great potential of
utilizing large-scale trapped-ion systems for ultimate
quantum computing.
Be+
Mg+
Ca+
Sr+
Ba+
Yb+
2S1/2
2D5/2
40Ca+
88Sr+
Ion qubit.—Despite hundreds of atomic species that
exist in nature, hydrogen-like ions are preferred for
trapped-ion quantum computing due to their relatively
simple atomic structures. Alkaline-earth ions like
[377], [393], [378, 379, 381, 394], [395],
[396, 397], and rare-earth ions like [380, 398–400]
are the most frequently utilized in current research. A
qubit can be encoded into a pair of energy levels of a
single ion, and a representative encoding scheme
employs a combination of levels belonging to the ground
manifold and the long-live metastable manifold
. This scheme is the prior choice for even-mass
isotopes, such as [381] and [395], and these
qubits have energy gaps on the order of optical frequen-
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-11
cies; therefore, they are denoted as optical qubits.
Although the typical lifetimes of metastable levels can
reach a few seconds or so, they would ultimately limit
the coherence time of optical qubits.
2S1/2
T1
T2
T2
43Ca+
171Yb+
138Ba+
For odd-mass isotopes, the encoding scheme is to
utilize the hyperfine splittings of the ground manifold
induced by non-zero nuclear spins. The lifetime of
ground-state hyperfine levels can approach the age of
the universe, resulting in an extremely long relaxation
time ( ) compared to that of the optical qubits. Mean-
while, a certain pair of hyperfine levels can form the so-
called “clock state” under a suitable external magnetic
field, and its energy gap is insensitive to the static
magnetic field to the first order, thus having a relatively
long coherence time ( ). It is experimentally observed
that the time of a single ion can reach 50 s
[372]. This record is then extended to 10 minutes in a
ion qubit by using DD pulses and sympathetic
cooling assisted by a ion [373]. Most recently, the
one-hour coherence time has even been approached by
further reducing the potential noise from the external
magnetic field and the leakage from microwave sources
[374]. Such a long coherence time allows systems to
execute millions of gate operations before losing quantum
features.
2S1/2 ↔2P1/2
10−4
The cycle transition of facilitates the
realization of extremely low state-preparation-and-
measurement (SPAM) errors on ion qubits. Qubit state
initialization is achieved by optical pumping techniques.
By choosing proper polarizations and frequencies of
pumping lasers, a certain energy level of the qubit can
be a dark state, and then the ion state would be pumped
to this level with high probability and high speed. A
typical initialization process can take a few microseconds,
and infidelity can be suppressed close to [372].
10−3
10−4
State measurement is implemented by resonating one
of the qubit levels to a short-live manifold and collecting
the corresponding fluorescent photons simultaneously.
Projected qubit states in a single shot can be distinguished
by determining whether the number of collected photons
reaches a certain threshold. Depending on the photon
collection rate, the measurement duration could vary
from several microseconds to milliseconds, while the
error can reach below [372, 375, 376]. This error
can be suppressed to around for ions with long-live
levels for state shelving [401]. Several other methods,
such as adaptive analysis or time-stamping of arriving
photons [375, 401, 402], are employed to further reduce
measurement infidelity or increase detection speed, and
meanwhile, machine learning methods can be utilized for
multiple-qubit detection to reduce crosstalk errors [403,
404].
Quantum gates.—Quantum gates on ion qubits are
typically performed by interacting ions with external
laser or microwave fields, depending on the qubit encoding
schemes. For example, single-qubit rotations on optical
10−4
10−6
qubits can be applied through optical quadrupole transi-
tions induced by a narrow linewidth laser, and that on
hyperfine qubits can be realized by using microwaves or
stimulated two-photon Raman transitions. Error rates
below have been reached for either quadrupole
transitions on optical qubits or Raman transitions on
hyperfine qubits [377, 378]. Error rate of was even
achieved on microwave-driven hyperfine qubits [372].
8×10−4
9Be+
9×10−4
43Ca+
40Ca+
6×10−4
Although high-fidelity single-qubit rotations are readily
accessible experimentally, the qualities of current quantum
processors are mainly limited by the performance of
entangling operations. The first proposal of the two-
qubit gate on ion qubits, the Cirac–Zoller gate [6], is
challenging to scale up due to the stringent requirements
for ground-state cooling and sensitivity to thermal exci-
tation on motional modes. However, this proposal
inspired the idea of using collective motional modes of
ion-chain to engineer effective qubit–qubit couplings.
The entangling schemes used today can be categorized
into Mølmer–Sørensen gates [411, 412] and light-shift
gates [413, 414], which both rely on the notion of state-
dependent forces. These schemes show excellent perfor-
mance in experimental demonstrations. The error rate of
the Mølmer–Sørensen gate below is achieved
with two ions [377], and that of the light-shift gate
below is realized on two ions [378].
Recently, light-shift gates on optical qubits are theoreti-
cally investigated and then experimentally demonstrated
in a two ions system [379, 415]. The gate infidelity,
as low as , is approached, representing the best
entangling gate achieved ever.
98.5%
99.7%
Laser fields are mostly utilized to drive entangling
gates on ion qubits due to their large spatial gradient of
electric fields to provide efficient ion-motion couplings.
However, microwave-driven entangling gates are also
pursued due to the extreme stability of long-wavelength
microwaves [416, 417]. Ion-motion coupling induced by
microwave fields can be achieved by placing magnetic
field-sensitive qubits into static magnetic fields with
large spatial gradients or by exploiting near-field oscillating
microwaves. The former scheme suffers from short
coherence times induced by fluctuating magnetic fields,
which can be overcome by utilizing microwave-dressed
qubits [418, 419], while the latter one requires
microwave sources close to the ions; therefore, the
crosstalks should be taken care of. Experiments have
demonstrated gate fidelities of about [398] and
for each scheme [420, 421] respectively. Moreover,
a recent experimental work has shown that with the
microwave-driven laser-free gate, an almost perfect
symmetric Bell state has been generated [393]. These
advances promise a scalable way to achieve ion-based
quantum computing with full microwave control [422].
However, most experimentally implemented entangling
operations so far are relatively slow, usually on the order
of tens to hundreds of microseconds, thus limiting the
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21308-12 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
99.8%
core speed of ion-based quantum processors. Therefore,
fast gate implementation becomes one important topic
of research in recent studies. The straightforward way to
speed up entangling gates is to increase the laser power
to enhance the laser-ion coupling strength. Along this
routine, an entangling gate with a duration of 1.6 μs is
achieved while the fidelity is still maintained at
[423]. However, the gate fidelity drastically drops to
60%
76%
around when the gate duration further reduces to
480 ns, due to the breakdown of the Lamb–Dicke (L–D)
approximation. It might be solved by considering high-
order qubit-motion couplings [424]. Another way to
achieve fast gates is employing a sequence of ultrafast
laser pulses to impose ultrafast state-dependent kicks on
ion qubits [425–427], which does not require the ions to
remain in the L–D regime. A Bell state with fidelity
Fig.4(a) Three-dimensional Paul trap and captured long ion chain. Compared to the conventional four-rod trap, electrodes
shown here are transformed into blade-shape to enhance optical accessibility. A one-dimensional chain of ions is trapped
along the null line of the radiofrequency field, and a tightly focused laser beam array individually controls ion qubits.
(b) QCCD architecture, reproduced from Ref. [405]. Large numbers of ions are distributed in large chip-type traps with
multiple trapping zones. Ions can be manipulated independently in different functional zones to realize logical operations,
storage, or readout. Quantum information can be interchanged by transporting ions between zones. (c) Remote-ion entangle-
ment, reproduced from Ref. [405]. Ions in different traps can be heraldedly entangled by generating ion-photon entanglements
and then applying Bell measurement on photons. It paves the way for large-scale distributed systems. (d) Surface trap fabricated
by Sandia National lab, reproduced from Ref. [405]. (e) Integrated photonic system to deliver laser beams to ions position.
Here we show the surface trap with multi-wavelength integration done by the MIT group, which is reproduced with permission
from Ref. [406], Springer Nature Limited. (f) On-chip detection of ion qubit. Several groups have been successfully demonstrated
integrated single photon detector on the fabricated surface traps [407–409].
Table2Selected state-of-the-art performance on ion qubitsa.
Qubit type Hyperfine qubit Optical qubit
T2
50 s [372]
5500 s [374]b0.2 s [410]
SPAM error
6.9×10−4
[376]
8.7×10−5
[401]
1Q gate Duration 1–10 μs typical 1–10 μs typical
Fidelity 0.99996 [377] 0.99995 [410]
2Q gate
Duration 10–100 μs typical 10–100 μs typical
Fidelityc0.9991 [379]
0.999 [378] 0.9994 [379]
Maximally entangled qubits 24 [381]
Environment
<10−11
Ultra-high vacuum Torr
a Here we only include data from peer-reviewed publications.
b With dynamical decoupling.
c Two-qubit gate fidelities are estimated from the fidelities of the prepared Bell states.
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-13
is prepared within 1.96 μs in a recent experimental
demonstration, and the main infidelities come from the
imperfect kick control and off-resonant coupling to unde-
sired energy levels [428]. High repetition-rate pulsed
lasers can be helpful to further improve the gate speed
[429]. Although recent implementations of fast gates still
have limited fidelities, these schemes all show well scala-
bility.
40Ca+
171Yb+
Scalability.—A straightforward way to scale up ion-
based quantum processors is to trap multiple ions in a
linear array, as illustrated in Fig. 4(a). By exploiting the
collective motional modes of the entire ion chain, entangling
gates can be applied to any two ion qubits by coupling
to single or multiple motional modes. For the latter case,
time-modulated state-dependent forces are required to
decouple multiple motional modes from ion qubits simul-
taneously to guarantee high-fidelity operations [430–436].
Along this route, up to 14 ion qubits were first
used to generate the Greenberger–Horne–Zeilinger (GHZ)
states [437], and then the qubit number was increased to
24 in a recent report [381]. Meanwhile, a programmable
trapped-ion quantum processor was implemented in
2016 [438], consisting of 5 individually controlled
ion qubits, and later this system was extended to 11
qubits in 2019 [380]. So far, multiple research groups
around the world have realized their quantum processors
with long ion chains [395, 399].
One distinct advantage of using an ion chain is the so-
called full connectivity [439], which, as already
mentioned, allows ion qubits confined in the same potential
to be directly entangled even if they are not spatially
adjacent. This feature makes the decomposition of quantum
circuits more efficient and makes it possible to realize
multi-qubit entangling gates. Several theoretical works
have pointed out that multi-qubit gates might bring
polynomial or even exponential speedups to running
quantum tasks [440–443]. Therefore, researchers have
been eager to explore scalable ways to achieve multi-
qubit gates in recent years [399, 444, 445]. However, this
linear-chain architecture also has several drawbacks,
making it hard to reach a large scale. For example, the
laser power required to entangle the ion qubits in a
chain would increase as the size of the chain enlarges.
The long chain's cooling also becomes imperfect, while
gate operations become more sensitive to external noises.
To further scale up to larger numbers of qubits, we
can trap multiple ion chains in several independent
potentials and construct some link channels for intercon-
nections. One representative architecture is the quantum
charged-couple device (QCCD) proposed in 2002 [445]
[see in Fig. 4(b)]. Links between chains are achieved by
modifying local electric potentials to redistribute ions
between trap regions physically. To achieve this goal,
shuttling operations [446] like linear transport, splitting
or merging of ion-chain, and position swap should be
included together with quantum gates in local chains as
the basic operations of quantum processors. These oper-
ations must be performed fast enough so that they do
not become processing speed bottlenecks. Several fast
shuttling methods have been fully investigated and
demonstrated to simultaneously satisfy these two
requirements [447, 448], promising to construct a reliable
highway of ion qubits to enable a large-scale QCCD
architecture.
However, the complexity of the ion trap is significantly
increased compared to that used for a single linear chain.
An ion trap with numerous independent control electrodes
is required to realize multiple trap regions and precise
control of the ion shuttle. Microfabricated chip traps are
a satisfactory solution to these complexities [449, 450].
Recently, a series of 1D chip traps and 2D traps with X-
type [451], Y-type [452], or T-type [453] junctions have
been presented. Utilizing these well-controlled traps, 4
qubits GHZ state has been prepared in a shuttling-based
way [394], and quantum gate teleportation has also been
demonstrated [454]. Moreover, high-quality quantum
processors based on QCCD architecture have shown
excellent performance according to quantum volume
measurement [455]. However, towards large-scale surface
traps, there are still several challenging issues that
should be addressed. One critical problem is the anomalous
heating on motional modes of ion chains [456–458]. It
would destabilize ion chains and become one of the main
error sources for quantum gates and ion shuttling.
Although many efforts have been made to reveal the
origins of this heating effect, the problem is still not well
solved. Other issues such as radio frequency (RF) potential
barrier [459] in the junction transport and relatively low
trap depth would also plague the development of scalable
QCCD-based quantum processors. Nevertheless, QCCD
architecture is still an outstanding approach for large-
scale trapped-ion quantum computing.
94%
Another natural choice to link ion qubits in different
regions is utilizing photons [460]. By exciting ion qubits
to a short-live ancilla atomic level, the polarization of
the spontaneously emitted photon would entangle with a
decayed ion state. By applying Bell measurement to the
photon pair from different ions, a heralded entanglement
can be generated between ion qubits depending on the
measurement outcomes, as shown in Fig. 4(c). This
remote-ion entangling method enables the feasibility of
entangling ion qubits in different vacuum systems even
if they are far away, leading to the paradigm of
distributed quantum computing. The generating rate of
the remote entanglement should be fundamentally deter-
mined by the scattering rate of the ancilla level.
However, in practice, this rate is mainly limited by the
collecting rate of the emission photons. In recent experi-
mental demonstrations, a generation rate of 4.5 Hz [461]
is first realized and then improved to 182 Hz [462], while
the best fidelity of the heralded qubit entangling state is
. This value is much slower than the gate speed of
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21308-14 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
directly entangling qubits in the same trap. Several
methods have been proposed to further increase the
generation rate, such as enlarging the numerical aperture
of the photon collecting system and increasing the quantum
efficiency of the single photon counter. Significant
improvement might be achieved if we place ion qubits
into a high finesse micro-cavity, enhancing the spontaneous
emission through the Purcell effect [463]. Moreover, the
conversion of a single photon from visible regime to tele-
com-wavelength has been demonstrated recently [464],
although with quite low converting efficiency, making it
possible to build distributed quantum systems with ultra-
low optical loss. Practical remote-ion entanglement
would facilitate the construction of large-scale quantum
computing platforms and also quantum networks.
9Be+
25Mg+
40Ca+
43Ca+
43Ca+
88Sr+
171Yb+
138Ba+
9Be+
9Be+
40Ca+
Here we briefly talk about hybrid-ion systems. When
we generate remote entanglement of two traps or apply
mid-circuits measurement to certain ion qubits belonging
to an ion register, we want to only excite targeted ion
qubits without disturbing others resonantly. One solution
is that the measured ion and others belong to different
species, so the resonance frequency is quite different,
significantly suppressing crosstalk. Consequently, entan-
gling gates on ion qubits of mixed species are required in
hybrid-ion systems. High-fidelity entangling operations
on mixed-species ion qubits are well displayed on such
as – [454], – [465], – [466],
– [467, 468] and even a long chain of
– – [469]. Meanwhile, mixed-species
systems make it feasible to apply sympathetic cooling,
allowing to cool down ion register without destroying
stored quantum information [470–472]. It is valuable for
suppressing motional excitation during ion shuttling in
the QCCD architecture. Meanwhile, proposals for hybrid
encoding, by utilizing multiple energy levels of a single
ion to encode different qubit types, have been made
recently to construct hybrid systems with single ion
species [473]. The interconversion of different qubit
encoding on a single ion has been experimentally demon-
strated [474]. It might open a new way for scalable
trapped-ion quantum computing.
System integration.—System integration is inevitable
in building scalable large-scale trapped ion quantum
computers. One typical example is the microfabricated
surface traps mentioned above. In the past decade,
researchers have done several important works to
promote the integration of trapped ion systems further,
and on-chip integrated optics is one of them. Laser
beams can be delivered and tightly focused at the location
of the ions by embedding optical waveguides beneath
the surface traps and fabricating properly designed gratings
coupler at the end of each waveguide. Optical integration
from single to multiple wavelengths has been well-
demonstrated [406, 475]. Single qubit rotations and two-
qubit entangling gates have been implemented using
laser beams delivered through waveguides [475, 476],
showing extreme robustness against vibrational noises.
Furthermore, the integration of single photon detectors
on chip traps has been successfully demonstrated
recently [407–409], showing a scalable way for high-
fidelity readout of multiple ion qubits on large-scale
quantum processors. Conventional analogue voltage
sources are also integrated on-chip [477], enabling an
expandable approach to control numerous ion trap elec-
trodes and laying the technical foundation for circuit
integration in large-scale QCCD architectures.
Outlook.—This section briefly reviews the significant
advancements in trapped ion quantum computing over
the past decades, from excellent control of several ion
qubits to demonstrations of scalable architectures. With
fully controllable trapped-ion processors, several great
advances have been made recently in QEC [410, 469,
478–481]. However, we still strive to explore specific
scalable ways to achieve a truly fault-tolerant logical
qubit encoding and ultimately build an applicable ion-
based quantum processor. One of the fundamental
requirements is that the size of the system increases
without compromising the quality of the control
[482–484]. Therefore, developing system integration-
related technologies would be critical for scaling-up ion
quantum processors in the following decades. Meanwhile,
by merging the techniques developed for trapped-ion
quantum computing, we might also gain better perfor-
mance in ion-based precision measurement. With contin-
uous development, trapped-ion systems would remain an
important platform and tool for future quantum infor-
mation applications.
5Semiconductorspinqubits
Introduction.—Spin qubits in semiconductors have made
tremendous progress over the past few decades.
Although most toolboxes have been built based on GaAs
quantum dots [491], this field had a revival after the
host material steered to silicon. Further momentum is
gained after some recent vital breakthroughs, such as
fault-tolerant quantum gates [492–494], RF-reflectometry
spin readout [495–498], spin-photon strong coupling
[499–503], hot qubits [504–506], and cryo-CMOS control-
ling chip [507, 508]. Combined with its inherent scalability
from the semiconductor industry and its small footprint,
spin qubits are now well poised for the following mile-
stones —quantum advantages over classical supercom-
puters, prototype machines of fault-tolerant quantum
computing and QEC, and hybridization between classical
and quantum electronics.
Qubit construction.—Semiconductor qubits are
defined on the charge and spin degrees of freedom of the
carriers trapped in quantum dots or dopants. There are
several types of qubits, such as spin qubits [10, 11, 509,
510], charge qubits [511, 512], exchange-only qubits [43,
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-15
Fig.5Representative semiconductor qubit systems. All the devices are presented with two panels, where a top panel
shows the top view of the device, and a bottom panel shows the lateral structure corresponding to a white line cut of the
active region in the top panel. Heterostructure quantum dots include (a) Si-MOS [485], (b) Si/SiGe [486], and (c) Ge/SiGe
[487] systems, where Si-MOS and Si/SiGe are mainly used for electron qubits, and Ge/SiGe is a hole qubit platform. System
(a–c) belong to the gate-defined quantum dot category. In (a) Si-MOS, quantum dots are formed close to the silicon-oxide
interface, with fabricated top gates providing lateral Coulomb confining potentials. On this device, an electron spin driving
ESR antenna and a spin readout single electron transistor are integrated as well. (b) Si/SiGe quantum dots are formed in the
middle silicon quantum well layer and sandwiched between SiGe layers on both sides. (c) Ge/SiGe is a hole spin platform,
using a Ge well to form a two-dimensional hole gas and combined with top gates to form hole quantum dots. (d) CMOS
nanowire field-effect transistor [488], where quantum dots are formed in the silicon nanowire sitting on a buried silicon oxide
layer (BOX), and the surrounding gates are fabricated using industrial microelectronics technology. It is a hole-spin system,
and the dot potential well is formed at the valance band top. (e) and (f) are the donor qubits in silicon. They are electron
spin systems, where the host donor nuclear spin is also a key resource to encode qubits. (e) represents a donor in MOS [489]
system, where a phosphorus atom is implanted in a fabricated MOS device which has a 28Si layer to prolong electron spin
coherent time. (f) Donor device with STM lithography technique [490], where the donors can be placed with atomic precision
and in-plane gates are formed by dense conducting phosphorus atoms in a lithographical manner. Top panel of (a) is reproduced
with permission from Ref. [485], Springer Nature Limited. Top panel of (b) is reproduced with permission from Ref. [486],
American Association for the Advancement of Science. Top panel of (c) is reproduced with permission from Ref. [487],
Springer Nature Limited. Top panel of (e) is reproduced with permission from Ref. [489], Springer Nature Limited.
FRONTIERS OF PHYSICS
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21308-16 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
|0⟩=|↓⟩
|1⟩=|↑⟩
|↓⟩
|↑⟩
|S⟩= (|↑↓⟩ − |↓↑⟩)/√2
|T0⟩= (|↑↓⟩ +|↓↑⟩)/√2
513, 514], hybrid qubits [515], and singlet-triplet qubits
[516, 517]. A spin qubit is usually defined by the spin
states of a single electron or hole trapped in a semiconductor
quantum dot or a dopant in silicon (see Fig. 5) [10, 11],
as and , where and denote spin
down and spin up, respectively. Chosen states of an
interacting multi-spin system can also be defined as a
qubit, such as the singlet-triplet qubit (i.e., the singlet
state and the unpolarized triplet
state of two exchange-coupled
spins). Here we will focus on the spin qubits in this
review, as they are becoming the central topic in the
field in recent years. We will also briefly introduce other
types of qubits. Interested readers can find more infor-
mation in the relevant references [491, 509, 510,
518–520].
T∗
2
Decoherence.—The development of semiconductor
host materials and related fabrication technologies
underpins progress in this field. GaAs/AlGaAs
heterostructure quantum well has been the key substrate
for gate-defined quantum dots [529], where this field has
accumulated many of its foundational building blocks,
such as single charge sensing, single-shot spin readout,
qubit operations, and interactions, to mention only a few.
Nevertheless, the nucleus of the host GaAs forms a fluc-
tuating magnetic field, namely the Overhauser field,
which limits the spin dephasing time in GaAs to
around the range of tens of nanoseconds [491, 510, 529].
Although GaAs quantum dots still stand out nicely as a
demonstration platform for quantum simulation [530,
531] and quantum physics research [532, 533], the
limited spin dephasing time hinders the development of
high-fidelity quantum gates for quantum computing.
T1
T1
T2
2T1≥T2
T1
T2
T2
T∗
2
T1
THahn
2
A host material consisting of zero nuclear spin
isotopes is necessary to embrace a long dephasing time.
In his seminal paper 20 years ago, Kane [10] emphasized
that group IV materials are ideal options as they feature
stable zero-nuclear spin isotopes with high natural abun-
dance. The nuclear-spin free isotope 28Si, for example,
could be accessed via purifying the natural silicon.
Another advantage of the silicon substrate is the long
spin times, which comes from the different spin relax-
ation behavior compared with the group III–V materials
[510]. Since sets an upper bound on for a spin
system via the relation , a long is the prerequisite
for having a long . Therefore, silicon spin qubits have
become the working horse in this field, and the mainly
explored platforms include silicon metal-oxide-semicon-
ductor (MOS) [489, 523, 525], silicon-on-insulator (SOI)
[488, 534], the dopant in silicon [490, 525, 535], silicon-
germanium heterostructures based Si/SiGe [492, 524,
536, 537], and Ge/SiGe [538–541]. After extensive stud-
ies, competitive and compared to the other major
quantum computing platforms are demonstrated in silicon
spin qubits. Up to now, values ranging from 160
milliseconds [522] to 30 seconds [542] and values
ranging from 99 microseconds [524] to ~1 second [489]
are observed in silicon dopant devices, Si/SiGe gate-
defined systems, and Si-MOS systems (for detailed
comparison, please refer to Table 3). Other silicon spin
Table3Comparison of reported values of different qubits in silicon.
Qubit type Si-MOS Si-SiGe P donor n P donor e
T1
2.6
s [521]
160
ms [522]
39
min [489]
30
s [489]
T∗
2
120
μs [523]
20
μs [524]
600
ms [489]
268
μs [489]
THahn
2
1.2
ms [523]
100
μs [524]
1.75
s [489]
0.95
ms[489]
Tsingle
2.4
μs [523]
20
ns [524]
24
μs [493]
150
ns [525]
Ttwo
1.4
μs [485]
103
ns [492]
1.89
μs [493]
0.8
ns [490]
F1RB
(%)
99.957(4)
[526]
99.861(5)
[524]
99.99
[527]
99.95
[527]
F2RB
(%)
98.0(3)
[485]
99.51(2)
[492]
99.37(11)
a [493]
86.7(2)
b [490]
Q1
c
50
1000
25000
1800
Q2
c
86
194
302d
3.4×105
NQ
e2 [485] 6 [528] 2 [493] 2 [490]
NE
f2 [485] 3 [528] 2 [493] 2 [490]
Env
B∼1.4
T
B∼0.5
T
B∼1
T
B∼1
T
T < 1.5
K
T < 1.5
K
T < 1.5
K
T < 1.5
K
Flying qubit N/A N/A N/A N/A
Footprint size
∼100
nm
∼100
nm
∼3
nm
∼100
nm
√SWAP
Q1≡T∗
2/Tsingle
Q2≡T∗
2/Ttwo
Tsingle
Ttwo
T∗
2
NQ
NE
a CZ gate.
b gate.
c and , where and are the time for the single-qubit and the two-qubit operations.
d = 570 μs [489] is used here for a P nuclear spin with a bounded electron.
e is the demonstrated number of qubits with individual control.
f is the number of entangled qubits.
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systems also exhibit typical long coherence times, such
as hole spins in SOI [488] and Ge/SiGe gate-defined
quantum dots [539]. Long coherence sustains even at
1.1–4.2 K in MOS [504, 505] and SOI devices [506].
99%
Gates.—Over the past years, prominent progress has
been made for quantum gate fidelities in semiconductor
qubits. A single-qubit operation could be performed
utilizing a few different mechanisms for different qubits.
For singlet-triplet qubits [517, 543], exchange-only
qubits [43, 513, 514], and hybrid qubits [515], exchange
coupling plays the key role. In comparison, electron spin
resonance (ESR) [525, 544, 545] and electrical dipole
spin resonance (EDSR) [524, 538, 546, 547] are the main
driving mechanisms for a spin qubit. They control the
spin rotation by either oscillating magnetic or electrical
fields. After years of persistent quest for fault-tolerant
operations, single-qubit gate fidelities well beyond
have all been demonstrated in the donor [489], Si-MOS
[485], and Si/SiGe [486, 514, 524, 548] (see Table 3).
fR
99.8%
FCZ = 99.65%
FCZ = 99.81%
FCZ = 99.37%
Compared to single-qubit gate operations, two-qubit
gates are more daunting. Exchange coupling, capacitive
coupling, or hyperfine coupling can be utilized to realize
a two-qubit gate. In the singlet-triplet qubit, capacitively
coupled two-qubit gates have been shown [549, 550].
Capacitively coupled two-qubit gates have been realized
in charge qubits as well [512]. In spin qubits, the well-
established two-qubit gate protocols, such as SWAP
[490, 516], C-Phase [523], and C-ROT [486], all need the
existence of exchange coupling. Since either the capacitive
or exchange coupling hinges on the charge degrees of
freedom, charge noise couples into the system and
renders a challenge for high-fidelity gate operations.
Different methods have been pursued to realize high-
fidelity two-qubit gates. A fixed exchange coupling was
used for Si/SiGe quantum dots [492], where the
unwanted rotation of the off-resonant states was
removed by carefully matching the Rabi frequency
and the exchange J relation, and a fidelity two-
qubit gate was realized. The other two teams both
utilized the tunability of the exchange coupling to
perform CZ gates in the Si/SiGe platform. After detailed
calibration and pulse optimization, two-qubit gate fidelities
of [494] and [551] were demon-
strated. In the silicon donor system, a two-qubit CZ
gate with is shown on two donor nuclei
with a shared electron [493] using a geometric gate and
hyperfine coupling.
Despite of the milestone breakthroughs in the fault-
tolerant qubit gates, the semiconductor qubits still have
much room to improve the qubit operation fidelities.
Especially for the charge noise issue [524, 526, 552, 553],
qubit host material engineering is necessary to have a
purer environment, such as fewer nuclear spins, charge
traps, and defects. Moreover, more sophisticated control
methods or encoding could be combined, such as dressed
qubits [554], global control [555], and profile optimized
pulses [556]. Besides, the sweet spot in the qubit energy
[552, 557] and composite pulse sequences [558] could also
help against the charge noise. We stay optimistic about
further improvements in gate fidelities from optimized
device fabrication and control-level engineering.
98.4%
FM= 99.8%
Readout and initialization.—Reducing SPAM errors is
as important as improving gate fidelities for pushing the
fault-tolerant quantum computing. Single-shot spin
readout is vital, as some state initialization protocols
can be performed by just performing a readout. The key
is to find a suitable state-to-charge conversion process,
where Pauli spin blockade [534, 559], hence the related
latching mechanism [560], and Elzerman type spin-
dependent tunneling process [561] are explored. Hyperfine
coupling is a key anchor for the nuclear spin readout in
the dopant system where the state information can be
converted to a spin ESR signal [562]. Next, the corre-
sponding charge signal shall be correlated to the capacitive
difference, picked up by a single-electron transistor (SET)
or quantum point contact (QPC), and amplified further
[563]. In singlet-triplet qubits [559], a readout fidelity of
is reported. For single spins, the readout fidelity is
pushed to , beyond the fault-tolerant level
[542].
98%
µ
Single-lead RF-reflectometry spin readout [564] is a
pivotal technology to reduce the gate density for spin
readout and is compatible with surface code scalable
architectures for the fan-out issue. It was demonstrated
nearly simultaneously by four groups [495–498]. Using
this technique, readout fidelities above were shown
[497, 498], and a readout time of 6 s was proved [498],
comparable to the gate operation speed. To remedy the
broadening of the Fermi surface and related obstacles for
the Elzerman protocol at high temperatures (1–4 K), the
Pauli blockade, latching mechanism [539, 560], and
double SET readout [565] are valuable approaches. Also,
to improve the signal-to-noise ratio (SNR), quantum
noise-limited JPA [566] and other amplification methods
were combined. Multiple spin readout has been shown
for a single electron box [567] and done by frequency
multiplexing [568]. Several teams have already demon-
strated enhanced readout fidelities for S–T qubits and
spin qubits in a non-demolition method [569, 570], which
shall find their importance in fault-tolerant computing
and in studying spin state collapse problems for funda-
mental quantum mechanics. Moreover, cascade readout
[571], dispersive spin readout [499, 534], triple-dot cavity
dispersive readout [569], and ramped spin measurement
[572] have been shown. In general, the spin readout
techniques are more mature and ready for the future
scaling-up stage. Further progress will focus on high-
level multiplexing and integrating with the qubit design
in a scalable manner. Potential readout signal crosstalk
also needs further investigation and engineering design.
To facilitate spin initialization, except the usual applied
readout-assisted state initialization, a hot spot on the
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21308-18 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
T1
energy level due to valley mixing [521, 574] and spin-
orbital coupling could also be used to enhance the
relaxation rate.
T > 1.5
Scalability.—A unique advantage for silicon qubits
comes from its industry backbone, very large-scale inte-
gration (VLSI) technology. This advantage becomes
more significant as integrated cryo-CMOS and hot
qubits techniques emerge. An industry-level CMOS-
compatible hybrid quantum computing chip with classical
control unit and quantum processing unit is becoming
possible. Along with this spirit, progress has been made
in recent years, such as Intel’s horse-ridge cryo-CMOS
chip demonstrating qubit control fidelities rivaling room
temperature bulky control instruments [507]. A proof-of-
principle experiment shows a low-temperature classical
control unit bonded to a quantum unit [508]. Also, on
the quantum side, CMOS compatible “hot” qubits opera-
ted at increased temperature ( K) for the electron
spin [504, 505] and hole spin systems [506] are realized
nearly simultaneously. On the industry side, advanced
fabrication technologies are touching down on qubit
physics with the industrial foundry’s massive production
process [575]. CEA-Leti [576], Imec [577], and Intel all
processed silicon qubits with 300 mm technology [578],
and Intel has shown promising quantum dot uniformity
and basic spin qubit operations.
In the quest for fault-tolerant computing, scaling up
would be an inevitable technological hurdle to overcome
[579]. Several scaling architectures have been designed
for phosphorus donor qubits [557, 580], Si/SiGe gate-
defined dots [581], and MOS quantum dots [582, 583].
Also, research teams have realized multi-qubit devices
and few-qubit algorithms, such as three qubits [584] and
six qubits in the Si/SiGe systems [528], where entanglement
states were shown. Similarly, a four-qubit in the Ge/
SiGe hole spin system also made its debut, where a four-
qubit GHZ state was demonstrated [539]. Meanwhile,
methods for multiple quantum dots tuning using virtual
gates [585] and automated machine learning were developed
[586, 587] and multiplexed quantum dots readout was
realized [568, 588]. Toward QEC, a three-qubit phase
error correction algorithm was carried out by two groups
[25, 26]. Moreover, qubit networks could be a remedy for
the dense packaging problem and could reduce the fan-
out overhead [579]. Therefore, the coupling of spin
qubits at a distance is necessary. The effective spin-spin
coupling could be realized by using microwave cavities
[502, 503], mediated big quantum dot [589], spin array
state transfer [532], and surface acoustic waves [590].
For the cavity approach, strong coupling [591] and
photon-mediated spin-spin interactions are demonstrated
[502, 503]. The next step would be to realize cavity-
Fig.6(a) Physical system and level structure of the NV center. (b) NV center and the nuclear spins nearby form a multi-
qubit quantum information processor [614]. (c) NV center with the nuclear spins as memory qubits form a node in the quantum
network [615]. (d) NV center as a quantum sensor [616].
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-19
mediated two-qubit gates for spins at a distance and
hence spin networks in a distributed manner.
Conclusion.—High-quality materials and advanced
fabrication technologies are the cornerstones of semicon-
ductor qubits. To realize a higher gate fidelity, interfaces
with low charge noise are critical, which could be
achieved by importing industrial techniques of integrated
circuits and encouraging a transfer from lab-level engi-
neering to foundry-level fabrication. The typical overheads
for scalable quantum computing, such as crosstalk, gate
heating, and frequency crowding, should also be carefully
considered. Subsequently, semiconductor spin qubits will
join the other sophisticated quantum computing platforms
for next-level applications, such as fault-tolerant operations
and quantum simulations on intermediate-scale multi-
qubit devices. In summary, semiconductor qubits, espe-
cially the silicon spin qubits, are well-positioned for scaling
up with all those above-mentioned breakthroughs and
technological improvements. We are confident that by
embracing its industrial advantages, silicon spin systems
will speedily scale up their qubit numbers and join the
next-level quantum computing endeavor together with
superconducting and ion trap platforms.
6NVcenters
ˆz
S= 1
|ms= 0⟩
|ms=−1⟩
Introduction.—NV center is a point defect in the
diamond, where a vacancy and a nitrogen atom substitute
for two carbon atoms along the quantization axis
(assumed to be the axis), as shown in Fig. 6(a). The
negative charge state NV– is of greatest interest, where
there are five unpaired electrons originating from the
nitrogen atom and three carbon atoms, respectively,
together with an additional electron captured from the
environment. This six-electron system is equivalent to
an electron with spin projection , whose spin state
can thus be employed as a qutrit, or a qubit if only the
and energy levels are considered. The
NV center is a promising candidate for the quantum
computer by virtue of the following merits. First, a
single NV center can be optically resolved and located,
and the polarization and measurement can also be
achieved with a laser pulse. Second, the NV center has
an excellent coherence property even at room tempera-
ture. At low temperature, it can be resonantly excited to
enable efficient single-shot readout. Third, the nuclear
spins near NV centers serve well as abundant available
memory qubits for solid-state quantum information
processors.
13
Qubits and coherence.—The exceptional lifetime of
NV electron spins even under ambient conditions is
experimentally favorable for quantum computation. The
inhomogeneous magnetic fluctuations due to the C
spin bath are the main noise source responsible for the
dephasing of NV electron spins. However, with the
T∗
2
T2
widely used DD technique, the dephasing time can be
extended from the order of microseconds ( ) to millisec-
onds ( ), where the quasi-static noise is mostly
suppressed.
T2= 2
T2= 1
T1>6
T2∼
T1
T2
0.6
77
T2
1.5
T1
In addition to the central electron spin, nearby
nuclear spins are a rich resource for memory qubits. The
relatively low gyromagnetic ratio (around three orders of
magnitude less than that of electron spin) of nuclear
spins is mainly responsible for their extraordinarily long
coherence time, up to s at room temperature [592].
In addition to the 14N and 13C nuclear spins [593] (up to
27 spins nowadays [594]), researchers have been endeav-
oring to explore more available qubits in diamond,
including P1 centers [595–597] and long-lived carbon
nuclear spin pairs [598] ( min and min at 4
K [599]). Further improving the coherence time of NV
electron spins compared to the timescale of these
memory qubits is desired. Since ms at room
temperature is limited by the spin relaxation time , a
straightforward solution is to lower the temperature,
where has been extended to s at K [600]. At
4 K, however, has reached s with carefully-
designed sequences decoupling unwanted interactions,
and has exceeded 1 h [598].
|ms= 0⟩
99.9%
98%
Initialization and readout.—NV center can be optically
initialized and readout due to the spin-dependent inter-
system crossing (ISC) [601, 602] [see Fig. 6(a), illustrated
by grey dashed lines]. Thus, the electron spin can be
initialized to under continuous optical pumping.
On the other hand, ISC leads to a nonradiative transition
through singlet states, which enables the discrimination
of the spin states according to the fluorescence difference.
Furthermore, at low temperature, the resonant optical
excitation allows high-fidelity single shot readout of NV
electron spins [603]. The preparation and readout
fidelity have achieved [604] and [605], respec-
tively.
≥1/T∗
2∼102
∼500
The approach to initializing nuclear spins is less
straightforward and varies with the different coupling
strengths. Specifically, for 14N and some strongly-
coupled 13C nuclear spins ( kHz) [606, 607],
applying a well-aligned magnetic field Gauss leads
to the excited state level anti-crossing (esLAC) and the
polarization of nuclear spins [608]. However, esLAC fails
to enable the efficient polarization if the quantization
axis of the hyperfine interaction in the excited state
differs from that in the ground state, or if the nuclei-
electron quantization axis differs from that of the NV
itself [607]. On the other hand, the strongly-coupled
nuclear spins are less abundant than the weakly-coupled
ones, and hence researchers have focused more on
weakly-coupled 13C nuclear spins. The initialization of
the weakly-coupled 13C nuclear spins employs a swap-
like gate constructed by the DD sequences (see below)
[28].
In addition to the approaches discussed above, several
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21308-20 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
13
strategies exist for different purposes. For example,
dynamic nuclear spin polarization (DNP) is designed to
polarize the whole C spin bath by imposing the Hartmann
–Hahn double resonance [609], and has been improved to
be more robust [610]. Projective measurement-based
initialization is also preferred, especially in the case of
simultaneous multiqubit or multidegree-of-freedom
initialization with high fidelity [597, 611]. The single-
shot readout associated with projective measurements
not only provides an efficient way to polarize nuclear
spins, but also enables the direct test of non-classical
correlations and active feedback in QEC protocols.
Single-shot readout of 14N [612], weakly-coupled nuclear
spins [613], nuclear spin pairs [599], and P1 centers [597]
have been realized, respectively.
0.99995
0.992
700
∼10
π/2
π
0.95 ±0.01
0.99 ±0.016
Gates.—The control techniques have been well devel-
oped to implement quantum logic gates with high precision
as well as narrow pulse widths, where quantum opti-
mization algorithms have been exploited. Combining the
composite pulses with a modified gradient ascent pulse
engineering (GRAPE) algorithm has yielded the records
of the fidelity of single-qubit gate and two-qubit gate
being and [617], respectively, which almost
hits the threshold value required by QEC. Remarkably,
this highly accurate two-qubit gate has a duration of
ns, which is three orders of magnitude smaller than the
coherence time. Meanwhile, typical single-qubit gates are
in the order of ns, and gigahertz Rabi oscillations
are also possible with proper design [618, 619]. Moreover,
optimized ultra-fast single-qubit gates beyond the rotating
wave approximation have been realized with the applica-
tion of chopped random basis (CRAB) quantum opti-
mization algorithm, with fidelity for and pulses
being and , respectively [620].
Manipulating multiple qubits while maintaining coher-
ence is a crucial task for quantum computing. In partic-
ular, in hybrid systems, where the timescale of each
component may differ, it is desirable to implement all
the control sequences before any of the components
decohere. Instead of counting on the isotopically purified
samples [621], an active way to extend the coherence
time is the well-known DD techniques, during which the
quasi-static noise is flipped and canceled, and hence can
also be construed as a frequency filter [622]. In this sense,
the DD sequence applied to NV electron spins enables
the detection of the resonant frequencies corresponding
to surrounding interactions [623–625]. Consequently,
conditional two-qubit gates have been designed based on
DD sequences to control 14N [626], where an RF pulse
acts equivalently as a transverse hyperfine coupling to
drive the nuclear spin flip conditioned on the state of
the electron spin. Similarly and subsequently, universal
DD-based gates on weakly-coupled nuclear spins have
been achieved [28, 627], through which the nuclear spins
can be initialized and measured via swap-like gates.
Recently, an active phase compensation scheme named
DDrf has freed the dependence of interpulse delay on
hyperfine parameters and enabled the optimization of
the interpulse delay to protect electron coherence, even-
tually entangling up to seven nuclear spins in a ten-
qubit register [628].
An alternative is to utilize decoherence-protected
subspaces, where the evolutions of quantum states are
purely unitary [629, 630]. Moreover, geometric gates are
also intrinsically noise-resilient, where the dynamical
phase vanishes. Both non-adiabatic [631–633] and adiabatic
[634] universal (non-Abelian) [633] geometric gates have
been demonstrated. Nevertheless, towards large-scale
quantum technologies, QEC is expected to be a more
essential strategy, with respect to encoding the physical
qubits subjected to the deleterious noise from the envi-
ronment into reliable logical qubits. Armed with state-of-
the-art multi-qubit control techniques, QEC protocols
[27, 28] and a related work deploying a robust coherent
feedback control [635] have been realized. Most recently,
fault-tolerant operations on the logical-qubit level have
been achieved on a seven-qubit NV quantum processor,
indicating a major step toward fault-tolerant quantum
information processing [636].
Quantum simulation and quantum algorithm.—Sophis-
ticated control of spins in diamond promises rich appli-
cations in diverse fields. Various exotic physical
phenomena have been simulated, such as the emulations
of tensor monopoles [638] and quantum heat engines
[639], opening avenues for the exploration of fundamental
physics. NV center quantum simulator also expands the
scope of experimental investigations on quantum topo-
logical phases [640, 641]. Ref. [642] proposed a feasible
and universal approach to investigate the non-Hermitian
Hamiltonian in Hermitian quantum systems and
observed parity-time symmetry breaking in an NV quan-
tum simulator [637, 643]. Besides, simulations of non-
Markovian dynamics of open systems [644], many-body
localized discrete time crystals [614] and emergent
hydrodynamics [645] are distinguished from other artificial
platforms due to the real quantum nature and shed light
on condensed-matter physics.
On the other hand, many quantum algorithms have
also been demonstrated, including the Deutsch–Jozsa
algorithm [646], adiabatic quantum factorization [647],
Grover’s search algorithm with very high efficiency [648],
quantum-enhanced machine learning [648], and resonant
quantum principal component analysis [650].
>1
Quantum network.—With the help of flying photons,
two remote NV nodes ( km) with memory qubits can
be entangled [605, 651, 652]. Combining entanglement
distillation [653] and deterministic entanglement delivery
with more experimentally-favorable single-photon
scheme [654], a three-node quantum network with more
than one long-lived memory qubits has been realized
[615, 655].
Quantum sensing.—Owing to the robustness and
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Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023) 21308-21
micro- or nanoscale features of diamond, NV centers
have demonstrated the potential for high-sensitivity
magnetic sensing in condensed-matter physics [656–661]
and biophysics [662–664]. Moreover, NV sensors are also
capable of detecting electric field [665–668], temperature
[669, 670] and pressure [671]. Recently, the boundaries of
NV quantum sensing have been pushed into some
special or extreme-condition areas, such as at zero or
low magnetic field [619, 672, 673], under high pressure
[616, 674–678], and at high temperatures [679]. In paral-
lel, quantum optimization algorithms [680] and QEC
[681–683] have also been incorporated into quantum
metrology so as to improve the sensitivity in the presence
of noise.
Outlook.—Scalability is an inescapable question to
which every candidate physical system for quantum
computers should give an answer. There are two main
challenges for the NV center-based quantum computing,
namely device fabrications and multi-qubit control tech-
niques. On the one hand, the deterministic schemes
yielding NV centers with satisfactory precision are
desired to produce large-scale functional devices [684,
685]. Meanwhile, since the underlying destruction and
noises induced by the implantation may affect the coher-
ence of the spin qubits [686], a trade-off solution for
controllable production of NVs while preserving the
coherence time should be developed. Accordingly, the
demands in the fabrication arrays of nanostructures,
such as nanopillars [687] and parabolic reflectors [688],
which significantly enhance the collection efficiency, are
also stringent. Additionally, it is also inspiring to
explore the photocurrent-based electric readout of NV
signals [689, 690]. On the other hand, with the growth of
qubit number, the mechanism of noises in the system
becomes more and more complicated. There is a need for
techniques that integrate high-precision multi-qubit
control techniques with decoupling techniques that
suppress errors and crosstalk between multiple qubits.
7NMRsystem
Introduction.—NMR spectroscopy is a powerful and
widely used analytical tool for the structural characteri-
zation of various organic matter. For nearly eighty years,
it has spawned numerous scientific and technological
applications in diverse areas of physics, chemistry, and
life science. At the end of the twentieth century, motivated
by a strong interest in quantum information science,
there arose the idea of using liquid-state NMR to
construct a quantum computer [7–9]. It was found that
NMR is capable of emulating many of the capabilities of
quantum computers, including unitary evolution and
coherent superpositions. Actually, NMR quantum
computing soon became one of the most mature tech-
nologies for implementing quantum computation [691,
692]. For instance, as early as 2001, researchers at IBM
reported the first successful implementation of Shor's
algorithm on a 7-qubit liquid-state NMR quantum
computer [693]. Based on its well-established experimental
technologies, NMR has now achieved universal control
of up to 12 qubits [694–696], and allows investigation of
a wide range of quantum information processing tasks,
such as quantum simulation, quantum control, quantum
tomography, and quantum machine learning. In the
following, we briefly introduce the basics of NMR quantum
computation and its impressive achievements.
Basic Principle.—In order to physically realize quantum
information, it is necessary to find ways of representing,
manipulating, and coupling qubits to implement non-
trivial quantum gates, prepare a useful initial state, and
readout the answer.
|↑⟩
|↓⟩
Qubit.—NMR quantum computation uses spin-1/2
nuclei in molecules to encode qubits. Due to the Zeeman
effect, a spin-1/2 placed in an external magnetic field
has two possible orientations, spin up and spin down
, which naturally offers a two-level system, or a qubit.
In choosing a sample to be a quantum register, one
Table4Reported values on the NV center quantum platform.
Property Parameter Qubit Value Condition Reference
Coherence
T⋆
2
e36 μs506 G, 300 K Ref. [637]
N 25.1 ms
403 G, 3.7 K Ref. [628]
13C17.2 ms
13C–13C pair 1.9 min Ref. [599]
T2
(echo)
e 1.8 ms 690 G, 300 K Ref. [621]
N 2.3 s 403 G, 3.7 K Ref. [628]
13C770 ms
T1
e
>1
h 403 G, 3.7 K Ref. [598]
13C
>6
min Ref. [628]
Gate time Single-qubit e
<10
ns 850 G, 300 K Ref. [618]
Two-qubit e-N 700 ns 513 G, 300 K Ref. [617]
Gate fidelity Single-qubit e 99.995% 513 G, 300 K Ref. [617]
Two-qubit e-N 99.2%
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property must be satisfied, that is, the spins should be
distinguishable in frequency to allow individual qubit
addressability. For heteronuclear molecules, because
different types of nuclei have different gyromagnetic
ratios, they can be easily distinguished. In the case of
homo-nuclear molecules, although the nuclei have the
same Zeeman splitting, they may stay in different electron
environments, and the resulting nuclear shielding effect
could induce different amounts of frequency shifts. In
reality, precession frequencies for the nuclear spins can
vary substantially, so it is best to choose such nuclei to
form the quantum register. Figure 7 shows the
schematic of the NMR spectrometer and the commonly
used molecules to encode qubits.
|0⟩
Initialization.—Conventionally, quantum computation
starts from a pure state with all qubits initialized to the
computational basis vector . However, due to the low
polarization of the NMR spin ensemble at room temper-
ature, it is practically rather difficult to get a genuine
pure state. Alternatively, one can use the concept of
pseudo-pure state (PPS) as a substitute [7]. PPS is a
mixture of the maximally mixed state and a pure state,
thus having similar behavior to that pure state under
quantum gates and quantum measurements. To prepare
PPS from the thermal equilibrium state, it would be
necessary to involve non-unitary operations, which can
be realized by applying gradient field pulses or utilizing
relaxation effects. Currently, there exist a number of
methods for PPS preparation, such as spatial averaging
[698], line selective [699], and labeled-PPS [694, 700], etc.
Ref. [701] analyzed and compared the efficiencies of
these methods based on the theory of optimal bounds on
state transfer under quantum channels. Overall, PPS
has proven to be a convenient and useful tool for small-
scale NMR quantum computation, yet when the number
of qubits grows, there is a significant scalability chal-
lenge, i.e., the achievable purity of PPS scales very unfa-
vorably. Approaches that attempt to address the scalability
issue include algorithmic cooling [702] and parahydrogen-
induced polarization [703], which have demonstrated the
ability to prepare NMR spin systems with purities even
above the entanglement threshold.
J
J
π
J
Operation.—One-qubit gates are just rotations on the
Bloch sphere, which can be easily implemented in NMR
with soft radio-frequency pulses. Soft pulses are usually
of predefined shapes, such as the Gaussian waveform
[704]. They contain energy only within a limited
frequency range, and thus can selectively excite those
spins that locate in this range. Therefore, a natural way
to implement a single-qubit gate is to use a resonant,
rotating Gaussian pulse with sufficient selectivity. But
one should be careful that, when going back to the lab
frame, there may be some phase errors that must be
compensated to get the genuine target gate [705]. In
NMR, a two-qubit gate is realized by making use of the
natural -coupling between the nuclei. In the case of
multi-qubit gates, since all the -couplings between the
spins are evolving, one has to design refocusing schemes
that are composed of a sequence of pulses to effectively
turn off the unwanted -coupling terms [706]. The usual
way to quantify the level of coherent control is the
randomized benchmarking protocol. Using randomized
Fig.7(a) Schematic of a high-field liquid-state NMR spectrometer [697], which can be used for quantum computation.
(b) A list of some commonly used molecules for nuclear spin quantum registers, ranging from two-qubit to twelve-qubit
samples. The labels indicate nuclei 13C, 1H, or 19F (all having spin number 1/2) that are chosen as qubits candidates.
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4.7±0.3×10−3
55.1%
benchmarking, an average error rate for one- and two-
qubit gates of on a three-qubit system
was reported [707]. Another work has used a unitary 2-
design and twirling protocol to estimate the average
fidelities of Clifford gates on a seven-qubit NMR proces-
sor, finding an average experimental fidelity of
[708]. NMR has also explored other types of quantum
gate implementation such as geometric quantum compu-
tation [709, 710]. Finally, we remark that NMR also
provides non-unitary control means such as gradient
field pulse and phase cycling, which are modeled as
random unitary channels and can be used to destroy
unwanted coherences.
Measurement.—NMR measurement is implemented
by observing the free induction decay (FID) of the
transversal magnetization of the spins by a detection
coil wound around the sample. The recorded time-
domain FID signal is Fourier transformed to obtain a
frequency-domain spectrum, which is then fitted to
obtain information about the spin state. Unlike projective
measurements in other quantum systems, we can
directly measure the expectation value of a single coherent
Pauli observable in NMR, which is, in fact, an ensemble
system. In order to measure the Pauli operators other
than the directly observable single quantum coherences,
it is necessary to apply an appropriate readout pulse to
the spin state before acquiring the FID signal. To estimate
an unknown quantum state, that is, to perform quantum
state tomography, one needs to measure a complete set
of basis operators. However, this is generally a challenging
task since the number of degrees of freedom to be deter-
mined grows exponentially with system size.
NMR quantum control.—Over 50 years of develop-
ment, researchers have developed abundant pulse
control techniques in the NMR spin system, such as
frequency selective pulses, composite pulses, refocusing
schemes, and multiple pulse sequences, to name a few.
These pulse techniques originated in demand for precise
spectroscopy of complex molecules, and continue to be
useful for NMR quantum information processing experi-
ments [706]. Despite this, it is still desirable to further
improve NMR control techniques to realize sufficiently
high-fidelity gates that fulfill the fault-tolerant quantum
computation requirement. Therefore, the interdisciplinary
field of NMR and quantum control theory naturally
arose, resulting in novel and more efficient pulse design
and optimization techniques. For small-sized systems,
one can employ time optimal control theory to reduce
gate time so as to reduce decoherence effects [711, 712].
For relatively larger systems, it is usually hard to derive
analytical control solutions, and then one needs to resort
to numerical means. One of the most successful
approaches in this regard is the GRAPE technique
developed in Ref. [713], which is flexible, easy to use,
and can produce smooth, optimal, and robust shaped
pulses. GRAPE and its many variants have found broad
applications not just in NMR but also in other experimental
platforms. However, these numerical approaches are
intrinsically unscalable. It would be a rather resource-
consuming task to simulate controlled quantum evolution
with a classical computer, even for a system with over
tens of qubits. One possible approach to overcome this
problem is to use subsystem-based quantum optimal
control [705]. Another promising strategy is the hybrid-
classical version of GRAPE, which employs a quantum
simulator to efficiently simulate the controlled evolution
[714]. This is essentially a closed-loop strategy, and has
been experimentally tested first on a seven-spin system
[714] and later on a twelve-spin system [696] to create
multiple-correlated spin states. Besides scalability, noise
is another major obstacle for high-fidelity quantum
control, which could be addressed by robust control or
open quantum system control. For example, more
advanced DD sequences were put forward on solid-state
NMR, resulting in much-improved robustness against
different types of experimental errors while retaining
good decoupling efficiency [715, 716]. It is worth
mentioning that, the above mentioned control methods,
such as composite pulse, GRAPE, spin echo, and DD,
though developed from NMR firstly, are not at all
restricted to NMR. Actually, many of these methods
have already been successfully applied to other physical
systems. Therefore, it is fair to say that NMR is an
excellent platform and testbed for developing quantum
control methods [705].
NMR quantum processor.—The NMR field has well-
established quantum control methods and experimental
technologies, enabling a series of influential fundamental
or applicative researches in quantum computing, quantum
simulation, quantum cloning [717–719], QEC [720, 721],
quantum thermodynamics [722–725], quantum contextu-
ality [726], etc. In the following, for short, we only
review a few developments related to quantum algo-
rithms, quantum simulation, and quantum learning.
Quantum algorithm.—Since the early stage of NMR-
based quantum computing, there have been reported
experimental realizations of some of the well-known
quantum algorithms, such as Deutsch-Jozsa algorithm
[727] on a two-qubit carbon-13 labeled chloroform
molecule, Grover’s search algorithm [728] on another
two-qubit sample partially deuterated cytosine, QFT
algorithm on a three-qubit sample [729], and Shor's
quantum factoring algorithm [693] on a seven-qubit
system.
Quantum simulation.—NMR has been used as a quan-
tum simulator to explore a variety of interesting quantum
phenomena, ranging from quantum many-body physics,
quantum chemistry, biology, and even cosmology. Simu-
lating the equilibrium and non-equilibrium dynamics of
quantum many-body systems is one of the most fascinating
topics in the field of quantum simulation, and NMR
seems to be very suitable for this task. For example, a
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21308-24 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
three-spin frustrated magnet was simulated with NMR,
in which the phase of the system as a function of the
magnetic field and temperature was explored [730]. The
phase diagram of the ground state of the Hamiltonian
with three-body interactions was simulated [731] and the
phase transition of the long-range coupling model was
first observed by monitoring Lee–Yang zeros [732]. In
another research, the authors employed a four-qubit
NMR simulator to explore the use of out-of-time order
correlators to probe quantum information scrambling
[733] and equilibrium or dynamical quantum phase tran-
sitions [734] in a chaotic Ising chain model. NMR has
also found applications in various chemistry problems by
directly simulating molecules or chemical reactions, such
as computing the ground-state energy of a hydrogen
molecule [735], finding the energy spectrum of a water
molecule [736], and exploring the prototype laser-driven
isomerization chemical reaction dynamics [737]. Besides,
the NMR quantum simulator can also be used to investigate
topological orders by simulating the ground state of a
topological Hamiltonian [738–741].
9
6
Quantum machine learning.—NMR has been one of
the experimental platforms where quantum machine
learning algorithms can be demonstrated, and it is in the
initial stages of exploring the use of quantum machine
learning to directly process classical image information.
For instance, a hand-written image recognition task to
discriminate between and is realized by implementing
a quantum support vector machine on a four-qubit
NMR processor [742]. The boundary that separates
different regions of an image is detected experimentally
by implementing a quantum image processing algorithm
[743]. Quantum principal component analysis, an impor-
tant tool for pre-processing data in machine learning,
has also been experimentally implemented on NMR for
the first time for small-scale human face recognition
tasks [744].
Outlook.—Primary challenges for liquid-state NMR
quantum computation include a lack of appropriate
molecules to serve as quantum registers, unavailability
of high-purity quantum states and quantum resources
such as entanglement, and difficulty in achieving scalable
and high-fidelity control on large spin systems. One
approach that may overcome some of these limitations is
to shift to solid-state NMR. Solid-state NMR has
already been used in demonstrating quantum heat
engine [745], exploring many-body localization [746, 747],
and observing prethermalization [748, 749]. Some other
promising approaches that are closely related to NMR
are the silicon-based nuclear spin quantum computer,
which is a hybrid between the quantum dot and the
NMR [10], and the recent technology of nuclear electric
resonance [750]. Finally, while NMR has certain intrinsic
difficulty in becoming a scalable route to large-scale
quantum computation, the many lessons learned in the
past decades’ research are very likely to be relevant for
advancing the development of other quantum technolo-
gies.
8Neutralatomarrays
Over the past two decades, deterministically prepared
neutral atom arrays have emerged as a promising platform
for quantum computing and quantum simulation
[751–755]. Controlled interactions between atomic qubits
are mediated by the long-range dipole-dipole interactions
via Rydberg states. These long-range Rydberg interactions
allow creating specific quantum Hamiltonians and easy
analog quantum simulations. They are also the workhor-
se for constructing digital gates and realizing any physi-
cal models. Most experiments to date focus on alkali ato-
ms Rb and Cs, which have single valence electrons and
can be simply laser-cooled and manipulated. In recent
years, with two valence electrons, alkaline-earth(-like)
elements Sr and Yb have attracted growing attention
Fig.8Schematic of a neutral atom quantum computer. In
a microscopic tweezer array, single atoms are rearranged into
defect-free arbitrary patterns. Atomic qubits can be encoded
in electronic spin states or nuclear spin states. Single-qubit
operations are performed through microwave or optical spec-
troscopy. Two-qubit gates and entanglement are realized
based on long-range Rydberg interactions. EMCCD: Electron
Multiplying Charge-Coupled Device. SLM: Spatial Light
Modulator. 2D AOD: Two-Dimensional Acousto-Optic
Deflector.
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due to their appealing features, such as narrow and ultra-
narrow optical transitions and magic-wavelength optical
traps for Rydberg states [756–759]. The schematic
diagram of atomic qubits and their pros and cons are
summarized in Fig. 2(e).
80%
90%
Scalability.—A neutral atom quantum computer is
based on an array of single atoms localized in optical
tweezers, as depicted in Fig. 8. Quantum information is
encoded in electronic spin states of alkali atoms or
nuclear spin states of alkaline-earth(-like) atoms.
Neutral atom platforms have a notable advantage for
scalability. It is relatively easy to expand the number of
atomic qubits. In a microscopic optical tweezer, either
one atom or zero atoms can be trapped each with a
probability of roughly 50% due to light-assisted collisions
[760, 761]. The stochastic loading efficiencies have been
enhanced to – for alkali species and 96% for alka-
line-earth(-like) species by accurately tuning parameters
under blue-detuning lasers and using artful cooling tech-
niques [762–766]. After probabilistic loading in tweezers,
single atoms can be rearranged into defect-free arbitrary
patterns using a real-time control system and dynamically
moving tweezers [767–769]. Up to date, large-scale plat-
forms consisting of more than 100 neutral atoms have
been created, such as an array with an average number
of 110 133Cs atoms [770], defect-free square and triangular
arrays of 196 and 147 87Rb atoms [771], and a defect-free
programmable array of up to 256 87Rb atoms [772]. The
rearrangement process takes a total time of hundreds of
milliseconds and results in a high filling fraction of up to
6000 s
∼37%
99% [771, 772]. In the rearrangement of larger arrays,
atom losses from tweezers will limit filling fractions, as
the rearrangement time will increase with the system
size. The typical trap lifetime is about tens of seconds
due to collisions with background gas in a vacuum
chamber. In order to reduce the residual gas pressure,
optical tweezers can be placed in a cryogenic environment
at a temperature of a few kelvins. The trapping of single
Rb atoms in cryogenic arrays of optical tweezers has
been demonstrated with a measured lifetime up to ,
a 300-fold improvement compared to the room-temperature
setup [773]. In this cryogenic experimental setup, large
arrays consisting of more than 300 87Rb atoms have
been realized with an unprecedented probability of
to prepare defect-free arrays [774]. We anticipate
that the number of atomic qubits in a neutral atom
processor will be increased from hundreds to thousands.
Furthermore, several individual processors will be
coupled together with atom-photon interconnects.
F > 99.5%
Initialization and readout.—Atomic qubits encoded in
the internal states can be initialized using optical pumping
techniques. The estimated preparation fidelity for single
alkalis is [775]. A widely used technique for
qubit readout relies on the state-selective ejection of
neutral atoms. When illuminating with a resonant laser
pulse, any atoms in one state are pushed out of the optical
tweezers, whereas atoms in the other state are not influ-
enced and remain trapped. Subsequently, trapped atoms
are detected by collecting laser-induced fluorescence
which is not state-selective. The typical measurement
T1
T∗
2
T′
2
F1
F2
t1
t2
Nd
Na
Table5Reported state-of-the-art performance of neutral-atom qubits. is the spin relaxation time. and refer to the
inhomogeneous and homogeneous dephasing time. ( ) and ( ) are the gate fidelity and the operation time of single(two)-
qubit manipulation. and refer to qubit numbers in digital quantum processors and analog quantum simulators, respec-
tively.
Property
Alkali atom Alkaline-earth(-like) atom
Electronic spin Nuclear spin
85,87Rb/133Cs 87Sr 171Yb
Coherence
T1
4s
[782, 791]
≫10 s
[794]
10−100 s
[766]
T∗
2
4ms
[782, 791]
21 s
[794] 3.7 s [766]
T′
2
∼1s
a [782, 791]
40 s
b [794] 7.9 sb [766]
Gate time
t1
0.1−10 μs
0.7μs
[766]
t2
0.4−2μs
0.9μs
[788]
Gate fidelity
F1
∼99.97%
c [782]
99.48%
d [766]
F2
97.4%
e [787] 83%e [788]
Qubit number
Nd
6f [791], 24g [782]
Na
289 [795]
Environment
P∼10−11 Torr
B∼10 G
Ultra-high vacuum , magnetic field
a XY8 and XY16 dynamical decoupling sequences.
b Spin echo process.
c This is estimated from the scattering limit which is consistent with an accumulated error. Randomized benchmarking will be applied
in the future.
d Randomized benchmarking.
e Bell state fidelity.
f GHZ state based on individual addressing of single atoms.
g Toric code state based on coherent transport of entangled atoms.
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F > 98%
fidelity is [775]. Atom losses in this technique
prevent from measuring qubits in the middle of quantum
circuit execution. As an alternative, for a lossless fluores-
cent state detection, only a small number of photons
should be scattered to minimize the atom heating. This
approach has been demonstrated for one qubit and
multiple qubits in optical tweezers [776–779]. In the
future, atom losses due to heating will be reduced to a
level that allows implementing repetitive QEC for quantum
computation.
F1= 99.83%
3×10−4
5.2×10−3
Gates.—We summarize the state-of-the-art gates
implemented in Table 5. Single-qubit operations are
performed through microwave or optical spectroscopy.
In large atom arrays, a universal approach for single-site
addressing is the focused laser beam scanning or the
application of static field gradients. The gate operation
time is t1 = 0.1–10 μs. Single-qubit gate errors are
caused by the fluctuations of the pulse amplitude and
detuning [775, 780]. An average fidelity of
was measured in the 133Cs experiment using randomized
benchmarking [781]. Recently, the gate fidelities for 87Rb
atoms have been enhanced by using composite pulse
sequences, which make gate errors highly insensitive to
pulse errors [782]. The estimated gate errors are about
. In the platform of alkaline-earth(-like) atoms,
the average single-qubit gate error of has been
extracted [766].
t2= 0.4
2µs
F2≥97.4%
F2= 83%
>99.5%
Controlled interactions between neutral atoms are a
fundamental requirement for entangling particles. One
strategy is to implement local entangling operations via
ultracold spin-exchange interactions, which has been
demonstrated with two individual atoms in movable
tweezers [783]. However, it is a great challenge to overlap
the atomic wavefunctions in neutral atom arrays.
Another strategy is based on the strong and long-range
dipole-dipole interactions between Rydberg atoms.
When two atoms are in close proximity to each other,
two-qubit gates and entanglement have been realized via
the Rydberg blockade effect [784, 785]. Most researchers
give attention to the latter one owing to its feasibility.
The corresponding gate time is – . For two-
qubit gates via Rydberg interactions, the dominant
sources of gate errors are the ground-Rydberg Doppler
dephasing, the spontaneous emission from the intermediate
state in the two-photon excitation process, and the exci-
tation laser phase noise [775]. In the 87Rb atom arrays,
the fidelities of the two-qubit entanglement operations
have been extracted to be by suppressing
Rydberg laser phase noise via a reference cavity [786,
787]. In an array of 171Yb atoms, a two-qubit gate with
the fidelity of has been firstly demonstrated in
[788]. In this experiment, the gate error is attributed to
Raman scattering from the gate beam and autoionization
from a small Rydberg population. For two individually
trapped 88Sr atoms, a Bell state has been created with a
high fidelity of , in which qubits are encoded in a
metastable state and a Rydberg state [789].
In the Rydberg excitation process, we must consider
the effect of the different trapping potentials for both
ground and Rydberg levels. In the experiments with
single 87Rb or 133Cs atoms, the tweezers are turned off
for a short duration to mitigate anti-trapping of the
Rydberg states. Atom losses and heating limit the
Rydberg excitation time and the number of excitation
loops. When using 0.5 μs drops for each two-qubit gate,
hundreds of drops can be made before atom loss
becomes significant [782]. It should be noted that the ion
core polarizability of the alkaline-earth(-like) atoms can
be used to trap Rydberg states in conventional, red-
detuned optical tweezers. The Rydberg states of single
174Yb atoms have been stably trapped by the same red-
detuned optical tweezer that also confines the ground
state [790]. Therefore, the interaction time of Rydberg
states for alkaline-earth(-like) atoms can be extended.
T1∼4s
T∗
2∼4ms
T′
2∼1s
225 ms
T∗
2∼1.5s
T1≫10 s
T∗
2= 21 s
T′
2= 40 s
Coherence.—Neutral atoms are well isolated from the
environment and exhibit long coherence times. For Rb
and Cs atoms [782, 791], the hyperfine qubit relaxation
time is about , which is limited by the spontaneous
Raman scattering of photons from the trapping laser.
The inhomogeneous dephasing originates from the
energy distribution of trapped atoms. The typical
dephasing time is . For the homogeneous
dephasing, common mechanisms are the intensity fluctu-
ations of the trapping laser, magnetic field fluctuations,
and heating of atoms. The homogeneous dephasing time
of has been observed using XY8 and XY16 DD
sequences [716, 792]. By analyzing these mechanisms, we
observe that the differential light shift of qubit states is
the root of the dephasing. A magic-intensity trapping
technique allows mitigating the differential light shift.
The coherence time has been enhanced to , where
the extracted inhomogeneous dephasing time is
[793]. In comparison with the electronic spin
qubits, nuclear spin qubits in alkaline-earth(-like) atoms
are robust to perturbation by the optical tweezers. The
estimated coherence time of single 87Sr atoms are
, , and in the spin echo
process [794]. In addition, coherence properties of single
171Yb atoms have been measured [766, 788], as listed in
Table 5.
Digital quantum operations.—Digital gate-based
circuits on programmable neutral atom processors were
demonstrated by two experimental groups. In Ref. [791],
researchers at Wisconsin employed an architecture based
on individual addressing of single atoms with tightly
focused beams. Quantum circuits were decomposed into
global microwave rotations, local phase rotations and
local two-qubit CZ gates. Scanning Rydberg excitation
beams enabled coherent and simultaneous addressing of
pairs of atoms. In this platform, researchers demonstrated
the preparation of GHZ states with up to 6 qubits,
quantum phase estimation algorithm for a chemistry
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problem, and QAOA for the maximum cut graph prob-
lem. In Ref. [782], researchers at Harvard employed
another architecture based on the coherent transport of
entangled neutral atoms. Two-qubit CZ gates were
implemented in parallel by two global Rydberg laser
beams. Subsequently, entangled qubits were coherently
transported to change the connectivity and perform the
next layer of quantum operations. This architecture was
used to generate a 12-qubit cluster state, a 7-qubit
Steane code state, topological surface, and toric code
states. Finally, researchers realized a hybrid analogue-
digital evolution and measured the entanglement
entropy. These results represent a key step toward realizing
a quantum computer with neutral atoms.
7×7
Zn
Analog quantum operations.—Combined with the
wide tunability of array geometry, Rydberg atom arrays
are suitable to implement various Hamiltonians
[753–755]. When the spin states are encoded in the
ground level and the Rydberg level, the quantum Ising-
like models are obtained. In a one-dimensional chain
with up to 30 atoms and a atoms array, the excitation
dynamics and the pair correlation functions of quantum
Ising models were explored after suddenly switching on
the Rydberg excitation pulse [796, 797]. The similar
quench dynamics were also studied in the linear and
zigzag chains [798]. For three-dimensional arrangements
of Rydberg atoms, quantum Ising Hamiltonians mapped
on various connected graphs were constructed with tens
of spins [799, 800]. Sweeping the Rydberg excitation
detunings allows probing more abundant many-body
dynamics. Quantum phase transitions into ordered
phases and the critical dynamics were demonstrated in a
one-dimensional chain with tunable interactions [801,
802]. Antiferromagnetically ordered states were further
explored in two-dimensional arrays with up to hundreds
of atoms [771, 772, 803]. Although the Rydberg interactions
generally lead to thermalization in many-body systems,
it was realized that quantum many-body scars avoided
rapid thermalization when preparing the two-dimensional
atoms array in the antiferromagnetic initial state [804].
Besides the Ising-like models mentioned above, recent
works include observing topological phases in a quantum
dimer model and a Su-Schrieffer-Heeger model [805, 806],
engineering the XXZ spin model using a periodic external
microwave field [807], and investigating quantum opti-
mization algorithms for solving the maximum independent
set problem [795].
Outlook.—Recent breakthroughs in Rydberg atom
arrays exhibit the ability to study many-body physics
and realize highly programmable and scalable quantum
computing. Primary challenges for this platform are
higher fidelity of two-qubit gates, quantum nondemolition
measurements, and low crosstalk between individual
qubits in a large array. Two-qubit gate fidelity can be
improved by further cooling alkali atoms via Raman
sideband cooling [808, 809] or using alkaline-earth(-like)
elements that have exhibited excellent gate performance.
Combined with lossless fluorescent state detection, intro-
ducing a second atomic element allows monitoring quantum
processors via quantum nondemolition couplings to
auxiliary qubits [810]. In addition, a dual-element platform
enables low crosstalk manipulations of the homonuclear
and heteronuclear interactions when increasing system
size [811].
9Photonicquantumcomputing
Introduction.—Amongst all platforms for quantum
computing, photon has several unparalleled advantages:
i) one of the best candidates for room-temperature quantum
computing, owing to the inherent advantage that it
weakly couples with the surrounding environment, and ii)
natural interface for distributed quantum computing,
which acts as a flying qubit to connect many quantum
nodes, and iii) compatible with CMOS technologies,
bringing optical quantum computing to a new cutting-
edge stage.
Optical quantum computing can be dated back to
2001, when Knill, Laflamme, and Milburn (KLM)
pointed out that it is possible to create universal quantum
computing solely with linear optical elements [812]. This
landmark work opens a way for linear optical quantum
computing. However, the daunting resource overhead
makes the KLM scheme extremely hard to implement.
In 2010, a much more feasible and intermediate
model—boson sampling—was proposed and analyzed by
Aaronson and Arkhipov [813]. Compared to the KLM
scheme, boson sampling is a much easier linear optical
quantum computing model which can beat all classical
computers with only 50–100 photons, but at a cost that
it's no longer universal. In 2017, a variant called Gaussian
boson sampling (GBS) was developed by Hamilton et al.
[814], in which the input is replaced as single-mode
squeezed states rather than single photons. It is a new
paradigm that not only can provide a highly efficient
approach to large-scale implementations but also can
offer potential applications in graph-based problems and
quantum chemistry.
In the past two decades, we have witnessed great
progress in linear optical quantum computing [815, 816],
especially on single-photon sources, linear optical
networks, and single-photon detectors. These achievements
have enabled a series of essential experimental results in
the preparation of large-scale entangled states [817–819],
and quantum computational advantage through boson
sampling [16, 18, 820].
Photon qubit.—Photon has the richest degrees of freedom
to encode as a qubit. In the following, we summarize
several frequently-used bases for photonic qubits.
● Polarization: The qubit can be encoded on the two
orthogonal geometrical orientations of electromagnetic
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field. It is widely used in linear optical quantum
computing.
● Path: Two transmission paths of single photons can
be a qubit. Usually, the phase of the path in free space
is unstable, while it’s perfect for integrated photonics.
● Time bin: The former and latter arrival time of
single photons is encoded as a qubit.
● Frequency bin: Frequency-bin qubit refers to the
superposition state of a photon with two different
frequencies (colors).
● Photon number: Vaccum and single-photon states
encode qubit’s 0 and 1, respectively.
Lz=mℏ
m
● Orbital angular momentum (OAM): OAM describes
the spatial distribution of light. In quantum theory,
OAM of a photon has a value of . Any two
OAM states with different values form a photonic
qubit.
Quantum light sources.—Single-photon source is one
kind of quantum light source that emits one and only
one photon at a certain time, in a well-defined polarization
and spatial-temporal mode. Specifically, the single
photons should possess the same polarization, spatial-
temporal mode and transform-limited spectral profile for
a high-visibility Hong-Ou-Mandel-type quantum inter-
ference [821].
Spontaneous parametric down-conversion (SPDC)
sources [822, 823] play a vital role in many fundamental
quantum optics experiments. Notably, this year’s
physics Nobel prize is partially “for experiments with
entangled photons”. However, SPDC is intrinsically
probabilistic and unavoidably mixed with multiphoton
components. The single-photon efficiency is typically as
low as ~1% to suppress unwanted two-photon emission.
To overcome this issue, an alternative way is to multiplex
many SPDC sources to boost the efficiency of single-
photon sources [824]. Another approach is directly
generating high-quality single photons from a two-level
system. Amongst all platforms [825–832], semiconductor
quantum dots [833] provide state-of-the-art single-
photon sources with an overall efficiency of 57% [834].
This mainly benefits from a polarized microcavity devel-
oped by Wang et al. [835], which has a polarization-
dependent Purcell enhancement of single-photon emission
so that the overall efficiency can surpass the 50%. In
near future, the single-photon efficiency can be improved
over 70% by better sample growth and boosted collection
efficiency, which should surpass the efficiency required
for universal quantum computing [836].
In quantum optics, another quantum light source is
squeezed state, which refers to a quantum state that the
uncertainty of the electric field strength for some phases
is smaller than that of a coherent state. Such a state is
commonly generated by strongly pumping nonlinear
mediums [837]. It was shown that continuous-variable
(CV) quantum computing can be constructed [838] by
using squeezed states and simple linear optical elements,
such as beam splitters and phase shifters. Then, Gottes-
man, Kitaev, and Preskill (GKP) [839] proposed a
robust QEC code over CVs to protect against diffusive
errors. Until now, the record squeezing is 15 dB from
type I optical parametric amplifier [840], and many
quantum experiments were executed towards large-scale
CV quantum computing [841–843]. Compared to
discrete-variable (DV) quantum computing, CV has a
valuable feature that the entanglement can deterministi-
cally emerge by mixing two squeezed states with a
simple beam splitter, while it is hard to obtain in DV
case since such a nonlinear interaction is so weak that
we have to generate entanglement by a conditional fash-
ion, namely as post selection. Nevertheless, CV quantum
information processing can never be perfect, because the
quality of entanglement strongly depends on the amount
of squeezing that it’s extremely sensitive to the loss.
2(n−1) −1
n(n−1)
2
n
n−1
n2−2n+2
2
Linear optical networks.—The interferometer acts as a
unitary transformation on the single-photon Fock state
or the single-mode squeezed state. In 1994, Reck et al.
[844] showed that a universal unitary transformation can
be realized by beam splitters and phase shifters arranged
in a triangular configuration. In this scheme, the optical
depth is , the number of beam splitters is
, where is the number of modes. In 2016,
Clements et al. [845] demonstrated that an interferometer
with a rectangular configuration is equivalent to a trian-
gular one. Moreover, the optical depth is reduced to
, and the number of beam splitters is reduced to
. This is a more compact and robust design with
a symmetry configuration.
For boson sampling, the linear optical network should
combine high transmission, Haar randomness, high
spatial and temporal overlap simultaneously. There are
many different implementation approaches, such as
micro-optics [16, 820, 846, 847], time-bin loops [18, 848],
and integrated photonic circuits [849–851]. For universal
quantum computing, it should further be programmable.
Micro-optics possesses the highest transmission efficiency,
while it lacks the demonstration of programmability.
Time-bin loops and integrated on-chip circuits are
programmable but suffer from serious losses. How to
reduce the losses meanwhile programmable is a long-
sought goal in the future.
n
m
m
≫
n
Boson sampling.—In 2011, Aaronson and Arkhipov
[813] argued that a passive linear optics interferometer
with single-photon state inputs cannot be efficiently
simulated. This model is so-called boson sampling, a non-
universal quantum computing model much easier to
build than universal quantum computing. In boson
sampling, identical bosons are sent into an -mode
( ) Haar-random interferometer and sampling the
output distribution in the photon number basis. Because
of the bosonic statistics, the probability amplitudes of
the final state are related to the permanent of submatri-
ces, a problem known to be in the complexity class of
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#P-complete. It is strongly believed that a moderate-
size boson sampling machine, even an approximate one
with a multiplicative error, will be intractable to be
simulated with state-of-the-art classical computers [813].
More importantly, boson sampling is a strong candidate
to demonstrate quantum computational supremacy [852],
an important milestone in the quantum computing field.
In 2013, simultaneously, four groups reported the
small-scale proof-of-principle boson sampling experiments
[849–851, 853]. Indeed, the final photon distribution is
proportional to the square of permanent modulus.
However, all these experiments are based on SPDC
sources, which were intrinsically probabilistic and mixed
with multi-photon components. In an attempt to solve
the intrinsic probabilistic problem of SPDC, scattershot
boson sampling was proposed in 2014 by Lund et al.
[854] and was first demonstrated in 2015 by Bentivegna
et al. [855]. Then, Zhong et al. [817] improved the
photon number to five. Though scattershot Boson
sampling is theoretically beautiful, it is hard to realize
quantum advantage experimentally due to the extensive
challenges of ultra-high heralding efficiency, fast optical
switches, and an excess of SPDC sources required.
A direct way to solve the problems of SPDC is
directly using on-demand single photon sources based on
coherently driving a quantum two-level system. In 2017,
Wang et al. [846] successfully performed the first five-
1014
photon boson sampling experiment using an actively
demultiplexed quantum-dot single-photon source and an
ultra-low loss photonic circuit, and showed a high
sampling rate that is 24000 times faster than all previous
experiments, beating early classical computers —
ENIAC and TRADIC. In 2019, Wang et al. [847]
demonstrated a boson sampling with 20 input photons
and a 60-mode interferometer. Finally, at most 14
photons are detected at the output, and the output state
Hilbert space reaches up to 3.7 × dimensions, which
is over 10 orders of magnitude larger than the previous
works.
1043
1010
A more efficient way to demonstrate quantum compu-
tational advantage is through GBS, thanks to the single-
mode squeezed state inputs, of which more than one-
photon components are allowed while its computational
complexity is as hard as original boson sampling. In
2020, a landmark experiment was executed by Zhong et
al. [16] and successfully demonstrated quantum compu-
tational advantage. Then, GBS was improved with 50
single-mode squeezed-state inputs and a 144-mode inter-
ferometer, and up to 113-click coincidences are detected
[820] (see Fig. 9). These rudimentary photonic quantum
computers, named as Jiuzhang, in honor of an ancient
Chinese mathematical classic — “The Nine Chapters on
the Mathematical Art”, yield an output state space
dimension of and a sampling rate that is faster
Fig.9Experimental setup of Jiuzhang. It is mainly composed of five parts. At the upper left region, high-intensity trans-
form-limited pulse laser with wavelength of 775 nm are prepared to pump 25 two-mode squeezed state (TMSS) sources (at
the left region, labeled in orange). Meanwhile, continuous-wave 1450-nm laser are guided and co-propagates with the
25 TMSS sources. The 1550-nm two-mode squeezed light is collected into temperature-insensitive single-mode fiber, of which
5-m bare fiber is winded around a piezo-electric cylinder to control the source phase (at the center region). In the center-right
region, by using optical collimators and mirrors, 25 TMSSs are injected into a photonic network and 25 corresponding light
beams (colored in yellow) with wavelength of 1450 nm and intensity power of about 0.5 μW are collected for phase locking.
The 144 output modes are distributed into four parts using arrays of tunable periscopes and mirrors. Finally, the output
modes are detected by 144 superconducting nanowire single-photon detectors and registered by a 144-channel ultra-fast electronics
coincidence unit.
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than using the state-of-the-art simulation strategy and
supercomputers.
Technically, Jiuzhang is partially programmable owing
to the precisely controlling the phases of input TMSSs.
To make boson sampling fully programmable, a notable
way is to encode photonic modes to time bins [856]. In
this approach, the splitting ratio and phase of every
time bin can be changed as will in real time. In 2017, He
et al. [848] reported the first time-bin-encoded boson
sampling combining an on-demand quantum dot single
photon source. It is worth noting that this time-bin
multimode network is fully electrically programmable,
and fully connected (without zero elements). Recently,
combining the aforementioned GBS approach, Madsen
et al. [18] reported quantum advantage by building a
GBS machine with time-bin loops, where 216 single-
mode squeezed state inputs and 16 photon-number-
resolving detectors are used, and finally up to 219
photons are detected. Noting that this machine is
partially programmable, and most of the matrix
elements are zeros, namely partially connected.
Because the output probability of GBS is related to
hafnian, which corresponds to the number of perfect
matching of a graph, it links to several potentially practical
applications [857–860]. Next, GBS will naturally be
developed as a special-purpose photonic platform to
investigate these real-world applications, as a step
toward NISQ processing [861].
m
1/(m−1)
Scalability.—For large-scale quantum information
processing, the photonic platform faces two major
hurdles in the current stage — the loss and the ultra-
weak interaction between independent photons. In prin-
ciple, the loss is both locatable and detectable, it should
be much easier to be solved than computational errors
(such as X, Z errors). For DV quantum computing,
theoretical analysis [862, 863] suggests that at most 50%
loss is allowed for scalable quantum computing, which is
much less stringent and restrictive than a threshold of ~
1% for surface code to address computational errors. In
one-way quantum computing, some works pointed out
that for -photon cluster state, at least a fusion success
probability of is required for universal quantum
computation [864], while the upper bond needs further
works in the future. For instance, the upper percolation
threshold of the 3-photon clusters is bonded by 0.5898
[864]. For CV quantum computing, loss will cause the
squeezed states to move closer to the vacuum state,
meanwhile losing their quantum feature. The exciting
thing is that the fault-tolerant quantum computing with
GKP qubits only requires a squeezing level of ~ 10 dB
[865], which allows a high loss threshold for scalable
quantum computing. In summary, losses in both DV
and CV photonic quantum computing can be handled
with a relatively large threshold.
Photon–photon interaction at a single-photon level is
a fundamental question both in photonic quantum
χ(2)
computing and quantum optics. It is strongly believed
that nonlinear interactions are needed to deterministically
generate entanglement between photons [866]. Over the
last two decades, several approaches were developed to
address this issue, such as electromagnetically induced
transparency [867], atom-cavity interaction [868], atom-
atom interaction [869] and atoms in chiral waveguide
[870]. In 2016, Hacker et al. [871] reported a photon-
photon gate with an efficiency of 4.8% and a fidelity of
76.2%, which suffers from inefficient photon storage and
retrieval during the whole process, and the gate fidelity
is limited by the precision of spin characterization. The
same issues happen to several recent experiments utilizing
Rydberg blockade [872–874]. By storing single photons
in a long-lived Rydberg state, the efficiency of single-
photon storage and retrieval has been improved to 39%
[873]. In the future, by harnessing the strong nonlinearity
in mediums and cQED systems, photon-photon
gates can be realized with both high fidelity and efficiency
which surpass the thresholds required for fault-tolerant
quantum computing. In this case, the photonic platform
will provide a perfect stage and a potential leading platform
working at room temperature for fault-tolerant quantum
computing.
10Outlookandconclusion
We have reviewed prominent quantum computing plat-
forms that have seen significant advances over the last
decade. Currently, these quantum platforms are in
different stages of maturity, where each system exhibits
both advantages and limitations. To achieve better
control fidelity and scalability of the various quantum
platforms, challenges must be addressed to match the
requirements for large-scale quantum computing on
different platforms. For solid-state quantum systems,
high-quality materials and advanced fabrication tech-
nologies are essential for the quality of qubits. For
example, low-charge noise interfaces are critical for semi-
conductor qubits, which could be improved by importing
industrial techniques of IC and encouraging a transfer
from laboratory-level engineering to foundry-level fabri-
cation. For photonic atom-based qubit systems, funda-
mentals and advanced techniques in precise control of
individual atoms and atom–photon interconnectors
between multiple processors could be developed. In the
cases of NMR systems, one approach that may overcome
difficulty in achieving scalable and high-fidelity control
on large spin systems in liquid NMR quantum computation
is to shift to solid-state NMR. While for qubits based on
NV centers, the deterministic and controllable production
of NV centers while preserving the coherence time needs
to be further developed.
For all quantum computing platforms, the typical
overheads for scalable quantum computing, such as
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crosstalk, gate heating, and frequency crowding, should
also be considered carefully. The techniques that integrate
high-fidelity control techniques, such as DD schemes
that suppress errors and crosstalk between multiple
qubits, are necessary for high-fidelity multi-qubit control
with the growing number of qubits, where system size
should increase without compromising control quality.
By merging the techniques developed for quantum
computing, we might also gain better performance in
quantum metrology and quantum simulations. In addition
to technological advances to address the challenges,
efforts in integrating multiple quantum platforms may
also be essential for utilizing the advantages of different
platforms for future applications. For instance, tasks for
computation, communication, and storage, may be allo-
cated to different units. Moreover, developments of
hybrid quantum-classical algorithms are also critical for
developing “killer applications” for near-term quantum
computers. Proper integration of different quantum plat-
forms and hybrid classical and quantum computing may
provide significant advantages in real-world applications.
In the future, significant breakthroughs, such as fault-
tolerant quantum operations, and quantum algorithms,
will be achieved in medium-sized quantum systems. It is
also desirable to achieve quantum advantage for applica-
tions in quantum chemistry, quantum machine learning,
etc. Further developments could include specialized
quantum machines, quantum clouds, and applications of
quantum computing systems in quantum sensing and
simulations. Finally, as quantum platforms develop
toward fully scalable fault-tolerant quantum computing,
we anticipate the emergence of broad real-world applica-
tions of quantum computing.
AcknowledgementsWe thank Fei Yan for his contributions to the
superconducting qubits section, and thank valuable discussions with
Xiaodong He, we thank Chao-Yang Lu, Andrea Morello and Lieven M.
K. Vandersypen for valuable comments, and we also thank Jiasheng
Mai for figure polishing. This work was supported by the National
Natural Science Foundation of China (Grant Nos. U1801661, 12174178,
11905098, 12204228, 12004165, 11875159, 12075110, 92065111,
12275117, 11905099, 11975117, 12004164, 62174076, 92165210,
11904157, 11661161018, 11927811, and 12004371), the National Key
Research and Development Program of China (Grant Nos.
2019YFA0308100 and 2018YFA0306600), the Key-Area Research and
Development Program of Guangdong Province (No. 2018B030326001),
the Guangdong Innovative and Entrepreneurial Research Team
Program (Nos. 2016ZT06D348 and 2019ZT08C044), the Guangdong
Provincial Key Laboratory (No. 2019B121203002), the Guangdong
Basic and Applied Basic Research Foundation (Grant Nos.
2021B1515020070 and 2022B1515020074), the Natural Science Founda-
tion of Guangdong Province (No. 2017B030308003), the Science, Tech-
nology and Innovation Commission of Shenzhen, Municipality (Grant
Nos. KYTDPT20181011104202253, KQTD20210811090049034,
K21547502, ZDSYS20190902092905285, KQTD20190929173815000,
KQTD20200820113010023, JCYJ20200109140803865 and
JCYJ20170412152620376), Shenzhen Science and Technology Program
(Nos. RCBS20200714114820298 and RCYX20200714114522109), the
Shenzhen-Hong Kong Cooperation Zone for Technology and Innovation
(HZQB-KCZYB-2020050), the Anhui Initiative in Quantum Information
Technologies (Grant No. AHY050000), the Innovation Program for
Quantum Science and Technology (Grant No. 2021ZD0303205),
Research Grants Council of Hong Kong (GRF No. 14308019), the
Research Strategic Funding Scheme of The Chinese University of Hong
Kong (No. 3133234). F.N. is supported in part by: Nippon Telegraph
and Telephone Corporation (NTT) Research, the Japan Science and
Technology Agency (JST) [via the Quantum Leap Flagship Program
(Q-LEAP), and the Moonshot R&D Grant Number JPMJMS2061],
the Japan Society for the Promotion of Science (JSPS) [via the Grants-
in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134],
the Asian Office of Aerospace Research and Development (AOARD)
(via Grant No. FA2386-20-1-4069), and the Foundational Questions
Institute Fund (FQXi) via Grant No. FQXi-IAF19-06.
Author contributionsM.-H.Y., Y.H., X.-H.D., J.L., D.L., B.-C.L.,
P.H., and Y.L. wrote the abstract and introduction. M.-H.Y. and B.C.
wrote the quantum algorithms section. X.G., Y.Z., and F.N. wrote the
superconducting qubits section. Y.L. wrote the trapped-ion qubits
section. Y.H., P.H., and G.H. wrote the semiconductor spin qubits
section. D.L. and C.Q. wrote the NV centers section. J.L., T.X., and
X.-H.P. wrote the NMR system section. S.Y. wrote the neutral atom
arrays section. H.W. wrote the photonic quantum computing section.
X.-H.D., P.H., Y.H., and M.-H.Y. wrote the outlook and conclusion.
The manuscript was revised by X.-H.D., P.H., J.Z., S.Z., F.N. and D.
Y. with input from all other authors. D.Y. supervised the review
project.
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21308-62 Bin Cheng, et al., Front. Phys. 18(2), 21308 (2023)
