No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Circuit QED: Implementation of the three-qubit refined Deutsch-Jozsa quantum algorithm
Abstract
We propose a protocol to construct the 35 $f$-controlled phase gates of a
three-qubit refined Deutsch-Jozsa (DJ) algorithm, by using single-qubit
$\sigma_z$ gates, two-qubit controlled phase gates, and two-target-qubit
controlled phase gates. Using this protocol, we discuss how to implement the
three-qubit refined DJ algorithm with superconducting transmon qutrits
resonantly coupled to a single cavity. Our numerical calculation shows that
implementation of this quantum algorithm is feasible within the present circuit
QED technique. The experimental realization of this algorithm would be an
important step toward more complex quantum computation in circuit QED.
... Detailed explanations of how the Deutsch-Jozsa algorithm can be extended to three or more qubits may be found in [200,202,206,214,215,218]. In addition to single-qubit gates, this requires different types of controlled-phase gates [214]. ...
... Detailed explanations of how the Deutsch-Jozsa algorithm can be extended to three or more qubits may be found in [200,202,206,214,215,218]. In addition to single-qubit gates, this requires different types of controlled-phase gates [214]. A controlled-phase gate has already been demonstrated for photons [331]. ...
The development of integrated quantum photonics is integral to many areas of quantum information science, in particular linear optical quantum computing. In this context, a diversity of physical systems is being explored and thus versatility and adaptability are important prerequisites for any candidate platform. Silicon oxynitride is a promising material because its refractive index can be varied over a wide range. This dissertation describes the development of silicon oxynitride waveguides for applications in the field of integrated quantum photonics. The project consisted of three stages: design, characterisation, and application. First, the parameter space was studied through simulations. The structures were optimised to achieve low-loss devices with a small footprint at a wavelength of 900 nm. Buried channel waveguides with a cross-section of 1.6 μm x 1.6 μm and a core (cladding) refractive index of 1.545 (1.505) were chosen. Second, following their fabrication with plasma-enhanced chemical vapour deposition, electron beam lithography, and reactive ion etching, the waveguides were characterised. The refractive index was shown to be tunable from the silica to the silicon nitride regime. Optimised tapers significantly improved the coupling efficiency. The minimum bend radius was measured to be less than 2 mm. Propagation losses as low as 1.45 dB cm-1 were achieved. Directional couplers with coupling ratios ranging from 0 to 1 were realised. Third, building blocks for linear optical quantum computing were demonstrated. Reconfigurable quantum circuits consisting of Mach-Zehnder interferometers with near perfect visibilities were fabricated along with a four-port switch. The potential of quantum speedup was illustrated by carrying out the Deutsch-Jozsa algorithm with a fidelity of 100 % using on-demand single photons from a quantum dot. This dissertation presents the first implementation of tunable Mach-Zehnder interferometers, which act on single photons, based on silicon oxynitride waveguides. Furthermore, for the first time silicon oxynitride photonic quantum circuits were operated with on-demand single photons. Accordingly, this work has created a platform for the development of integrated quantum photonics.
... Thus, the operation time required for the gate implementation is independent of the number N of qubits, this type of controlled gate with N target qubits is useful in quantum information processing. For instance, it has applications in entanglement preparation [20,21,22], error correction [23], Grover search algorithm [24,25], quantum discrete Fourier transform [26,27], Deutsch-Jozsa algorithm [28,29], quantum dense coding [30,31], and quantum cloning [32,33]. The unitary operator representing this type of multiqubit gate is given by [29] ...
... For instance, it has applications in entanglement preparation [20,21,22], error correction [23], Grover search algorithm [24,25], quantum discrete Fourier transform [26,27], Deutsch-Jozsa algorithm [28,29], quantum dense coding [30,31], and quantum cloning [32,33]. The unitary operator representing this type of multiqubit gate is given by [29] ...
We propose an effective way for realizing a three quantum logic gates (NTCP gate, NTCP-NOT gate and NTQ-NOT gate) of one qubit simultaneously controlling N target qubits based on the qubit-qubit interaction. We use the superconducting qubits in a cavity QED driven by a strong microwave field. In our scheme, the operation time of these gates is independent of the number N of qubits involved in the gate operation. These gates are insensitive to the initial state of the cavity QED and can be used to produce an analogous CNOT gate simultaneously acting on N qubits. The quantum phase gate can be realized in a time (nanosecond-scale) much smaller than decoherence time and dephasing time (microsecond-scale) in cavity QED. Numerical simulation under the influence of the gate operations shows that the scheme could be achieved efficiently within current state-of-the-art technology.
Superconducting quantum circuits are promising candidates for realizing quantum computation due to their intrinsic scalability. Here, we report the implementation of the refined Deutsch–Jozsa algorithm in a 3D superconducting transmon qutrit system. Such 3D superconducting systems usually preserve a relatively long coherence time. A qutrit can take advantage of the third level of a superconducting artificial atom and extends the Hilbert space. Compared with the two-qubit system used for realizing Deutsch–Jozsa algorithm, the employment of qutrit simplifies the hardware design. This simplification and the intrinsic long coherence time jointly lead to a higher fidelity than in previous transmon implementation [DiCarlo et al., Nature 460 (7252), 240–244 (2009)].
Quantum error correction (QEC) is an essential step towards realising
scalable quantum computers. Theoretically, it is possible to achieve
arbitrarily long protection of quantum information from corruption due to
decoherence or imperfect controls, so long as the error rate is below a
threshold value. The two-dimensional surface code (SC) is a fault-tolerant
error correction protocol} that has garnered considerable attention for actual
physical implementations, due to relatively high error thresholds ~1%, and
restriction to planar lattices with nearest-neighbour interactions. Here we
show a necessary element for SC error correction: high-fidelity parity
detection of two code qubits via measurement of a third syndrome qubit. The
experiment is performed on a sub-section of the SC lattice with three
superconducting transmon qubits, in which two independent outer code qubits are
joined to a central syndrome qubit via two linking bus resonators. With
all-microwave high-fidelity single- and two-qubit nearest-neighbour entangling
gates, we demonstrate entanglement distributed across the entire sub-section by
generating a three-qubit Greenberger-Horne-Zeilinger (GHZ) state with fidelity
~94%. Then, via high-fidelity measurement of the syndrome qubit, we
deterministically entangle the otherwise un-coupled outer code qubits, in
either an even or odd parity Bell state, conditioned on the syndrome state.
Finally, to fully characterize this parity readout, we develop a new
measurement tomography protocol to obtain a fidelity metric (90% and 91%). Our
results reveal a straightforward path for expanding superconducting circuits
towards larger networks for the SC and eventually a primitive logical qubit
implementation.
We demonstrate enhanced relaxation and dephasing times of transmon qubits, up
to ~ 60 \mu s by fabricating the interdigitated shunting capacitors using
titanium nitride (TiN). Compared to lift-off aluminum deposited simultaneously
with the Josephson junction, this represents as much as a six-fold improvement
and provides evidence that previous planar transmon coherence times are limited
by surface losses from two-level system (TLS) defects residing at or near
interfaces. Concurrently, we observe an anomalous temperature dependent
frequency shift of TiN resonators which is inconsistent with the predicted TLS
model.
We present a way to realize a $n$-qubit controlled phase gate with superconducting quantum interference devices (SQUIDs) by coupling them to a superconducting resonator. In this proposal, the two logical states of a qubit are represented by the two lowest levels of a SQUID. An intermediate level of each SQUID is utilized to facilitate coherent control and manipulation of quantum states of the qubits. It is interesting to note that a $n$-qubit controlled phase gate can be achieved with $n$ SQUIDs by successively applying a $\pi /2$ Jaynes-Cummings pulse to each of the $n-1$ control SQUIDs before and after a $\pi$ Jaynes-Cummings pulse on the target SQUID. Comment: 9 pages, 4 figures, 1 table, RevTeX, Resubmitted to Phys. Rev. A
We propose an approach to realize an $n$-qubit controlled-$U$ gate with superconducting quantum interference devices (SQUIDs) in cavity QED. In this approach, the two lowest levels of a SQUID represent the two logical states of a qubit while a higher-energy intermediate level serves the gate manipulation. Our method operates essentially by creating a single photon through one of the control SQUIDs and then performing an arbitrary unitrary transformation on the target SQUID with the assistance of the cavity photon. In addition, we show that the method can be applied to implement an $n $% -qubit controlled-$U$ gate with atomic qubits in cavity QED. Comment: 13 pages, 7 figures, 1 table, RevTeX, Accepted by Physical Review A
We present a scheme to achieve maximally entangled states, controlled
phase-shift gate, and SWAP gate for two
superconducting-quantum-interference-device (SQUID) qubits, by placing SQUIDs
in a microwave cavity. We also show how to transfer quantum information from
one SQUID qubit to another. In this scheme, no transfer of quantum information
between the SQUIDs and the cavity is required, the cavity field is only
virtually excited and thus the requirement on the quality factor of the cavity
is greatly relaxed.
We propose a way for realizing a two-qubit controlled phase gate with
superconducting quantum interference devices (SQUIDs) coupled to a
superconducting resonator. In this proposal, the two lowest levels of each
SQUID serve as the logical states and two intermediate levels of each SQUID are
used for the gate realization. We show that neither adjustment of SQUID level
spacings during the gate operation nor uniformity in SQUID parameters is
required by this proposal. In addition, this proposal does not require the
adiabatic passage or a second-order detuning and thus the gate is much faster.
A quantum computer will require quantum bits (qubits) with good coherence that can be coupled together to form logic gates. Superconducting circuits offer a novel solution because qubits can be connected in elaborate ways through simple wiring, much like that of conventional integrated circuits. However, this ease of coupling is offset by coherence times shorter than those observed in molecular and atomic systems. Hybrid architectures could help skirt this fundamental trade-off between coupling and coherence by using macroscopic qubits for coupling and atom-based qubits for coherent storage. Here, we demonstrate the first quantum memory operation on a Josephson-phase qubit by transferring an arbitrary quantum state to a two-level state (TLS), storing it there for some time, and later retrieving it. The qubit is used to probe the coherence of the TLS by measuring its energy relaxation and dephasing times. Quantum process tomography completely characterizes the memory operation, yielding an overall process fidelity of 79%. Although the uncontrolled distribution of TLSs precludes their direct use in a scalable architecture, the ability to coherently couple a macroscopic device with an atomic-sized system motivates a search for designer molecules that could replace the TLS in future hybrid qubits.
Hybrid quantum circuits combine two or more physical systems, with the goal
of harnessing the advantages and strengths of the different systems in order to
better explore new phenomena and potentially bring about novel quantum
technologies. This article presents a brief overview of the progress achieved
so far in the field of hybrid circuits involving atoms, spins and solid-state
devices (including superconducting and nanomechanical systems). We discuss how
these circuits combine elements from atomic physics, quantum optics, condensed
matter physics, and nanoscience, and we present different possible approaches
for integrating various systems into a single circuit. In particular, hybrid
quantum circuits can be fabricated on a chip, facilitating their future
scalability, which is crucial for building future quantum technologies,
including quantum detectors, simulators and computers.
Quantum computers could be used to solve certain problems exponentially faster than classical computers, but are challenging to build because of their increased susceptibility to errors. However, it is possible to detect and correct errors without destroying coherence, by using quantum error correcting codes. The simplest of these are three-quantum-bit (three-qubit) codes, which map a one-qubit state to an entangled three-qubit state; they can correct any single phase-flip or bit-flip error on one of the three qubits, depending on the code used. Here we demonstrate such phase- and bit-flip error correcting codes in a superconducting circuit. We encode a quantum state, induce errors on the qubits and decode the error syndrome--a quantum state indicating which error has occurred--by reversing the encoding process. This syndrome is then used as the input to a three-qubit gate that corrects the primary qubit if it was flipped. As the code can recover from a single error on any qubit, the fidelity of this process should decrease only quadratically with error probability. We implement the correcting three-qubit gate (known as a conditional-conditional NOT, or Toffoli, gate) in 63 nanoseconds, using an interaction with the third excited state of a single qubit. We find 85 ± 1 per cent fidelity to the expected classical action of this gate, and 78 ± 1 per cent fidelity to the ideal quantum process matrix. Using this gate, we perform a single pass of both quantum bit- and phase-flip error correction and demonstrate the predicted first-order insensitivity to errors. Concatenation of these two codes in a nine-qubit device would correct arbitrary single-qubit errors. In combination with recent advances in superconducting qubit coherence times, this could lead to scalable quantum technology.
Superconducting quantum circuits based on Josephson junctions have made rapid progress in demonstrating quantum behavior and scalability. However, the future prospects ultimately depend upon the intrinsic coherence of Josephson junctions, and whether superconducting qubits can be adequately isolated from their environment. We introduce a new architecture for superconducting quantum circuits employing a three-dimensional resonator that suppresses qubit decoherence while maintaining sufficient coupling to the control signal. With the new architecture, we demonstrate that Josephson junction qubits are highly coherent, with T2 ∼ 10 to 20 μs without the use of spin echo, and highly stable, showing no evidence for 1/f critical current noise. These results suggest that the overall quality of Josephson junctions in these qubits will allow error rates of a few 10(-4), approaching the error correction threshold.
We describe the fabrication and measurement of microwave coplanar waveguide
resonators with internal quality factors above 10 million at high microwave
powers and over 1 million at low powers, with the best low power results
approaching 2 million, corresponding to ~1 photon in the resonator. These
quality factors are achieved by controllably producing very smooth and clean
interfaces between the resonators' aluminum metallization and the underlying
single crystal sapphire substrate. Additionally, we describe a method for
analyzing the resonator microwave response, with which we can directly
determine the internal quality factor and frequency of a resonator embedded in
an imperfect measurement circuit.
The Toffoli gate is a three-quantum-bit (three-qubit) operation that inverts the state of a target qubit conditioned on the state of two control qubits. It makes universal reversible classical computation possible and, together with a Hadamard gate, forms a universal set of gates in quantum computation. It is also a key element in quantum error correction schemes. The Toffoli gate has been implemented in nuclear magnetic resonance, linear optics and ion trap systems. Experiments with superconducting qubits have also shown significant progress recently: two-qubit algorithms and two-qubit process tomography have been implemented, three-qubit entangled states have been prepared, first steps towards quantum teleportation have been taken and work on quantum computing architectures has been done. Implementation of the Toffoli gate with only single- and two-qubit gates requires six controlled-NOT gates and ten single-qubit operations, and has not been realized in any system owing to current limits on coherence. Here we implement a Toffoli gate with three superconducting transmon qubits coupled to a microwave resonator. By exploiting the third energy level of the transmon qubits, we have significantly reduced the number of elementary gates needed for the implementation of the Toffoli gate, relative to that required in theoretical proposals using only two-level systems. Using full process tomography and Monte Carlo process certification, we completely characterize the Toffoli gate acting on three independent qubits, measuring a fidelity of 68.5 ± 0.5 per cent. A similar approach to realizing characteristic features of a Toffoli-class gate has been demonstrated with two qubits and a resonator and achieved a limited characterization considering only the phase fidelity. Our results reinforce the potential of macroscopic superconducting qubits for the implementation of complex quantum operations with the possibility of quantum error correction.
The von Neumann architecture for a classical computer comprises a central processing unit and a memory holding instructions and data. We demonstrate a quantum central processing unit that exchanges data with a quantum random-access memory integrated on a chip, with instructions stored on a classical computer. We test our quantum machine by executing codes that involve seven quantum elements: Two superconducting qubits coupled through a quantum bus, two quantum memories, and two zeroing registers. Two vital algorithms for quantum computing are demonstrated, the quantum Fourier transform, with 66% process fidelity, and the three-qubit Toffoli-class OR phase gate, with 98% phase fidelity. Our results, in combination especially with longer qubit coherence, illustrate a potentially viable approach to factoring numbers and implementing simple quantum error correction codes.
Superconducting circuits based on Josephson junctions exhibit macroscopic
quantum coherence and can behave like artificial atoms. Recent technological
advances have made it possible to implement atomic-physics and quantum-optics
experiments on a chip using these artificial atoms. This review presents a
brief overview of the progress achieved so far in this rapidly advancing field.
We not only discuss phenomena analogous to those in atomic physics and quantum
optics with natural atoms, but also highlight those not occurring in natural
atoms. In addition, we summarize several prospective directions in this
emerging interdisciplinary field.
The ability to generate particles from the quantum vacuum is one of the most
profound consequences of Heisenberg's uncertainty principle. Although the
significance of vacuum fluctuations can be seen throughout physics, the
experimental realization of vacuum amplification effects has until now been
limited to a few cases. Superconducting circuit devices, driven by the goal to
achieve a viable quantum computer, have been used in the experimental
demonstration of the dynamical Casimir effect, and may soon be able to realize
the elusive verification of analogue Hawking radiation. This article describes
several mechanisms for generating photons from the quantum vacuum and
emphasizes their connection to the well-known parametric amplifier from quantum
optics. Discussed in detail is the possible realization of each mechanism, or
its analogue, in superconducting circuit systems. The ability to selectively
engineer these circuit devices highlights the relationship between the various
amplification mechanisms.
We propose how to realize a multiqubit tunable phase gate of one qubit
simultaneously controlling $n$ qubits with four-level quantum systems in a
cavity or coupled to a resonator. Each of the $n$ two-qubit controlled-phase
(CP) gates involved in this multiqubit phase gate has a shared control qubit
but a {\it different} target qubit. In this propose, the two lowest levels of
each system represent the two logical states of a qubit while the two
higher-energy intermediate levels are used for the gate implementation. The
method presented here operates essentially by creating a single photon through
the control qubit, which then induces a phase shift to the state of each target
qubit. The phase shifts on each target qubit can be adjusted by changing the
Rabi frequencies of the pulses applied to the target qubit systems. The
operation time for the gate implementation is independent of the number of
qubits, and neither adjustment of the qubit level spacings nor adjustment of
the cavity mode frequency during the gate operation is required by this
proposal. It is also noted that this approach can be applied to implement
certain types of significant multiqubit phase gates, e.g., the multiqubit phase
gate consisting of $n$ two-qubit CP gates which are key elements in quantum
Fourier transforms. A possible physical implementation of our approach is
presented. Our proposal is quite general, and can be applied to physical
systems such as various types of superconducting devices coupled to a resonator
and trapped atoms in a cavity.
Traditionally, quantum entanglement has been central to foundational discussions of quantum mechanics. The measurement of correlations between entangled particles can have results at odds with classical behaviour. These discrepancies grow exponentially with the number of entangled particles. With the ample experimental confirmation of quantum mechanical predictions, entanglement has evolved from a philosophical conundrum into a key resource for technologies such as quantum communication and computation. Although entanglement in superconducting circuits has been limited so far to two qubits, the extension of entanglement to three, eight and ten qubits has been achieved among spins, ions and photons, respectively. A key question for solid-state quantum information processing is whether an engineered system could display the multi-qubit entanglement necessary for quantum error correction, which starts with tripartite entanglement. Here, using a circuit quantum electrodynamics architecture, we demonstrate deterministic production of three-qubit Greenberger-Horne-Zeilinger (GHZ) states with fidelity of 88 per cent, measured with quantum state tomography. Several entanglement witnesses detect genuine three-qubit entanglement by violating biseparable bounds by 830 ± 80 per cent. We demonstrate the first step of basic quantum error correction, namely the encoding of a logical qubit into a manifold of GHZ-like states using a repetition code. The integration of this encoding with decoding and error-correcting steps in a feedback loop will be the next step for quantum computing with integrated circuits.
Minimizing phase and other errors in experimental quantum gates allows higher fidelity quantum processing. To quantify and correct for phase errors in particular, we have developed a new experimental metrology --- amplified phase error (APE) pulses --- that amplifies and helps identify phase errors in general multi-level qubit architectures. In order to correct for both phase and amplitude errors specific to virtual transitions and leakage outside of the qubit manifold, we implement "half derivative" an experimental simplification of derivative reduction by adiabatic gate (DRAG) control theory. The phase errors are lowered by about a factor of five using this method to $\sim 1.6^{\circ}$ per gate, and can be tuned to zero. Leakage outside the qubit manifold, to the qubit $|2\rangle$ state, is also reduced to $\sim 10^{-4}$ for $20\%$ faster gates. Comment: 4 pages, 4 figures with 2 page supplemental
We propose how to realize a three-step controlled-phase gate of one qubit simultaneously controlling $n$ qubits in a cavity or coupled to a resonator. The $n$ two-qubit controlled-phase gates, forming this multiqubit phase gate, can be performed simultaneously. The operation time of this phase gate is independent of the number $n$ of qubits. This phase gate controlling at once $n$ qubits is insensitive to the initial state of the cavity mode and can be used to produce an analogous CNOT gate simultaneously acting on $n$ qubits. We present two alternative approaches to implement this gate. One approach is based on tuning the qubit frequency while the other method tunes the resonator frequency. Using superconducting qubits coupled to a resonator as an example, we show how to implement the proposed gate with one superconducting qubit simultaneously controlling $n$ qubits selected from $N$ qubits coupled to a resonator ($1<n<N$). We also give a discussion on realizing the proposed gate with atoms, by using one cavity initially in an arbitrary state. Comment: 16 pages, 9 figures, 2 tables
We present the realization of a cavity quantum electrodynamics setup in which photons of strongly different lifetimes are engineered in different harmonic modes of the same cavity. We achieve this in a superconducting transmission line resonator with superconducting qubits coupled to the different modes. One cavity mode is strongly coupled to a detection line for qubit state readout, while a second long lifetime mode is used for photon storage and coherent quantum operations. We demonstrate sideband-based measurement of photon coherence, generation of n photon Fock states and the scaling of the sideband Rabi frequency with square root of n using a scheme that may be extended to realize sideband-based two-qubit logic gates.
A scheme is proposed for realizing quantum entanglement, information transfer,
CNOT gates, and SWAP gates with superconducting-quantum-interference-device
(SQUID) qubits in cavity QED. In the scheme, the two logical states of a qubit
are the two lowest levels of the SQUID. An intermediate level of the SQUID
is utilized to facilitate coherent control and manipulation of quantum states of
the qubits. The method presented here does not involve a real excitation of the
intermediate levels during the operations. Thus, decoherence due to energy relaxation of
intermediate levels is minimized. In addition, the present method does not require the
adjustment of the SQUID level spacings, which simplifies the operation significantly.
Quantum computers, which harness the superposition and entanglement of physical states, could outperform their classical counterparts in solving problems with technological impact-such as factoring large numbers and searching databases. A quantum processor executes algorithms by applying a programmable sequence of gates to an initialized register of qubits, which coherently evolves into a final state containing the result of the computation. Building a quantum processor is challenging because of the need to meet simultaneously requirements that are in conflict: state preparation, long coherence times, universal gate operations and qubit readout. Processors based on a few qubits have been demonstrated using nuclear magnetic resonance, cold ion trap and optical systems, but a solid-state realization has remained an outstanding challenge. Here we demonstrate a two-qubit superconducting processor and the implementation of the Grover search and Deutsch-Jozsa quantum algorithms. We use a two-qubit interaction, tunable in strength by two orders of magnitude on nanosecond timescales, which is mediated by a cavity bus in a circuit quantum electrodynamics architecture. This interaction allows the generation of highly entangled states with concurrence up to 94 per cent. Although this processor constitutes an important step in quantum computing with integrated circuits, continuing efforts to increase qubit coherence times, gate performance and register size will be required to fulfil the promise of a scalable technology.
We demonstrate time resolved driving of two-photon blue sideband transitions between superconducting qubits and a transmission line resonator. Using the sidebands, we implement a pulse sequence that first entangles one qubit with the resonator, and subsequently distributes the entanglement between two qubits. We show generation of 75% fidelity Bell states by this method. The full density matrix of the two qubit system is extracted using joint measurement and quantum state tomography, and shows close agreement with numerical simulation. The scheme is potentially extendable to a scalable universal gate for quantum computation. Comment: 4 pages, 5 figures, version with high resolution figures available at http://qudev.ethz.ch/content/science/PubsPapers.html
We investigate the experimental feasibility of realizing quantum information transfer (QIT) and entanglement with SQUID qubits in a microwave cavity via dark states. Realistic system parameters are presented. Our results show that QIT and entanglement with two-SQUID qubits can be achieved with a high fidelity. The present scheme is tolerant to device parameter nonuniformity. We also show that the strong coupling limit can be achieved with SQUID qubits in a microwave cavity. Thus, cavity-SQUID systems provide a new way for production of nonclassical microwave source and quantum communication.
The interaction of matter and light is one of the fundamental processes occurring in nature, and its most elementary form is realized when a single atom interacts with a single photon. Reaching this regime has been a major focus of research in atomic physics and quantum optics for several decades and has generated the field of cavity quantum electrodynamics. Here we perform an experiment in which a superconducting two-level system, playing the role of an artificial atom, is coupled to an on-chip cavity consisting of a superconducting transmission line resonator. We show that the strong coupling regime can be attained in a solid-state system, and we experimentally observe the coherent interaction of a superconducting two-level system with a single microwave photon. The concept of circuit quantum electrodynamics opens many new possibilities for studying the strong interaction of light and matter. This system can also be exploited for quantum information processing and quantum communication and may lead to new approaches for single photon generation and detection.
In the emerging field of quantum computation and quantum information, superconducting devices are promising candidates for the implementation of solid-state quantum bits (qubits). Single-qubit operations, direct coupling between two qubits and the realization of a quantum gate have been reported. However, complex manipulation of entangled states-such as the coupling of a two-level system to a quantum harmonic oscillator, as demonstrated in ion/atom-trap experiments and cavity quantum electrodynamics-has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux qubit (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic oscillator. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi oscillations of the coupled system.
Superconducting circuits are promising candidates for constructing quantum bits (qubits) in a quantum computer; single-qubit operations are now routine, and several examples of two-qubit interactions and gates have been demonstrated. These experiments show that two nearby qubits can be readily coupled with local interactions. Performing gate operations between an arbitrary pair of distant qubits is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the qubits to a 'quantum bus', which distributes quantum information among the qubits. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting qubits on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity-induced loss. Using fast control of the qubits to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the qubits. The cavity is also used to perform multiplexed control and measurement of the qubit states. This approach can be expanded to more than two qubits, and is an attractive architecture for quantum information processing on a chip.
Superconducting circuits are macroscopic in size but have generic quantum properties such as quantized energy levels, superposition of states, and entanglement, all of which are more commonly associated with atoms. Superconducting quantum bits (qubits) form the key component of these circuits. Their quantum state is manipulated by using electromagnetic pulses to control the magnetic flux, the electric charge or the phase difference across a Josephson junction (a device with nonlinear inductance and no energy dissipation). As such, superconducting qubits are not only of considerable fundamental interest but also might ultimately form the primitive building blocks of quantum computers.
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Quantum computers promise to solve certain problems exponentially faster
than possible classically but are challenging to build because of their
increased susceptibility to errors. Remarkably, however, it is possible
to detect and correct errors without destroying coherence by using
quantum error correcting codes. The simplest of these are the
three-qubit codes, which map a one-qubit state to an entangled
three-qubit state and can correct any single phase-flip or bit-flip
error of one of the three qubits, depending on the code used [1]. The
fidelity of a process in which errors can occur on all qubits, where
there is the possibility of an uncorrectable double or triple error,
should therefore decrease only quadratically with error probability. I
will first introduce how the three-qubit encoded state can be produced
in our superconducting architecture by employing interactions with
non-computational qubit states, as previously demonstrated [2]. I will
then discuss how these non-computational interactions can be generalized
to produce a novel three-qubit conditional-conditional NOT (CCNot) or
Toffoli gate, which implements the correcting step of an error
correction algorithm. Finally, I will explain how, by combining these
ingredients, we have performed a single pass of both quantum bit- and
phase-flip error correction and have demonstrated the predicted
first-order insensitivity to errors.[4pt] [1] M. A. Nielsen and I. L.
Chuang, Cambridge University Press, 2000.[0pt] [2] L. DiCarlo, et al.
Nature 467, 574 (2010).
Superconducting quantum circuits have made significant advances over the
past decade, allowing more complex and integrated circuits that perform
with good fidelity. We have recently implemented a machine comprising
seven quantum channels, with three superconducting resonators, two phase
qubits, and two zeroing registers. I will explain the design and
operation of this machine, first showing how a single microwave photon |
1 > can be prepared in one resonator and coherently transferred
between the three resonators. I will also show how more exotic states
such as double photon states | 2 > and superposition states | 0 >+
| 1 > can be shuffled among the resonators as well [1]. I will then
demonstrate how this machine can be used as the quantum-mechanical
analog of the von Neumann computer architecture, which for a classical
computer comprises a central processing unit and a memory holding both
instructions and data. The quantum version comprises a quantum central
processing unit (quCPU) that exchanges data with a quantum random-access
memory (quRAM) integrated on one chip, with instructions stored on a
classical computer. I will also present a proof-of-concept demonstration
of a code that involves all seven quantum elements: (1), Preparing an
entangled state in the quCPU, (2), writing it to the quRAM, (3),
preparing a second state in the quCPU, (4), zeroing it, and, (5),
reading out the first state stored in the quRAM [2]. Finally, I will
demonstrate that the quantum von Neumann machine provides one unit cell
of a two-dimensional qubit-resonator array that can be used for surface
code quantum computing. This will allow the realization of a scalable,
fault-tolerant quantum processor with the most forgiving error rates to
date. [4pt] [1] M. Mariantoni et al., Nature Physics 7, 287-293
(2011.)[0pt] [2] M. Mariantoni et al., Science 334, 61-65 (2011).
Quantum mechanics can speed up a range of search applications over unsorted data. For example, imagine a phone directory containing N names arranged in completely random order. To find someone`s phone number with a probability of 50{percent}, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of 0.5N times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O({radical}(N)) accesses to the database. {copyright} {ital 1997} {ital The American Physical Society}
Circuit quantum electrodynamics is a system, which allows us to do new experiments in quantum optics with a superconducting integrated circuit on a chip. In circuit QED, microwave photons are guided and confined by superconducting transmission lines and cavities, and can then be coherently coupled to a transmon qubit. This system leads to much stronger coupling of the ``light'' and ``matter'' than is possible with traditional atomic systems. Making use of that strong coupling it is possible to couple two qubits via the cavity[1]. I will show how one can use the cavity as a coupling bus which provides non-local and non-nearest neighbor coupling. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity-induced loss. The same cavity is also used to perform multiplexed control and read-out of the two qubits. The coupling is effectively switchable which allows for time domain transfer of the quantum states between the qubits. [1] Coupling superconducting qubits via a cavity bus, J. Majer, J. M. Chow, J. M. Gambetta, Jens Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf. Nature 449 443 (2007)
We have realized a quantum phase gate operating on quantum bits carried by a single Rydberg atom and a zero- or one-photon field in a high- Q cavity. The gate operation is based on the dephasing of the atom-field state produced by a full cycle of quantum Rabi oscillation. The dephasing angle, conditioned to the initial atom-field state, can be adjusted over a wide range by tuning the atom-cavity frequency difference. We demonstrate that the gate preserves qubit coherence and generates entanglement. This gate is an essential tool for the nondestructive measurement of single photons and for the manipulation of many-qubit entanglement in cavity QED.
The theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines. Quantum gates are the generalization of classical logic gates. A single type of gate, the univeral quantum gate, together with quantum 'unit wires', is adequate for constructing networks with any possible quantum computational property.
We present an experimentally realizable microwave pulse sequence that effects a controlled-NOT (C-NOT) gate operation on a Josephson-junction-based flux qubit/resonator system with high-process fidelity. We obtained a C-NOT gate process fidelity of 0.988 (0.980) for a two-(three-)qubit/resonator system under ideal conditions and a fidelity of 0.903 for a two-qubit/resonator system under the best, currently achieved, experimental conditions. Our simulations show that this gate is a feasible first step toward multiqubit quantum-information processing with flux qubit/resonator systems.
During the last few decades, an extensive development of the theory of
computing machines has occurred. On an intuitive basis, a computing
machine is considered to be any physical system whose dynamical
evolution takes it from one of a set of 'input' states to one of a set
of 'output' states. For a classical deterministic system the measured
output label is a definite function f of the prepared input label.
However, quantum computing machines, and indeed classical stochastic
computing machines, do not 'compute functions' in the considered sense.
Attention is given to the universal Turing machine, the Church-Turing
principle, quantum computers, the properties of the universal quantum
computer, and connections between physics and computer science.
We propose a method for realizing two-qubit quantum phase gate with 4-level systems in cavity QED. In this proposal, the two logical states of a qubit are represented by the two lowest levels of each system, and two intermediate levels of each system are utilized to facilitate coherent control and manipulation of quantum states of the qubits. The present method does not involve cavity-photon population during the operation. In addition, we show that the gate can be achieved using only two-step operations.
In the system with superconducting quantum interference devices (SQUIDs) in cavity, the quantum logic gates operation and entanglement can be achieved by using a quantized cavity field and classical microwave pluses, via Raman transition. In this scheme, no transfer of quantum information between the SQUIDs and cavity is required, the cavity field is only virtually excited and thus the cavity decay is suppressed during the gate operation and entanglement generations. The gate operation and entanglement generations are realized by using only the two lower flux states of the SQUID system and the excited state would not be excited. Therefore, the effect of docoherence based on the levels of the SQUID system is possible to minimize.
We propose the implementation of fast resonant gates in circuit quantum electrodynamics for quantum information processing. We show how a suitable utilization of three-level superconducting qubits inside a resonator constitutes a key tool to perform diverse two-qubit resonant gates, improving the operation speed when compared to slower dispersive techniques. To illustrate the benefit of resonant two-qubit gates in circuit quantum electrodynamics, we consider the implementation of a two-dimensional cluster state in an array of N×N superconducting qubits by using resonant controlled-phase and one-qubit gates, where the generation time grows linearly with N. For N=3, and taking into account decoherence mechanisms, a fidelity over 60% for the generation of this cluster state is obtained.
We propose a theoretical scheme to couple superconducting charge qubits (SCQs) via Raman transitions in a circuit QED architecture. Qubit array interacts a quantum data bus generated by a one-dimensional superconducting transmission line resonator (TLR). Based on the Raman transitions, the controllable and selective interqubit couplings mediated by the data bus can be obtained by just addressing the applied gate pulses, which provides the possibility for scaling up to many SCQs.
We propose a different scheme to realize holonomic quantum computation with rf superconducting quantum interference device (SQUID) qubits in a microwave cavity. In this scheme associated with the non-Abelian holonomies, the single-qubit gates and a two-qubit controlled-PHASE gate as well as a controlled-NOT gate can be easily constructed by tuning adiabatically the Rabi frequencies of classical microwave pulses coupled to the SQUIDs. The fidelity of these gates is estimated to be possibly higher than 90% with the current technology.
We measured the quality factor (Q) and hence the losses of thin film superconducting Nb coplanar waveguide resonators fabricated with processes and materials similar to those used for Josephson effect qubits, where such losses can cause significant decoherence. Intrinsic Q-values range from several thousand to almost 106 depending on the process details. Reactive ion etching appears to reduce the resonator Q-values and the lift-off process can also degrade the Q-value for some resists. The resistivity of the Si substrates affects the intrinsic Q at 1 K, where the resonators were measured. The Q-values obtained for optimized processing are sufficiently high as to suggest that qubits fabricated by a similar technique would not be limited by losses associated with the film or substrate.
A class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method. The quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation.
A quantum processor (QuP) can be used to exploit quantum mechanics to find
the prime factors of composite numbers[1]. Compiled versions of Shor's
algorithm have been demonstrated on ensemble quantum systems[2] and photonic
systems[3-5], however this has yet to be shown using solid state quantum bits
(qubits). Two advantages of superconducting qubit architectures are the use of
conventional microfabrication techniques, which allow straightforward scaling
to large numbers of qubits, and a toolkit of circuit elements that can be used
to engineer a variety of qubit types and interactions[6, 7]. Using a number of
recent qubit control and hardware advances [7-13], here we demonstrate a
nine-quantum-element solid-state QuP and show three experiments to highlight
its capabilities. We begin by characterizing the device with spectroscopy.
Next, we produces coherent interactions between five qubits and verify bi- and
tripartite entanglement via quantum state tomography (QST) [8, 12, 14, 15]. In
the final experiment, we run a three-qubit compiled version of Shor's algorithm
to factor the number 15, and successfully find the prime factors 48% of the
time. Improvements in the superconducting qubit coherence times and more
complex circuits should provide the resources necessary to factor larger
composite numbers and run more intricate quantum algorithms.
Part I. Fundamental Concepts: 1. Introduction and overview; 2.
Introduction to quantum mechanics; 3. Introduction to computer science;
Part II. Quantum Computation: 4. Quantum circuits; 5. The quantum
Fourier transform and its application; 6. Quantum search algorithms; 7.
Quantum computers: physical realization; Part III. Quantum Information:
8. Quantum noise and quantum operations; 9. Distance measures for
quantum information; 10. Quantum error-correction; 11. Entropy and
information; 12. Quantum information theory; Appendices; References;
Index.
In realizations of quantum computing, a two-level system (qubit) is often singled out from the many levels of an anharmonic oscillator. In these cases, simple qubit control fails on short time scales because of coupling to leakage levels. We provide an easy to implement analytic formula that inhibits this leakage from any single-control analog or pixelated pulse. It is based on adding a second control that is proportional to the time derivative of the first. For realistic parameters of superconducting qubits, this strategy reduces the error by an order of magnitude relative to the state of the art, all based on smooth and feasible pulse shapes. These results show that even weak anharmonicity is sufficient and in general not a limiting factor for implementing quantum gates.
We analyze a new scheme for quantum information processing, with superconducting charge qubits coupled through a cavity mode, in which quantum manipulations are insensitive to the state of the cavity. We illustrate how to physically implement universal quantum computation as well as multiqubit entanglement based on unconventional geometric phase shifts in this scalable solid-state system. Some quantum error-correcting codes can also be easily constructed using the same technique. In view of the gate dependence on just global geometric features and the insensitivity to the state of cavity modes, the proposed quantum operations may result in high-fidelity quantum information processing.
The quantum model of computation is a probabilistic model, similar to the probabilistic Turing Machine, in which the laws of chance are those obeyed by particles on a quantum mechanical scale, rather than the rules familiar to us from the macroscopic world. We present here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class. We thus offer compelling evidence that the quantum model may have significantly more complexity theoretic power than the probabilistic Turing Machine. In fact, drawing on this work, Shor (1994) has recently developed remarkable new quantum polynomial-time algorithms for the discrete logarithm and integer factoring problems