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# Faster quantum chemistry simulation on fault-tolerant quantum computers

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Quantum computers can in principle simulate quantum physics exponentially faster than their classical counterparts, but some technical hurdles remain. We propose methods which substantially improve the performance of a particular form of simulation, ab initio quantum chemistry, on fault-tolerant quantum computers; these methods generalize readily to other quantum simulation problems. Quantum teleportation plays a key role in these improvements and is used extensively as a computing resource. To improve execution time, we examine techniques for constructing arbitrary gates which perform substantially faster than circuits based on the conventional Solovay–Kitaev algorithm (Dawson and Nielsen 2006 Quantum Inform. Comput. 6 81). For a given approximation error , arbitrary single-qubit gates can be produced fault-tolerantly and using a restricted set of gates in time which is O(log ) or O(log log ); with sufficient parallel preparation of ancillas, constant average depth is possible using a method we call programmable ancilla rotations. Moreover, we construct and analyze efficient implementations of first- and second-quantized simulation algorithms using the fault-tolerant arbitrary gates and other techniques, such as implementing various subroutines in constant time. A specific example we analyze is the ground-state energy calculation for lithium hydride.
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Faster quantum chemistry simulation on
fault-tolerant quantum computers
N Cody Jones1,7, James D Whitﬁeld2,3,4, Peter L McMahon1,
Man-Hong Yung2, Rodney Van Meter5, Al ´
an Aspuru-Guzik2
and Yoshihisa Yamamoto1,6
1Edward L Ginzton Laboratory, Stanford University, Stanford, CA 94305-4088,
USA
2Department of Chemistry and Chemical Biology, Harvard University,
12 Oxford Street, Cambridge, MA 02138, USA
3NEC Laboratories, America, 4 Independence Way, NJ 08540, USA
4Physics Department, Columbia University, 538 W 120th Street, New York,
NY 10027, USA
5Faculty of Environment and Information Studies, Keio University, Japan
6National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku,
Tokyo 101-8430, Japan
E-mail: ncodyjones@gmail.com
New Journal of Physics 14 (2012) 115023 (35pp)
Published 27 November 2012
Online at http://www.njp.org/
doi:10.1088/1367-2630/14/11/115023
Abstract. Quantum computers can in principle simulate quantum physics
exponentially faster than their classical counterparts, but some technical hurdles
remain. We propose methods which substantially improve the performance
of a particular form of simulation, ab initio quantum chemistry, on fault-
tolerant quantum computers; these methods generalize readily to other
quantum simulation problems. Quantum teleportation plays a key role in
these improvements and is used extensively as a computing resource. To
improve execution time, we examine techniques for constructing arbitrary gates
which perform substantially faster than circuits based on the conventional
Solovay–Kitaev algorithm (Dawson and Nielsen 2006 Quantum Inform. Comput.
681). For a given approximation error , arbitrary single-qubit gates can be
7Author to whom any correspondence should be addressed.
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New Journal of Physics 14 (2012) 115023
1367-2630/12/115023+3533.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 produced fault-tolerantly and using a restricted set of gates in time which is O(log ) or O(log log ); with sufﬁcient parallel preparation of ancillas, constant average depth is possible using a method we call programmable ancilla rotations. Moreover, we construct and analyze efﬁcient implementations of ﬁrst- and second-quantized simulation algorithms using the fault-tolerant arbitrary gates and other techniques, such as implementing various subroutines in constant time. A speciﬁc example we analyze is the ground-state energy calculation for lithium hydride. Contents 1. Introduction 2 2. Fault-tolerant phase rotations 5 2.1. Phase kickback .................................. 5 2.2. Gate approximation sequences .......................... 7 2.3. Programmable ancilla rotation .......................... 8 2.4. Analysis of a single-qubit phase rotation ..................... 9 3. Simulating chemistry in second-quantized representation 10 3.1. Controlled phase rotations ............................ 13 3.2. Finite precision in pre-calculated integrals .................... 13 3.3. Jordan–Wigner transform using teleportation .................. 15 3.4. Resource analysis for ground-state energy simulation of lithium hydride . . . . 17 4. Simulating chemical structure and dynamics in ﬁrst-quantized representation 19 4.1. Quantum variable rotation ............................ 21 4.2. Improved parallelism in potential energy operator ................ 24 4.3. Resource analysis for ﬁrst-quantized molecular simulations ........... 25 5. Comparing simulation methods 27 6. Conclusions 29 Acknowledgments 30 Appendix A. Methods for calculating resources 30 Appendix B. Transforming the phase kickback register 31 Appendix C. Quantum circuits for potential and kinetic energy operators in ﬁrst- quantized molecular Hamiltonians 31 References 32 1. Introduction Simulating quantum physics from ﬁrst principles is arguably one of the most important applications of a quantum computer—a problem intractable to solve in many cases, yet valuable to science [1]. The objective of quantum simulation is to model natural physical systems with Hamiltonians that permit a compact representation [2,3]. Several different applications of a quantum physics simulator have been proposed, including: spin glasses [4] and lattices [5,6]; Bardeen–Cooper–Schrieffer Hamiltonians [7,8]; and quantum chemistry [9,10]. More recently, Jordan et al [11] proposed a variant of this approach for simulating relativistic quantum ﬁeld theories. In this investigation, we narrow our focus to quantum chemistry problems such as New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 3 calculating the eigenvalues of a molecular Hamiltonian [9,1214]. We aim to demonstrate constructively how quantum computers can simulate chemistry with an efﬁcient use of resources by representing the molecular system with a ﬁrst-principles Hamiltonian consisting of kinetic energy and Coulomb potential operators between electrons and nuclei. In doing so, we indicate how close the ﬁeld of quantum information processing is to solving novel problems for less computational cost than a classical computer. The motivation behind our study is that in order for computational physics on quantum computers to be useful as a scientiﬁc tool, it must have an efﬁcient implementation. Often general algorithmic complexity such as ‘polynomial time’ is taken as a by-word for efﬁcient, but we go deeper to show the substantial performance disparities between different polynomial-time algorithms, revealing which ones are signiﬁcantly less costly in space and time resources than their peers. By introducing algorithmic improvements and making quantitative analysis of the resource costs, we show that simulating quantum chemistry is feasible in a practical execution time. An example problem we analyze is calculating the ground-state energy of lithium hydride (LiH) in 5.6 h on a hypothetical fault-tolerant quantum computer with an execution time per error-corrected gate of 1 ms. This stands in contrast to previous results based on slower methods [6], which would require 3.8 years to complete the same task on the same machine. Quantum chemistry and band structure calculations account for up to 30% of the compu- tation time used at supercomputer centers [15], and ab initio chemistry is one of the two physics-simulation applications which dominate the use of supercomputing resources (the other being fusion-energy research). The most-employed techniques include density functional theory and polynomially tractable approximate quantum chemistry methods [16]. Despite the success of these methods, for example, in simulating the dynamics of a small protein from ﬁrst principles [17] or in predicting novel materials [18], they are still approximate, and much work is carried out in developing more accurate methods. Quantum simulators offer a fresh approach to quantum chemistry [19] as they are predicted to allow for the exact simulation (within a selected basis) of a chemical system in polynomial time. A quantum computer of a sufﬁcient size, say 128 logical quantum bits [9,20], would already outperform the best classical computers for exact chemical simulation. This would open the door to high-quality ab initio data for parameterizing force ﬁelds for molecular dynamics [21] or understanding complex chemical mechanisms such as soot formation [22], where a number of different chemical species must be compared. This tends to suggest that computational chemistry would be one of the ﬁrst novel applications of universal quantum computers. Several possible simulators have been proposed and studied [19,2326], but we focus on fault-tolerant circuit-model quantum simulation in this investigation [2,6,9,10,14,20,27, 28]. The reasons for these constraints are straightforward: quantum computers will probably be sensitive to noise and other hardware errors, thus requiring fault tolerance [29], and fault- tolerant quantum computing has been most successfully applied in the circuit model. Fault tolerance requires an additional overhead for the quantum computer; error correcting codes and the mechanisms they use to correct errors have been studied previously [2931]. We focus here on another matter critical to simulation algorithms, which is making arbitrary fault-tolerant gates. Arbitrary quantum operations, such as a single-qubit rotation of arbitrary angle around the σzaxis on the Bloch sphere, are typically constructed using a sequence of primitive error- corrected gates [30,32,33]. Quantum simulation depends sensitively on the execution time of arbitrary gates of this form, so one of the core contributions of this paper is to demonstrate efﬁcient constructions for such gates, which would allow simulation of more complex systems New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 4 U(τ)U(2 τ) k-2 U(2 τ) k-1 . . . . . . t(k-1) 0 { t(k-2) t(0) . . . Initialize State Preparation: 0 { In i t i a li ze S tate Preparat i on: QFT . . . Q FT Readout Figure 1. Schematic of a digital quantum simulation algorithm for energy eigenvalue estimation [2,9]. The three main steps are state preparation, simulated evolution, and readout; this investigation focuses on the middle process. After preparing an initial state |ψ0i, the system is evolved in simulated time by solving the time-dependent Schr¨ odinger equation. Note the system propagators U(2xδt) are controlled by qubits in a register representing simulated time. A quantum Fourier transform (QFT) on the time register provides an estimate of an energy eigenvalue. The accuracy of the simulation depends on suppressing errors in both state preparation and simulated-time evolution, which is why fault tolerance is an important consideration for quantum simulation algorithms. under a ﬁxed-resource constraint. Many of the fast circuits we use can be understood as a form of quantum teleportation, and prior efforts have established the importance of teleportation and entanglement as information-processing resources [3436]. A digital quantum simulation algorithm consists of three primary steps (ﬁgure 1): state preparation, simulated time evolution, and measurement readout. This paper focuses on the second step, evolving the system in simulated time, because this represents the core of the algorithm. Simulation of time evolution on a quantum computer is a sequence of quantum gates which closely approximates the evolution propagator U(t;t+δt)=Texp(i ¯ hRt+δt tH(τ )dτ ) of a desired Hamiltonian H, where Tis the usual time-ordering operator. In the case of a time-independent Hamiltonian, we have Ut)=exp(i ¯ hHδt), as in ﬁgure 1. The increment δtis a single time step of simulation, and a simulation algorithm often requires many time steps, depending on the desired result (e.g. energy eigenvalue). State preparation and measurement readout are necessary steps which are not discussed here, but details can be found in references [3,30,3740]. The quantum simulation problem we analyze is the ground-state energy calculation of LiH from ﬁrst principles. This was called the ‘chemist’s workbench’ and is an appropriate continuation of quantum computational applications of chemistry going beyond molecular hydrogen [10,14,41,42]. For some of the selected methods, the quantum circuit is compact enough to be tractable for classical computation. Still, this example is useful for two reasons. Firstly, the LiH simulation preserves the features of more complicated chemical simulations while permitting a simple analysis that illustrates the improved methods we propose. Secondly, with quantum computers still in early stages of development, a compact problem such as LiH would be a convenient choice for experimental demonstrations of quantum simulation in the New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 5 near term. A larger, more complex simulation problem was studied in [43] using the methods analyzed here. This paper provides constructive methods for simulating quantum chemistry efﬁciently using fault-tolerant quantum circuits. Section 2describes how to construct quantum circuits for arbitrary phase rotations, which are essential to simulation. Section 3develops a fault-tolerant simulation algorithm in second-quantized representation using phase rotations from the prior section; analysis of the computing resources required follows. Section 4demonstrates how to construct an efﬁcient chemistry simulation in ﬁrst-quantized form, and total quantum resources are analyzed. Section 5outlines how to determine the optimal simulation parameters for a given set of engineering constraints and performance objectives. The paper concludes by discussing the prospects for fault-tolerant quantum computers to solve novel simulation problems. 2. Fault-tolerant phase rotations The algorithms which simulate chemistry on a circuit-model quantum computer require many phase rotations, accurate to high-precision. A single-qubit rotation gate in general form is RZ(φ) =eiφ 2eiφ 2σz=1 0 0 eiφ,(1) where φis arbitrary and σzis the Pauli spin operator. Additionally, any arbitrary single- qubit gate can be produced using three distinct phase rotations and two Hadamard gates [30]. Quantum error correction constrains the available operations to a ﬁnite set of fundamental gates, so the arbitrary rotations needed to simulate Hamiltonian evolution must be constructed with a circuit consisting of these fundamental gates. Phase rotations are needed at every time step of simulation, so the performance of the simulation algorithm depends on the computational complexity of these arbitrary-gate circuits. In this section, we discuss three different approaches for implementing arbitrary phase gates efﬁciently: phase kickback [4446], which uses multi- qubit gates acting on an ancilla register; gate approximation sequences, such as those generated by the Solovay–Kitaev algorithm [30,32] or by Fowler’s algorithm [33], which are sequences of single-qubit gates; and programmable ancilla rotations (PARs), which compute ancillas in advance using one of the above methods to achieve very low circuit depth in the algorithm. 2.1. Phase kickback Phase kickback [44,45], also known as the Kitaev–Shen–Vyalyi algorithm [46], is an ancilla- based scheme that uses an addition circuit to impart a phase to a quantum register. Phase kickback relies on a resource state |γ(k)iwhich can be deﬁned by the inverse quantum Fourier transform (QFT) [30,47,48]: |γ(k)i = U QFT |ki=1 N N1 X y=0 e2πiky /N|yi.(2) The register |kicontains nqubits prepared in the binary representation of k, an odd integer. The state |γ(k)iis a uniform-weighted superposition state containing the ring of integers from 0 to N1, where N=2n, and each computational basis state has a relative phase proportional to the equivalent binary value of that basis state. This ancilla register must be produced fault- tolerantly. Kitaev et al [46] provide a method to prepare |γ(k)iusing phase estimation such New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 6 u ( ) Z R { (k) { (k) Figure 2. Controlled addition of the quantity udetermined by equation (4) is approximately equivalent to an arbitrary phase rotation RZ), but the former uses only fault-tolerant gate primitives and ancillas. The operation denotes unitary addition modulo 2n, where nis the number of qubits in the |γ(k)iregister; for illustration, n=3 in the circuits above. that kis a random odd integer; hence our analysis does not assume a value for k. If necessary, appendix Bprovides a technique to convert any |γ(k)iinto |γ(1)i. The circuit complexity for creating |γ(k)iis small, requiring perhaps a few thousand gates, so the cost of this initialization step is negligible compared to quantum algorithms we analyze later. One could also view the |γ(k)istate as a discretely sampled plane wave with wavenumber k. Consider then that |γ(k)iis an eigenstate of the unitary operation Uu|mi=|m+u(mod N)i for modular addition, so that Uu|γ(k)i = 1 N N1 X y=0 e2πik(uy)/N|yi=e2πik u/N|γ(k)i,(3) where denotes addition modulo Nand uis an integer. Moreover, the eigenvalue of modular addition on |γ(k)iis a phase factor proportional to the number uadded. Note that the addition operation Uuis readily implemented with a fault-tolerant quantum circuit [4953]. To determine the value of uin the addition circuit which approximates a phase rotation RZ(φ), one solves the modular equation ku Nφ 2π(mod N), (4) which always has a solution since kis odd and Nis a power of 2 (kand Nhave no common factors). The operation bxedenotes rounding any real xto the nearest integer; any arbitrary rule for half-integer values sufﬁces here. By proper selection of u, one can approximate any phase rotation to within a precision of ||62π 2n+1 radians, where || = |φ2π Nku |(mod 2π ). We can now understand how the method received its name: since |γ(k)iis an eigenstate of addition, when an integer uis added (using an addition circuit) to this register, a phase is ‘kicked back.’ This method is quite versatile, as several different types of phase gates are developed using phase kickback in this work. Single-qubit phase rotations using phase kickback are constructed with a controlled addition circuit, as shown in ﬁgure 2. Intuitively, a phase is kicked back to the control qubit if it is in the |1istate, which is equivalent to the phase rotation in equation (1). The accuracy of the phase gate and the quantum resources required depend on the number of bits in the ancilla state |γ(k)i. After solving equation (4), the integer uis added to |γ(k)iusing a quantum adder controlled by the qubit which is the target of the phase rotation. There are various implementations of quantum adder circuits which have tradeoffs in performance between circuit New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 7 Table 1. Universal set of fault-tolerant gates in this investigation. Symbol Name Matrix representation X,Y,ZPauli gates 0 1 1 0,0i i 0 ,1 0 01 HHadamard 1 21 1 11 Sπ/4 phase gate 1 0 0 i Tπ/8 phase gate 1 0 0 eiπ/4 CNOT Controlled-NOT (two-qubit gate) 1000 0100 0001 0010 depth, circuit size, and difﬁculty of implementation [4953]. Since |γ(k)iis not altered by phase kickback, the number of such registers required for a quantum algorithm is equal to the maximum number of phase rotations which are computed in parallel at any point in the algorithm. 2.2. Gate approximation sequences A gate approximation sequence uses a stream of fault-tolerant single-qubit gates to approximate an arbitrary phase rotation, such as that in equation (1). For context, a common set of fault- tolerant gates is listed in table 1. Such sequences must be calculated using a classical algorithm, and at least two options exist. The Solovay–Kitaev algorithm [30,32] is perhaps the best known method for generating arbitrary quantum operations, so it will serve as a benchmark in our analysis. A subsequently derived alternative, Fowler’s algorithm [33], offers shorter gate sequences for a given approximation accuracy, with some notable drawbacks in classical algorithmic complexity. The efﬁciency of a gate approximation sequence is determined by the accuracy of approximation (i.e. how close the composite sequence is to the desired gate) as a function of resource costs. Both the Solovay–Kitaev and Fowler algorithms produce better approximations if one can afford more quantum gates; however, quantum resources are expensive, so we must implement ﬁnite-length sequences which produce a sufﬁciently good approximation. We adopt the distance measure in [33] to determine approximation accuracy: distd(U,V)=sdtr(UV) d,(5) New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 8 X 1 20 ( ) 1 + ei((-1) ) ZMZ R Figure 3. Probabilistic rotation using an ancilla qubit. The state of the top qubit is teleported to the bottom qubit with a phase rotation applied. The measurement is in the computational (Z) basis. The circuit enacts either RZ(φ) or RZ(φ) with equal probability. The Xgate is classically conditioned on the measurement result. where dis the dimensionality of Uand V(e.g. d=2 for a single-qubit rotation). At the end of this section, we provide a quantitative analysis of resource costs to produce phase rotations. What is sufﬁcient for the moment is to know that, if we denote the approximation error as =dist2(U,Uapprox), the corresponding approximating sequence Uapprox has asymptotic length O(poly(log )), a result known as the Solovay–Kitaev theorem [30]. 2.3. Programmable ancilla rotation We introduce a third method for producing phase rotations, the PAR, which pre-computes ancillas before they are needed. Shifting the computing effort to a different point in the quantum circuit (assuming parallel computation) allows this method to achieve constant average depth in the algorithm for any desired accuracy of rotation, which can be as small as four quantum gates. The pre-calculated ancillas still require quantum circuits of similar complexity to the previously discussed methods, so this approach is best-suited to a quantum computer with many excess logical qubits available for parallel computing. The PAR is based on a simple circuit which uses a single-qubit ancilla to make a phase rotation, which is a ‘teleportation gate’ [34,35], as shown in ﬁgure 3. This circuit is probabilistic, so there is a 50% probability of enacting RZ(φ) instead of RZ(φ); in such an event, we attempt the circuit again with angle 2φ, then 4φif necessary, etc. This proceeds until the ﬁrst observation of a positive angle rotation, in which case we have enacted a rotation φtotal =2mφPm1 x=12xφ=φ. The circuit for the PAR is shown in ﬁgure 4. The programmed ancillas |ω(1)i = 1 2(|0i+ eiφ|1i),|ω(2)i = 1 2(|0i+ ei(2φ) |1i), etc are pre-computed using one of the methods above for a phase rotation. A very similar method was shown in [54], but we generalize here from φ=π 2kto arbitrary rotation angles. In practice, phase kickback may be preferable for producing the pre- computed ancillas since reusing the same |γ(k)iancilla does not introduce additional errors into the circuit. The cascading series of probabilistic rotations continues until the desired rotation is produced or the programmed ancillas are exhausted. For practical reasons, one may only calculate a ﬁnite number of the PAR ancillas, and if all such rotations fail, then a deterministic rotation using phase kickback or a gate approximation sequence is applied. The probability of having to resort to this backstop is suppressed exponentially with the number of PAR ancillas pre-computed. The average number of rounds of the circuit in ﬁgure 4before a successful rotation is simply given by P m=1m 2m=2. The Xgate in each round can be performed with a Pauli frame [43,55,56], so counting measurement as a gate, the number of gates per round is 2, and the average number of gates per PAR is 4. With a ﬁnite number of pre-computed ancillas M, New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 9 ?X ?X ?X(2 ) Z3 R (1) (2) (3) 0H 0H 0H Precomputed Ancillas () Z R (2 ) Z R (2 ) Z R2 Figure 4. PAR circuit. The bulk of the computing effort is shifted to an earlier part of the circuit, when the ancillas are produced. The programmed ancillas are used in multiple rounds of the circuit in ﬁgure 3, each of which succeeds with 50% probability. The cascading circuit above terminates after the ﬁrst success, as denoted by the ‘?’ decision gates. The average number of rounds required is 2, so by pre-computing the ancillas, this method contributes very few additional gates to an algorithm’s circuit depth. there is a probability 2Mof having to implement the considerably more expensive (in circuit depth) deterministic rotation. Nevertheless, if the computer supports the ability to calculate the programmed ancillas in advance, the PAR produces phase rotations that are orders of magnitude faster than other available methods, which also leads to faster execution of simulation algorithms. 2.4. Analysis of a single-qubit phase rotation We begin our quantitative analysis by examining fault-tolerant single-qubit phase rotations. We construct rotations using phase kickback, the Solovay–Kitaev algorithm, Fowler’s algorithm, and PARs. In each case, we determine the depth of the quantum circuit and the types of fault- tolerant gates required. The techniques developed here will be used in the more complicated phase rotations for the simulation algorithms in sections 3and 4. To assess the performance of quantum circuits, let us assume the following simpliﬁed quantum computing model. The hypothetical system uses fault-tolerant quantum error correction, so we presume the quantum gates are ideal. The quantum computer only has access to a limited set of ‘fundamental’ gates, which are summarized in table 1; this set of gates is typical for a fault-tolerant quantum computer [30,43,54,57]. The available set of quantum gates is restricted by error-correction codes [30]. In essence, errors cannot be corrected for the continuous set of arbitrary quantum gates, just as in classical analogue computing. We allow full parallelism so that gates can be applied to all qubits simultaneously, as long as the two-qubit (CNOT) gates do not overlap. Because the fundamental gate set has a ﬁnite number of members, phase kickback or gate approximation sequences are required to produce approximations to arbitrary gates. We should note that each logical gate with error correction will require many more physical operations to be implemented [29,43,54], but we purposefully avoid these details so that our present analysis is independent of hardware and error correction models. To make a connection to future quantum hardware, separate investigations ﬁnd that one error- corrected qubit requires about 1000 faulty physical qubits coupled into an encoded state [43,54]. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 10 Therefore, when we state that 100 encoded qubits are required, this implies the quantum processor will contain 105physical qubits. When benchmarking the performance of a phase rotation, the important ﬁgures of merit are the quantum resources consumed to achieve a given accuracy of approximation. Using the distance measure in equation (5), the approximation error is quantiﬁed as =dist2RZ(φ), Uapprox,(6) where Uapprox is the circuit approximating RZ). Figure 5reports two quantum resources for a single-qubit rotation: circuit depth, which is the minimum execution time in gates, and the total number of Tgates required (see table 1). These calculations are explained in more detail in appendix A.Tgates are signiﬁcantly more expensive to prepare fault-tolerantly than other fundamental gates in many prominent error-correcting codes [30,57], so they represent an important consideration for large-scale quantum computing [6,43,54]. It is apparent from ﬁgure 5that Solovay–Kitaev sequences are substantially more expensive than their counterparts in both circuit depth and Tgates. Fowler sequences are very compact and, in fact, optimal for an approximation sequence, but the classical algorithm to ﬁnd them requires a calculation time that appears to grow exponentially faster than the other methods: 6102requires minutes, 6103requires about an hour, and 6104requires about a day, for each rotation, on a modern workstation. For these reasons, phase kickback may be the method of choice when high-precision (6106) rotations are required. Phase kickback requires quantum resources comparable to Fowler sequences, but the quantum circuit depends on adders, which are trivial to compile. The methods we analyze for producing fault-tolerant phase rotations are summarized in table 2. 3. Simulating chemistry in second-quantized representation Simulation in the second-quantized form expresses the electronic Hamiltonian Hin terms of the creation operators a pand the wavefunction in terms of fermionic (or bosonic) modes |pi a p|0i(i.e., occupation number representation). In chemistry, the single-electron molecular orbital picture has provided a practical method for approximating an N-electron wavefunction. Using second-quantized algorithms, basis sets in computational chemistry can be imported directly into quantum computational algorithms. For this reason, both theoretical [9,12,14] and experimental [10,42] investigations in second-quantization have been performed. Following the standard construction (see e.g. [19]), an arbitrary molecular Hamiltonian in second-quantized form can be expressed as H=X p,q hpq a paq+1 2X p,q,r,s hpqr s a pa qaras,(7) where hpq = hp|(ˆ T+ˆ VN)|qiare one-electron integrals ( ˆ Tis the kinetic energy operator, and ˆ VN is the nuclear potential) and hpqr s = h pq|ˆ Ve|rs irepresent the Coulomb potential interactions between electrons. All of the terms hpq ’s and hpqr s ’s are pre-computed numerically with classical computers, and the values are then used in the quantum computer to simulate the Hamiltonian evolution through the operators Upq =eihpq (a paq+a qapt(8) New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 11 10 -1 10-2 10 -3 10 -4 10-5 T gates Approximation error () Circuit depth (gates) 100 101 102 103 104 105 106 Solovay-Kitaev Sequences Phase Kickback Fowler Sequences PARs 100 101 102 103 104 105 Figure 5. Quantum computing resources required to produce a fault-tolerant single-qubit phase rotation to accuracy =dist2RZ(φ), Uapproxusing various methods. (Top) Circuit depth for single-qubit rotations. (Bottom) Number of Tgates required for each rotation. There is variation in the resources required for Solovay–Kitaev sequences, Fowler sequences, and PARs; each point is the mean number of gates required, and where applicable, the bars show plus/minus one standard deviation. The Solovay–Kitaev data is averaged over 9534 random angles (φ), and the Fowler data is averaged over 98 random angles per point. Fowler sequences are numerically intensive to calculate, so curves ﬁt to the data are shown for 6103: depth = −24.9 log10 7.64 and Tgates = 9.75 log10 2.81. Phase kickback is implemented here with a ripple-carry adder [52]. PARs use six pre-computed ancillas. Solovay–Kitaev sequences were calculated using code written by Dawson and Nielsen [32]; Fowler sequences were calculated using code written by Fowler. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 12 Table 2. Summary of methods for producing fault-tolerant phase rotations. The quantity is the accuracy of an approximate rotation, and is deﬁned by equations (5) and (6). Method Description Advantages Disadvantages Phase Approximates Trivial to compile. Requires a kickback arbitrary phase Circuit depth is logical ancilla rotation via O(log ) or O(log log ), register controlled depending on consisting of addition applied adder circuit O(log ) qubits. to the |γ(k)iancilla Resource costs register are about 2–3× higher than Fowler sequences Solovay–Kitaev Approximates Polynomial-time Dramatically sequence arbitrary compiling more expensive rotation with a algorithm. No in quantum sequence of logical ancilla resources than fundamental states alternatives gates. Depth is O(logc), with c4 Fowler Approximates Minimal-depth Sequence- sequence arbitrary sequences. No determination rotation with a logical ancilla algorithm has sequence of states exponential fundamental complexity and gates. Depth is becomes O(log ) infeasible for high-accuracy rotations PAR Approximates Constant Requires logical arbitrary average depth ancillas which rotation with a (four gates) for must be probabilistic any phase pre-computed circuit using rotation ancilla and measurement and Upqr s =eihpq rs (a pa qaras+asaraqapt.(9) These operators are constructed with a Jordan–Wigner transform and an arbitrary controlled phase gate CRZ) [14], as shown in ﬁgure 6. The Jordan–Wigner transform requires H,S, and CNOT gates, which are often readily available in fault-tolerant settings, so we focus ﬁrst on the considerably more resource-intensive controlled phase rotations. We later show how to implement the Jordan–Wigner transform efﬁciently. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 13 H H ( ) Z R H H(- /2) X R t 1 2( ) Z R (- /2) X R (/2) X R (/2) X R Figure 6. Excitation operator eih12(a 1a2+a 2a1tencoded into a quantum circuit [14]. Above, θ=h12δt. The gate RX(π/2)=H·S·His available from the set in table 1. In this example, the control qubit |tiis used for phase estimation, and the qubits |χ1iand |χ2iare basis functions (e.g. molecular orbitals). The controlled phase rotations CRZ ) must be approximated using circuits of available fault-tolerant gates. 3.1. Controlled phase rotations As can be seen in ﬁgure 6, when Upq or Upq rs is implemented in a controlled operation (such as in energy eigenvalue estimation, see also ﬁgure 1), the core component of the circuit is a controlled phase rotation CRZ) = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 eiφ .(10) One way to implement the controlled rotation in equation (10) is to deconstruct the operation into CNOTs and single-qubit rotations [58], as shown in ﬁgure 7. Another method requires just one single-qubit rotation, as well as an ancilla |0i, as shown in ﬁgure 8. Nielsen and Chuang [30] (p 182) provide a circuit decomposition for the Toffoli gate into gates in table 1. We use the circuit in ﬁgure 8(requiring just one phase rotation) for the remainder of this paper, because the cost of one ancilla qubit is typically modest compared to a phase rotation. One can implement phase kickback, gate approximation sequences, or PARs to produce the single-qubit rotations, as in section 2.4. Additionally, the PAR construction can be modiﬁed to produce controlled rotations more directly. If the control qubit only controls other circuits between ancilla production and the time a controlled-PAR is needed, as is the case for phase estimation algorithms, one can create the ancillas (see ﬁgure 4) using controlled rotations with one of the above methods and produce a controlled-PAR with the same cascading circuit. The different methods of producing a controlled phase rotation are analyzed in ﬁgure 9. We have excluded Solovay–Kitaev sequences, which permits a linearly scaled vertical axis, showing that each of these methods has execution time linear in logor constant. As before, the values for Fowler sequences are extrapolated. We can see that Fowler sequences and phase kickback are separated by approximately a factor of 3 in execution time, and the choice between the two would be motivated by whether compiling the Fowler sequence is feasible or not. The PAR circuit requires one of the above methods to pre-compute ancillas. 3.2. Finite precision in pre-calculated integrals The execution time of a second-quantized simulation algorithm is proportional to the number of integral terms hpq and hpqr s , as indicated by equations (7)–(9). We now consider how to speed up New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 14 = (φ) Z R(φ/2) Z R (φ/2) Z R (-φ/2) Z R Figure 7. Decomposition of a controlled phase rotation into CNOTs and fault- tolerant single-qubit rotations. If the control qubit only controls other circuits, as in phase estimation algorithms, the third phase rotation commutes with the CNOTs. In such an event, the third single-qubit rotations from all decompositions of controlled rotations commute, and they can be combined into just one rotation prior to a non-commuting operation on this qubit (such as the QFT and measurement readout in ﬁgure 1). As a result, controlled rotations in phase estimation algorithms are effectively decomposed into two CNOTs and two single-qubit rotations with this circuit. = (φ) Z R (φ) Z R x y 0 x y 0 Figure 8. Controlled rotation CRZ) (see equation (10)) between qubits |xiand |yiusing two Toffoli gates, just one single-qubit rotation gate, and an ancilla |0i. The ancilla qubit is conditionally set to |1iusing a Toffoli gate, and a phase is imparted to this state with the rotation RZ(φ). A ﬁnal Toffoli gate returns the ancilla qubit to state |0i. the algorithm by omitting the integral terms that are negligibly small in magnitude. For a basis set consisting of Msingle-particle orbitals, the maximum number of integral terms is O(M4). In practice, however, the effort for evaluating these integrals often scales somewhere between O(M2)and O(M3)with modern implementations [59], because typically many integral terms may be neglected for being smaller in magnitude than a cutoff threshold. Consequently, the execution time of second-quantized simulation is determined by the number of pre-computed integrals of the form hpq and hpqr s of sufﬁciently large magnitude, as well as the efﬁciency of producing the corresponding arbitrary phase rotations in the quantum computer, such as CRZ(hp q δt)in the gate sequence for eihpq(a p(a)q+a qapt[14]. To illustrate how many integral terms are present in a typical chemical problem, we have calculated the integrals for a second-quantized simulation of LiH. We performed calculations in the minimal basis and in a triple-zeta basis, using the GAMESS quantum chemistry package [60,61], at a bond distance of 1.63 Å, with an integral term cutoff of 1010 in atomic units. We computed the number of integrals above cutoff using the STO-3G basis [62] containing 12 spin orbitals (6 spatial orbitals) and the TZVP basis [63] containing 40 spin orbitals (20 spatial orbitals). The cumulative number of integral terms as a function of cutoff in TZVP basis is plotted in ﬁgure 10. With the STO-3G basis, there were 231 non-zero molecular integrals, but only 99 of them were greater than 1010 atomic units in magnitude. This is an order of magnitude below what is expected from O(M4)scaling. Considering the larger, more accurate basis set (TZVP), there were 22 155 non-zero integrals, but only 10 315 were greater than the New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 15 0.5 0 50 100 150 200 250 300 350 400 10 -1 10 -2 10 -3 10 -4 10 -5 Circuit depth (gates) Approximation error () Phase Kickback Fowler Sequences PARs Figure 9. Circuit depth for controlled phase rotations using various methods. A desired controlled rotation CRZ) is approximated with a fault-tolerant circuit Uapprox with accuracy =dist4(Uapprox,CRZ)) using the method in ﬁgure 8. Solovay–Kitaev sequences are omitted here to permit comparison of the more efﬁcient schemes on a linear scale. The bars on Fowler sequence data indicate the standard deviation taken over 98 random-angle rotations. The controlled-PARs have a depth of four gates, on average, regardless of rotation accuracy. Phase kickback uses a ripple-carry adder since the addends have less than 16 bits [52]. If very high precision were desired, a carry-lookahead adder can achieve depth O(log log ) at the expense of additional qubits and parallel circuits (more Tgates) [53]. cutoff. Figure 10 shows that a higher cutoff, such as 104, can further reduce the number of integrals in the TZVP basis implemented in the simulation. As a result, the effective number of integral terms the quantum computer must implement as phase rotations is nearly two orders of magnitude less than the asymptotic analysis would suggest. This is an example of the over- estimation of the resource costs that can occur when using asymptotic estimates. This technique becomes particularly relevant in large molecules since distant particles interact weakly, and in such an event, many of the associated integral terms may be negligibly small. Raising the cutoff threshold impacts the accuracy of the simulation, so one must attempt to balance the resource costs of simulation with the usefulness of the result. 3.3. Jordan–Wigner transform using teleportation The second-quantized algorithm uses Jordan–Wigner transforms to implement operators such as eihpq (a paq+a qapt, and this section shows how to perform such transforms in constant time. As elaborated in [14], the circuits for Jordan–Wigner transforms often consist of ladders of CNOT New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 16 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100101 0 2000 4000 6000 8000 10000 12000 Cutoff value (atomic units) Number of integrals Figure 10. The number of integral terms implemented in a second-quantized simulation of LiH using a TZVP basis, as a function of cutoff threshold. Only integral terms with absolute value above the threshold are implemented in circuits, and the rest are neglected. As shown in the ﬁgure, a cutoff of 104 would require the algorithm to implement just over 9000 integral terms. BSM BSM BSM (a) (b) (c) Figure 11. Rearrangement of the CNOT ladder common in Jordan–Wigner transforms using teleportation. (a) The original CNOT ladder requires an execution time that grows with the extent of the simulation in qubits. (b) A conceptual diagram of what teleportation accomplishes. The qubits ‘move’ backwards in time. (c) A valid quantum circuit that uses teleportation to move qubits in a manner which allows parallel computation of the CNOTs. The BSM is the Bell state measurement which teleports the qubits; the result of this measurement indicates the Pauli errors which are tracked by the Pauli frame [43]. The Bell state |8+i = 1 2(|00i+|11i)can be prepared from |0iancillas using one Hgate and one CNOT gate. Similarly, the BSM can be implemented using one H, one CNOT, and measurement of the two qubits in the computational basis. gates, such as the one in ﬁgure 11(a). In a simulation with Mbasis states, these ladders can extend across the entire register of qubits corresponding to these basis states, which leads to the O(M5)asymptotic runtime quoted in [19] when there are at most O(M4)integral terms. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 17 The CNOT ladder is a sparse network of Clifford gates, so we show how it may be implemented in constant time using teleportation [34,35]. Figure 11(b) gives an intuitive picture for what will be accomplished. If the path of the qubits could be rearranged to somehow propagate backwards in time, the CNOT gates could be implemented simultaneously. Qubits cannot move backwards in time per se, but they can be moved arbitrarily using teleportation; notice how the conceptual (but unphysical) circuit in ﬁgure 11(b) is realized by a physical circuit in ﬁgure 11(c). Ancilla Bell states |8+i = 1 2(|00i+|11i)are used to teleport qubits in this rearranged CNOT ladder. Teleportation introduces a random Pauli error on the teleported qubit, but it is possible to track these errors and their propagation through CNOT gates using Pauli frames [43,55,56]. With this modiﬁcation, it is possible to implement the Jordan–Wigner transform in constant time, which removes one of the bottlenecks to high-speed second- quantized simulation. This method could be adapted to implement other Clifford-group circuits in constant time, at the expense of requiring enough ancilla Bell states. 3.4. Resource analysis for ground-state energy simulation of lithium hydride Using the hypothetical quantum computer from section 2.4, we examine the resources required to perform simulation in second-quantized form. Estimates of the number of qubits required for various instances of second-quantized chemical simulation have been reported previously [9,19], so we focus instead on the execution time and effort to prepare fault-tolerant gates (here we consider the number of Tgates). Figure 12 shows both the circuit depth and the number of Tgates required to simulate LiH in the STO-3G basis as a function of rotation accuracy threshold max, for 1023 simulated time steps. The precision in the readout is proportional the number of time steps simulated. The energy estimate in this simulation has 10 bits of precision, and in general, 2n1 steps are required for nbits of precision. If we assume that the duration of a single quantum gate is 1 ms (cf [43]), then the total execution time of the simulation ranges from 5.6 h using PARs to 3.8 years using Solovay–Kitaev rotations. The number of Tgates in ﬁgure 12 serves as an indication of the complexity demanded of the quantum computer. Although we do not delve into this matter, Jones et al [43] and Isailovic et al [54] discuss the importance (and difﬁculty) of producing these gates. What becomes apparent is that using PARs, while very fast, is also more expensive in the consumption of Tgates than directly implementing Fowler sequences or phase kickback. Choosing between such approaches depends on the capabilities of the quantum computer, and we discuss this matter in more detail in section 5. To provide an indication of how much execution time in second-quantized simulation is devoted to phase rotations, ﬁgure 13 shows the relative ratio of circuit depth devoted to implementing rotations versus all other gates for each of the methods considered when simulating LiH with rotation accuracy 6104. It is clear here that Solovay–Kitaev has such high circuit depth that it cannot be drawn to scale. We see also that Fowler and phase kickback sequences require execution times that are comparable, whereas PARs actually do not represent the majority of the circuit depth, unlike all of the prior methods. This is an encouraging result, because it shows that previous examinations that depended on Solovay–Kitaev sequences can be improved by orders of magnitude with more efﬁcient phase rotations [6]. We do not consider Solovay–Kitaev sequences further in this investigation. The techniques for improving New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 18 T gates Approximation error () max 10 -1 10-2 10 -3 10 -4 10-5 Solovay−Kitaev Sequences Phase Kickback Fowler Sequences PARs 107 108 109 1010 1011 107 108 109 1010 1011 1012 Circuit depth (gates) Figure 12. Total circuit depth and Tgates for a second-quantized simulation of LiH using the STO-3G basis, calculated for different constructions of controlled rotations as a function of accuracy max. For a given max, every controlled rotation CRZ) in the algorithm is approximated with a fault-tolerant circuit Uapprox with accuracy distance =dist4(Uapprox,CRZ )) such that 6max. An accuracy threshold max 6104is used in later analysis. This simulation implements all integral terms in the Hamiltonian (see equation (7)). (Top) Circuit depth using the gate set in table 1. In this plot, only the mean number of gates for PAR circuits is shown. (Bottom) Tgates required for each method. The controlled-PAR ancillas are produced using controlled rotations constructed using Fowler sequences; six controlled-PAR ancillas are pre-computed for each rotation, and only mean values are plotted. The sudden jump in Solovay–Kitaev resource costs is because many controlled rotations in this algorithm have a small angle φ0 that is approximated with identity gate at low precision, whereas the other methods are using a typical sequence length for arbitrary φ. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 19 1 1 1 1 7460 7.0 (0.24) Clifford Gates Approximate Phase Rotations Solovay-Kitaev: Fowler Sequences: PAR (in-place): 1 20.0 Phase Kickback: Figure 13. The relative amount of time (circuit depth) of a fault-tolerant, second-quantized simulation of LiH devoted to Clifford gates {X,Y,Z,H,S,CNOT} versus phase rotations that must be approximated. In this example, rotations are computed to an accuracy 6104. The relative circuit depth of rotations calculated by the Solovay–Kitaev algorithm is too large to be drawn to scale here. In the case of PAR, the ancillas must be pre-computed with a method such as Fowler sequences, but this can be carried out in parallel with other algorithm operations. Table 3. Summary of methods for efﬁcient second-quantized chemical simulation. The quantity Mis the number of basis functions used in the representation of the chemical problem; larger basis sets produce more accurate results at the expense of greater circuit complexity. Method Description Advantages Disadvantages Finite-precision Neglect to implement Second-quantized None if the cutoff cutoff in integral terms circuit complexity is threshold is second-quantized below a chosen reduced in both below gate integrals cutoff in the depth and approximation algorithm execution the number of Tgates accuracy Jordan–Wigner Use a Second-quantized Teleportation transform using teleportation circuit depth circuit requires teleportation circuit to reduces to at at most 3M4 implement most O(M4)qubits instead of Jordan–Wigner from O(M5)M(only during transform in Jordan–Wigner constant time transform) second-quantized simulation are summarized in table 3. The methods for determining resource costs are summarized in appendix A. 4. Simulating chemical structure and dynamics in ﬁrst-quantized representation The ﬁrst-quantized simulation algorithm is in some ways more complex than the second- quantized algorithm, but for problems in chemistry larger than a handful of particles, it is computationally faster. A ﬁrst-quantized simulation is essentially a ﬁnite-difference method for solving the Schr¨ odinger equation. Conﬁguration space is discretized into a Cartesian grid, and each particle (e.g. electron) has a wavefunction expressed in a quantum register that encodes a New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 20 probability amplitude at each coordinate on the grid. For example, let us imagine that we form a position-basis representation for a single electron on a 2p×2p×2pgrid, which requires only 3pqubits. Explicitly, the electronic wavefunction is represented as |ψei= 2p1 X x,y,z=0 c(x,y,z)|xi|yi|zi=X r c(r)|ri,(11) where c(x,y,z)is the complex probability amplitude for the electron to occupy the volume element centered at the position r(x,y,z). The rightmost part of equation (11) is shorthand that will be used throughout this section. The spin degree of freedom can easily be incorporated by including an extra qubit, and to describe a many-electron state, the wavefunction has to be properly anti-symmetrized [37,64]. To simulate the evolution of a time-independent molecular Hamiltonian Hfor problems in quantum chemistry, we adopt the method given in [3,20]. The complete Hamiltonian in ﬁrst-quantized form can be expressed as the sum of the kinetic ( ˆ T) and potential ( ˆ V) operators H=ˆ T+ˆ V= − X i ¯ h22 i 2mi +1 2X i6=j qiqj 4π0ri j ,(12) where the indices iand jrun over all particles (electrons and nuclei) of any given molecule. Here ri j rirjis the distance between particles iand j, which carry charges qiand qj respectively. Let us outline how ﬁrst-quantized simulation works before delving into details. The core of the algorithm is evolving the Hamiltonian in simulated time, achieved by applying the propagator U(t)=exp(iHt)(setting ¯ h=1 and assuming His time-independent), which solves the time-dependent Schr¨ odinger equation [2]. This process is readily achieved using the split operator approximation, a form of Trotter–Suzuki decomposition [19,27,65,66], where the kinetic and potential energy operators are simulated in alternating steps as U(t)=eiHtheiˆ Tδt/2eiˆ Vδteiˆ Tδt/2it δt.(13) The exponent t δtis the number of times the circuit corresponding to the expression in brackets is implemented, so it is always an integer. The operators eiˆ Vδtand eiˆ Tδtare diagonal in the position and momentum bases, respectively. One can switch the encoded conﬁguration space representation between these two bases by applying the QFT to each spatial dimension of the wavefunction (cf equation (11)), which can be efﬁciently implemented in a quantum computer [48]. Kassal et al [20] show how to construct quantum circuits for operators eiˆ Vδt and eiˆ Tδt, and in this section, we complement that work with analysis of fault-tolerant versions of these operators. To make an algorithm fault-tolerant, its constituent operations must be decomposed into circuits of fault-tolerant primitive gates such as those in table 1. Consider the potential energy propagator eiˆ Vδtas an example. Given a b-particle wavefunction in the position basis as ψ1,2,...,b=X r1,r2,...,rb c(r1,r2,...,rb)|r1r2. . . rbi,(14) where c(·)is the complex amplitude as a function of position in conﬁguration space and subscripts correspond to particles in the system, one calculates the phase evolution of the New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 21 potential operator eiˆ Vδtin three steps, as follows: X r1,...,rb c(r1,...,rb)|r1. . . rbi|000 . . .i X r1,...,rb c(r1,...,rb)|r1. . . rbi|V(r1,...,rb)i(15) X r1,...,rb eiV(r1,...,rbtc(r1,...,rb)|r1. . . rbi|V(r1,...,rb)i(16) X r1,...,rb eiV(r1,...,rbtc(r1,...,rb)|r1. . . rbi|000 . . .i.(17) Firstly, equation (15) calculates the potential energy as a function of position coordinates [20] (note that ˆ Vis diagonal in this basis) and stores the result in a quantum register |V(r1,r2,...,rb)ito some ﬁnite precision. Appendix Cdescribes how to implement this quantum circuit for molecular Hamiltonians. Secondly, equation (16) uses the |V(r1,r2,...,rb)iregister in a ‘quantum variable’ phase rotation that imparts a phase to each grid point of the wavefunction in position basis proportional to the potential energy at those coordinates. This section discusses how to implement the quantum variable rotation (QVR) using fault-tolerant quantum circuits. Finally, the quantum circuit from the ﬁrst step is reversed in equation (17) to reset the |V(r1,r2,...,rb)iregister to |000 . . .i, also known as ‘uncomputation’ [30]. The sequence of these three steps is equivalent to the operation eiˆ Vδt|ψi. The kinetic energy propagator eiˆ Tδtis calculated similarly in three steps, with the second also being a QVR. This operator is diagonal in momentum basis, so we transform the representation of the system wavefunction from position basis {x,y,z}to momentum basis {kx,ky,kz}by applying a QFT along each spatial dimension of the encoding in equation (11). This form permits efﬁcient calculation of the kinetic energy operator [20], which is described in appendix C. 4.1. Quantum variable rotation The phase rotation subroutine in the ﬁrst-quantized simulation algorithm imparts a quantum phase to each binary-encoded phase state in a superposition |θi=Pjcjφjstored in a quantum register (cjs are arbitrary complex amplitudes). Formally, it is the transformation X j cj|φji −X j e2πiξ φ jcj|φji,(18) which generalizes the operation in equation (16) using ξ, which is a scaling factor that varies with implementation, as explained below and in appendix C. Each 0 6φj<1 is a ﬁnite binary representation of a rotation on the unit circle encoded in a quantum register (the angle, in radians, divided by 2π). Equation (18) is the QVR, which is essential to ﬁrst-quantized simulation. We show how to implement this phase rotation subroutine using phase rotations from previous sections, as well as a new construction based on phase kickback. At the end of the section, we analyze the resource costs of these methods. To produce a QVR, various circuit manipulations are possible. The ﬁrst is to simply apply a single-qubit rotation to each qubit in register |θi, as shown in ﬁgure 14. Each individual rotation could be created using the techniques in section 2. Since a t-bit QVR requires t New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 22 q-1 q-2 q-3 q-4 (2 /2) Z R (2 /4) Z R (2 /8) Z R (2 /16) Z R . . . Figure 14. QVR decomposed into single-qubit rotations applied to each qubit in the |θiregister consisting of qqubits (see equation (18)). |θq1irefers to the most signiﬁcant bit in the register |θi, etc. separate bitwise rotations, we require that each rotation has accuracy /tto achieve accuracy in the QVR, where we have used the fact that the distance measure in equation (5) obeys the triangle inequality [33]. If the QVR is controlled by another qubit (e.g. if the propagator is controlled by a ‘simulated time’ qubit as in ﬁgure 1), then the gates in ﬁgure 14 are replaced with controlled rotations from section 3.1. In either case, one must know the quantity ξin advance to compile these gates; typically, ξis a product of physical constants and simulation parameters, as explained in appendix C. The QVR can also be produced in a more elegant manner using phase kickback. Rather than applying bitwise gates to the |θiregister, we instead use the entire register in a modiﬁed version of the phase kickback procedure. Firstly, we require a binary approximation to ξ, denoted [ξ]. Secondly, we deﬁne some quantities that describe this quantum circuit. Let mdenote the number of signiﬁcant bits in [ξ], minus the number of trailing zeros. Deﬁne w= blog2[ξ]c, or in other words, wis the largest integer such that 2w6[ξ]. Denote p=(m1)w, which is how many bits we must shift [ξ] up to produce an odd integer (if p<0, we shift down). Following equation (18) and the preceding text, let qbe the number of qubits in |θi. Deﬁne integers k[ξ]=(2p)[ξ] and uφ=(2qfor some arbitrary φ[0,1)represented using qbits. Thirdly, we construct a phase kickback ancilla register |γ(k[ξ])iof size n=p+qqubits, using techniques in appendix B. Finally, we perform phase kickback with an addition circuit between registers |θiand |γ(k[ξ])i(in-place addition applied to |γ(k[ξ])i), except this time the |θiregister is shifted in one of two ways, as shown in ﬁgure 15. If p>0, then the |θiregister is shifted down by pqubits, and the |θiregister is padded with plogical zeros at the most-signiﬁcant side of the adder input (ﬁgure 15(a)). If p<0, then |θiis shifted up by |p|qubits, so that the |p| most-signiﬁcant bits of |θiare not used in the adder (ﬁgure 15(b)). If n60, then all rotations are identity and no QVR circuit is constructed. We now conﬁrm that this procedure produces the intended QVR. Using equation (3), we see that the above procedure will implement a phase rotation of X j cjφjX j e2πik[ξ]uφj/2p+qcjφj.(19) Since k[ξ]=(2p)[ξ] and uφ=(2q, this is the same as X j cjφjX j e2πi[ξ]φjcjφj,(20) New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 23 p+q-1 q+1 q q-1 2 0 q-1 1 0 2 1 0 0 0 Adder Circuit [ ] (k ) tt n-1 n-2 n-3 3 2 3 1 0 2 1 0 Adder Circuit [ ] (k ) n-3 n-2 n-1 (a) (b) Figure 15. QVR using phase kickback. This circuit implements the operation in equation (18) with scaling factor [ξ], which has been ‘programmed’ into the phase kickback register |γ(k[ξ])i(see appendix B). A control qubit |tiis included for illustration. This ﬁgure shows how the bits in the adder are aligned for different cases. (a) The register |θiis shifted down pbits since p>0. θ0is the least-signiﬁcant bit in the |θiregister, etc. The input qubits above |θiare logical zeros. (b) The register |θiis shifted up |p|bits since p<0. In this case, the |p| most-signiﬁcant bits of |θiare not used in the adder. which is equivalent to equation (18) using our ﬁnite representation for ξ. As before, if we require a controlled-QVR, then the adder can be controlled by an external qubit, which is the conﬁguration shown in ﬁgure 15. This ‘quantum variable’ phase kickback uses substantially fewer Tgates than the bitwise approach, as shown in ﬁgure 16, while having comparable circuit depth. Moreover, since there is only one phase rotation instead of many, it does not have to be as accurate as the individual rotations in ﬁgure 14 must be to achieve the same total accuracy in the QVR. It may seem inefﬁcient to produce a different phase kickback register for each QVR operation, but three properties of the ﬁrst-quantized simulation algorithm make this approach efﬁcient. Firstly, there are only a polynomial number of such operations: for bparticles, there are bQVRs in the kinetic energy operator and 1 2b(b1)QVRs in the potential operator. Secondly, many of these QVRs have the same scaling factor ξ, so a phase kickback register can be reused many times without modiﬁcation. For example, the scaling factor in the kinetic energy operator is the same for all electrons and for all nuclei with the same mass. Third, the |γ(k[ξ])iregisters can be calculated independently of other operations in the algorithm, so the impact of this process on circuit depth is minimal. This phase kickback QVR has interesting applications to other useful quantum circuits. It can be used to make a fault-tolerant QFT; one replaces each block of controlled rotations with a controlled-QVR. As before, this approach uses substantially fewer Tgates than an equivalent circuit where each controlled rotation in the QFT is implemented individually with techniques New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 24 102 103 104 T gates 10−6 10−5 10−4 10−3 10−2 10−1 Approximation error () PAR (SingleQubit Bitwise) Phase Kickback (Single−Qubit Bitwise) Fowler (Single−Qubit Bitwise) Phase Kickback (Quantum Variable) Figure 16. Number of Tgates required to produce a QVR with various methods, assuming ξ=1 and number of signiﬁcant ﬁgures is chosen to satisfy the approximation error . The special-purpose ‘quantum variable’ phase kickback clearly requires the least circuit effort, and the asymptotic scaling of Tgates is linear in log for this approach and super-quadratic for the others. The circuit depth for Fowler or phase kickback approaches is equivalent to the comparable single-qubit rotation; however, the PAR must succeed across all individual rotations for this circuit to succeed, so the mean circuit depth increases slightly. In the above, ten rounds of PAR ancilla are pre-computed for each single-qubit rotation in the QVR. in section 3.1, and the same methods can be applied to an approximate QFT [67] by simply truncating the size of the |γ(1)iregister. The phase kickback QVR can also be used to efﬁciently produce ancillas for PAR if the particular rotation RZ(φ) is required frequently, which can have applications to second-quantized simulation. If we denote the state |+i=1 2(|0i+|1i), then an input state of |+i|+i|+i. . . will be transformed using QVR (with appropriate ξ) into the set of ancillas for PAR, but requiring only one addition circuit for the entire set instead of a phase kickback addition or Fowler sequence for each ancilla qubit, which can be seen by comparing ﬁgure 14 with the ancilla preparation in ﬁgure 4. Creating the necessary |γ(k[ξ])ifor this process is costly, so there is a net gain only if a certain rotation angle φis required often. 4.2. Improved parallelism in potential energy operator The majority of the circuit effort in ﬁrst-quantized simulation is devoted to calculating the potential energy [20]. We introduce here a technique to substantially speed up the calculation of the potential energy operator ˆ V, which is simply the sum of the Coulomb interactions ˆ Vi j =qiqj 4π0ri j between all pairwise combinations of the electrons and nuclei. Note that this operator is a function of the positions riof the system particles only, so it is diagonal in the New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 25 position basis |r1r2. . . rbi. This fact means that all terms ˆ Vi j commute with each other, so they may be calculated in any order. Moreover, there are many sets of the ˆ Vi j operators that are disjoint, which means that each particle in the system is acted on by just one operator in the set. Using this observation, for example, we may calculate the Coulomb interaction ˆ V12 between particles 1 and 2 at the same time as ˆ V34 between particles 3 and 4, and so on. In general, for a system of bparticles, there are 1 2b(b1)pairwise interactions, and we can perform bb 2cpairs in parallel, which means that a potential energy operator with O(b2)terms can be calculated in O(b)time. This parallelism can increase the speed of simulation signiﬁcantly since evaluation of the potential energy dominates resource costs [43]. The potential operator calculation can be further parallelized to achieve O(log b)or O(1) (constant) circuit depth. Exploiting the fact that all ˆ Vi j are diagonal in position basis (and hence commute), we use transversal CNOT gates to copy the data in the position-basis particle wavefunction onto multiple empty quantum registers. For a single particle, this process is 2p1 X x,y,z=0 c(x,y,z)|xi|yi|zi |000 . . .i|000 . . .i. . . 2p1 X x,y,z=0 c(x,y,z)(|xi|yi|zi) (|xi |yi|zi) (|xi|yi|zi). . . . (21) For bparticles, the copy operation is performed b2 times (for b1 total copies), which can be fanned out using a binary tree with depth dlog2(b1)e; constant depth can be achieved in some quantum computer architectures which support one-control/many-target CNOTs [43,57] or in general architectures using a teleportation circuit similar to those described in section 3.3. This approach is similar to that employed in [47] to produce a parallel circuit for the QFT. The system wavefunction is now expanded to the state ψexpand=X r1,...,rb c(r1,...,rb)(|r1i)(b1). . . (|rbi)(b1),(22) which requires O(b2)memory space. Note that this process is not cloning—the position- basis particle registers are still entangled to one another. With multiple accessible copies of each particle’s position-basis information, the particles are matched in all b(b1)possible pairings, and the potential energy operator applied to each pairing in parallel, which can be accomplished in constant time, but still requires O(b2)circuit effort. After each of the potential energy operators ˆ Vi j kicks back a phase, the excess copies of each particle wavefunction are uncomputed by reversing the tree of CNOTs above. The preceding example demonstrates that it is possible to calculate ˆ Vin time which is sub-linear in the number of particles, even if each ˆ Vi j is treated as a black box operator. In practice, more efﬁcient circuits can be produced by generating the internal ‘workspace’ registers of ˆ Vin parallel, rather than making copies of the input registers Pr1,...,rbc(r1,...,rb)|r1. . . rbi(see appendix C). 4.3. Resource analysis for ﬁrst-quantized molecular simulations The advantage of using the ﬁrst-quantized approach is that the approximation errors of the simulation are systematically improvable by increasing the spatial precision of the wavefunction New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 26 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 109 Number of p articles Circuit depth (gates) In-place potential calculations Fully-parallel potential calculations Figure 17. Circuit depth for two instances of ﬁrst-quantized simulation. The in-place calculation of potential energy computes each pairwise Coulomb interaction in sets of non-overlapping particle pairs, and both the depth and number of qubits required increase linearly with the number of particles. The fully parallel calculation creates many copies of the wavefunction to permit the potential energy to be determined in constant time, at the expense of requiring substantially more application qubits (quadratic in the number of particles). In both cases, the wavefunction precision along any spatial dimension is ten qubits, and the simulation uses 1023 time steps for 10 bits of precision, or 3 signiﬁcant ﬁgures. and the temporal precision of the time steps. However, calculating kinetic and potential energy interactions requires quantum arithmetic circuits and phase rotations, which together require substantial resources in terms of fault-tolerant gates and qubits. Figure 17 shows two versions of ﬁrst-quantized simulation using the techniques for parallel calculation of potential energy from the previous section. Although constant-depth evaluation of the Hamiltonian is possible, it requires a signiﬁcantly larger quantum computer to achieve the parallel calculations, so this implementation is probably best suited to large-scale quantum computers. Examining ﬁgure 17, note that the circuit depth at six particles (e.g. LiH) is comparable to that of the equivalent PAR-based second-quantized simulation in ﬁgure 12 while requiring many more qubits, indicating that ﬁrst-quantized simulation is more appropriate for larger molecules than LiH, since the circuit depth for ﬁrst-quantized simulation is asymptotically less than second-quantized as particle number is increased [19]. Moreover, these calculations have assumed that the spatial precision is ten qubits for any molecules with 2–20 particles. As the size of the molecule increases, the number of qubits for each dimension of the encoded wavefunction will have to increase as the molecule itself is spatially larger. One may also choose to increase spatial resolution to achieve a higher-precision simulation. Each of the methods we propose for improving ﬁrst-quantized simulation are summarized in table 4. Appendix Aexplains how the resources were calculated. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 27 Table 4. Summary of methods for efﬁcient ﬁrst-quantized chemical simulation. The quantity bis the number of particles in the chemical problem, which inﬂuences algorithm resource costs. Method Description Advantages Disadvantages QVR Use phase Reduces Not the minimal kickback to apply a complexity of depth achievable, fault-tolerant ﬁrst-quantized such as with PARs phase rotation simulation. to each element Circuit depth is in a superposition, essentially the proportional same as single-qubit to the binary-encoded phase kickback, value of that element but the QVR requires substantially fewer Tgates than the method in ﬁgure 14 Parallel Reduce potential Shorter circuit Concurrent evaluation operator circuit depth than computation of potential depth using calculating all requires more Tgates energy terms parallel computation 1 2b(b1)terms simultaneously individually Teleportation Use a Potential Circuit size in circuit expansion teleportation operator can be qubits increases for potential operator circuit to evaluated in a to O(b2)from O(b) ‘control-copy’ time which is position-basis independent of wavefunction in problem size constant time 5. Comparing simulation methods The prior sections illustrate that there exist numerous ways to simulate a molecular Hamiltonian, including choices between encoded representation in a quantum computer and the way fault- tolerant rotations are prepared. The ﬁnal result one desires to know is, which method is best? Determining an optimal approach is subjective to the quantum computing resources available, so in this section we describe how to make such a decision. To visually compare different implementations of a simulation algorithm, we plot the efﬁcient frontier for each method in a plane deﬁned by machine size (qubits) on the x-axis and execution time (circuit depth) on the y-axis. The efﬁcient frontier is the set of all points (size, depth) such that for each achievable machine size, the (achievable) depth is minimized, and vice versa. As an example, ﬁgure 18 shows the efﬁcient frontiers of various implementations of a LiH simulation. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 28 101102103 107 108 109 1010 1011 1012 Second-Quantized (Solovay-Kitaev) Second-Quantized (Phase-Kickback) Second-Quantized (Fowler) Second-Quantized (PAR) First-Quantized (Phase-Kickback QVR) Quantum computer size (application qubits) Algorithm runtime (circuit depth in gates) Figure 18. The efﬁcient frontiers for various implementations of simulating LiH ground state energy on a quantum computer. Each star data point corresponds to the equivalent method in ﬁgure 12, at rotation accuracy max 6104; similarly, ﬁrst-quantized simulations use QVRs with the same accuracy. The PAR frontier (purple) and ﬁrst-quantized frontier (brown) have adjustable parameters that reduce circuit depth through parallel computation at the expense of increased system size (application qubits). For example, the PAR-based algorithm only achieves the circuit depth shown in ﬁgure 12 when the system has 68 qubits, which is the yellow star here. To determine the optimal implementation, one speciﬁes a cost function g(x,y), which associates with any point (x,y)a ‘cost’ to implement simulation using these parameters. For example, cost could be the estimated engineering challenge to produce a quantum computer of size xqubits combined with a penalty for the execution time of ygates, which is a measure of performance. Minimizing the cost function along each efﬁcient frontier gives the optimal set of parameters for that particular method, and minimizing over all efﬁcient frontiers gives the best implementation that is known to be achievable. For the various implementations for a LiH simulation in ﬁgure 18, it seems likely that one would choose between the compact algorithm with Fowler gate sequences or the faster version with PAR sequences, which requires additional qubits to compute the necessary ancillas. First- quantized can potentially deliver the fastest execution time here, but for this problem the number of qubits required is substantially greater. Still, ﬁrst-quantized gains an appreciable performance advantage if the number of particles is increased or if one moves to simulating time-varying dynamics [19]. Naturally, future algorithm advancements could produce new frontiers that are more desirable for a given cost function. In general, one would like to make such comparisons, which can inform design decisions for quantum hardware, with full consideration of the cost to implement error correction, produce non-Clifford group gates (e.g. Tgates), and so forth. Future work building on this investigation and prior efforts, such as [6,43,54,68], should include such comprehensive system analysis. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 29 6. Conclusions This paper examines the methods required to simulate chemistry on a fault-tolerant quantum computer. A crucial operation in these algorithms is the production of phase rotations, and several approaches—phase kickback, gate approximation sequences, PARs, and QVRs—are analyzed. First, it should be clear that sequences generated by the Solovay–Kitaev algorithm are not nearly as efﬁcient as the alternatives, phase kickback and Fowler sequences. Fowler sequences are the shortest for a fault-tolerant single-qubit rotation, but the classical computing effort required to determine such sequences becomes intractable for high-precision (e.g.  < 106) rotations. Recently, Bocharov and Svore have presented an alternative algorithm for ﬁnding depth-optimal gate sequences which might feasibly provide high-precision rotations [69]. Phase kickback is a versatile technique that produces rotations comparable to Fowler’s algorithm in resource usage, with the former having circuit depth O(log ) or O(log log ) gates and requiring O(log ) Tgates. Furthermore, the underlying circuit for phase kickback is an adder, which can be determined using efﬁcient classical algorithms (unlike Fowler’s algorithm), and phase kickback can be extended more readily to QVRs. The PAR allows the quantum algorithm to achieve exceptionally low-circuit-depth rotations, at the expense of computing ancillas in advance (which is less efﬁcient in terms of Tgates). Finally, the QVR is particularly useful for ﬁrst-quantized simulation. The relative merits of the methods for producing phase rotations are compared in table 2. This investigation also examined two variants of the simulation algorithm, second- quantized and ﬁrst-quantized, whose primary difference is the way wavefunctions are encoded and operated upon. Generally speaking, second-quantized is a more compact representation, requiring fewer qubits, but it requires asymptotically longer execution times than ﬁrst- quantized, measured in circuit depth, as the problem size increases in terms of independent particles to simulate. Our results provide a more nuanced way to compare these methods by explicitly considering the possible ways to make the algorithms compatible with fault- tolerant quantum computing and the resulting resource costs incurred. We have also introduced several improvements to the simulation algorithms. In the second-quantized approach, one can neglect some of the integral terms smaller in magnitude than a cutoff threshold, implement the Jordan–Wigner transform in constant time, and use PARs to substantially reduce circuit depth, at the expense of requiring parallel production of the pre-computed PAR ancillas. In ﬁrst-quantized, we demonstrated how to produce QVRs with arbitrary scaling factor, as well as how to parallelize the calculation of the potential energy to time linear in system size (without increase in qubits) or to constant time (requiring a number of qubits that grows quadratically instead of linearly with the number of particles simulated). The methods we present for efﬁcient chemical simulation on quantum computers are summarized in tables 3and 4. Although we have focused on simulating quantum chemistry, these methods can be extended to simulating other Hamiltonians on quantum computers, such as spin lattice models [4], lattice gas automata [70] and lattice gauge theories [71], or quantum chaos theories [72]. Moreover, the fault-tolerant rotations could ﬁnd application in other quantum algorithms, including any which require a Fourier transform. This investigation provides a ﬂexible set of methods for making simulation algorithms practically realizable on fault-tolerant quantum computers. New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 30 Acknowledgments The authors would like to thank Kevin Obenland for suggesting improvements to quantum circuits; Paul Pham for providing assistance with Solovay–Kitaev code; Aram Harrow and Isaac Chuang for helpful discussions on Solovay–Kitaev; and Austin Fowler for providing code for minimum-length approximation sequences. This work was supported by the National Science Foundation CCF-0829694, the University of Tokyo Special Coordination Funds for Promoting Science and Technology, NICT, and the Japan Society for the Promotion of Science (JSPS) through its ‘Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program).’ NCJ was supported by the National Science Foundation Graduate Fellowship. JDW acknowledges support from National Science Foundation award number 1017244. MHY and AAG acknowledge support from the Air Force Ofﬁce of Scientiﬁc Research, award number FA8721-05-C-0002. AAG acknowledges support from the Alfred P Sloan Foundation, and the Camille and Henry Dreyfus Foundation. Appendix A. Methods for calculating resources We brieﬂy describe how the resources for quantum circuits were calculated. Beginning with single-qubit phase rotations, the Solovay–Kitaev algorithm is described in [32], while the Fowler algorithm is described in [33]. Gate sequences for rotations were generated using these algorithms, and the total circuit depth and number of Tgates were recorded. Phase kickback implements an adder circuit, and we select the CDKM adder (named for the authors of [52]). For reference, an n-qubit addition for phase kickback requires the n-qubit ancilla state |γ(k)i and nancilla qubits. The latter ancilla qubits (representing an input to the adder) are initialized to the binary representation of the rotation angle, conditioned on the qubit to which the rotation is applied, which is accomplished using 2nCNOT gates that can be implemented in constant time using the methods in section 3.3. The depth of the adder is 2n3 Toffoli gates and 5 CNOT gates. Our implementation of the Toffoli gate uses 11 time steps, but recent work shows many variations are possible [69]. The depth of our n-bit adder-based phase-kickback rotation is 22n26, which includes the parallel CNOT gates for setting the ancilla addend. Each Toffoli gate requires 7 Tgates, so the CDKM adder has 14n21 Tgates in total. Controlled phase rotations are built using single-qubit rotations. The Fowler and Solovay– Kitaev sequences require effectively two single-qubit rotations (see section 3.1); the rotations are inverses of each other, so the circuits for the two rotations are mirror images, and the total resources (depth and Tgates) are doubled. Phase kickback requires only the slightest modiﬁcation to become a controlled rotation. The nancilla states are initialized to rotation angle conditioned on both the control and target qubit for this rotation. This requires two Toffolis (one at the start and one at the end) and 2n2 additional CNOT gates (which can be implemented in a parallel fashion). The second-quantized algorithm is implemented using the methods described in [14] and each of these types for controlled rotations. The depth and Tgates are given by the number of integral terms times the controlled phase rotation resource requirements, plus circuits for each of the necessary Jordan–Wigner transforms. The ﬁrst-quantized simulation algorithm is implemented using arithmetic circuits and QFTs. Resource analysis for exact and approximate QFTs is given in [30,47,48]. Detailed explanation of the arithmetic operations for this algorithm is given in [20] and appendix C. Each arithmetic circuit is decomposed into CDKM adders. The phase rotations in the kinetic New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 31 operators, potential operators, and QFT are implemented using phase-kickback QVRs, each of which requires a single CDKM adder. Appendix B. Transforming the phase kickback register In some situations it is useful to change the k-value in |γ(k)i, the phase kickback ancilla register (see equation (2)). Without control over k, the QVR in section 4.1 would require solving equation (4) in a quantum circuit; this step would in turn require a multiplication operation, which can be expensive in terms of quantum gates. We deviate here from [46] and propose a simple way to avoid the expensive operations associated with modular multiplication. The speciﬁc ancilla state |γ(1)idoes not require an additional circuit to solve equation (4), so we create this state explicitly using a simple transform |γ(k)i→|γ(1)i. We begin by factoring the |γ(k)iregister into individual qubits (note that all such states are separable, i.e. not entangled): |γ(k)i = 1 N N1 X y=0 e2πiky /N|yi =1 N|0i+ e2πik/2|1i|0i+ e2πik/4|1i·· · |0i+ e2πik/2n|1i.(B.1) We convert this state into |γ(1)iwith a series of single-qubit phase rotations using the controlled addition circuit from ﬁgure 2. Since kis odd, the ﬁrst bit of our ancilla register is always 1 2(|0i|1i). The next bit must be rotated by the phase gate RZ (k1)), which is either identity or Z, depending on k. In general, the corrective gate applied to the mth bit is the phase rotation RZ(2πk1 2m1), which may be produced using the preceding m1 bits of the ancilla register and phase kickback. By iterating through all qubits in the register, we complete the transformation with circuit depth O(n2)gates or less, depending on the type of adder used in phase kickback. This procedure can be generalized to any transformation |γ(k)i→|γ(l)ifor odd integers 1 6k,l<2n, where nis the number of bits in the phase kickback register. Appendix C. Quantum circuits for potential and kinetic energy operators in ﬁrst-quantized molecular Hamiltonians First-quantized molecular simulation represents the simulated system wavefunction on a Cartesian grid, and the Hamiltonian is calculated with digital arithmetic acting on this coordinate space. Similar methods were discussed in the supplementary material of [20], but we update this analysis for the QVR introduced in this work. The potential energy operator is diagonal in position basis, and is the sum of Coulomb interactions between electrons and nuclei in the system ˆ V=1 2Pi6=jˆ Vi j , where ˆ Vi j =qiqj 4πε01 |rirj|(C.1) and qjis the charge of particle j. The prefactor on the rhs of equation (C.1) is a constant for any given pair of particles, and we can later encode this scaling factor into the QVR. What remains is to calculate 1 |rirj|over the position-encoded wavefunction. Each position register New Journal of Physics 14 (2012) 115023 (http://www.njp.org/) 32 can be decomposed in Cartesian components |ri=|xi|yi|zi, so for a pair of particles we calculate ri j 2=xixj2+yiyj2+zizj2E.(C.2) The required multiplication operations can be implemented using quantum adder circuits. Next, the quantity |1 ri j iis calculated using the Newton–Raphson method with the iterative equation an+1 =1 2an3an2ri j 2.(C.3) With a suitably chosen initial value a0, equation (C.3) converges within ﬁve iterations at 32-bit arithmetic, and typically less precision is required for simulation. The register |1 ri j iis used in a QVR with scaling factor ξ=qiqjδt 8π2ε0¯ hfrom above, where δtis the time step of this simulated evolution and an additional factor 1/2πcomes from equation (18). Note that each component of |1 ri j iis entangled to a position-basis component of the system wavefunction, so the QVR effectively kicks back a phase to the wavefunction. Each of the steps prior to the QVR is uncomputed, and the net effect of this sequence of operations is to implement the potential energy propagator ei¯ h1ˆ Vij δt, as in equations (15)–(17). The kinetic energy operator is calculated using a similar approach as the potential energy. 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It is important to understand that such algorithms that can be realized on a fault-tolerant quantum computers are needed (Jones et al. 2012).The most important point to be considered in quantum simulation of molecular systems is that exact first-principles calculations of molecular properties are presently not solvable. The reason has already been explained in terms of the increase in the computational cost exponentially with basis set size and the number of atoms. ... Article Full-text available The ideas of quantum simulation and advances in quantum algorithms to solve quantum chemistry problems have been discussed. Theoretical proposals and experimental investigations both have been studied to gauge the extent to which quantum computation has been applied to solve quantum chemical problems till date. The distinctive features and limitations of the application of quantum simulation on chemical systems and current approaches to define and improve upon standard quantum algorithms have been studied in detail. The possibility and consequences of designing an efficient quantum computer that can address chemical problems have been assessed. The experimental realization of quantum supremacy defies the conventional belief of chemists, that millions of qubits would be required to solve fundamental chemistry problems. It is predicted that quantum simulation of quantum chemistry problems will radically revolutionize this field. ... [36,[39][40][41][42] attain the optimal gate sequence length L scaling with O (log(1/ε)) under the Clifford +T basis set when the target unitary is specified to be a rotational single-qubit gate along the z-axis. Similarly, other quantum compilers [43][44][45][46][47][48][49] approach the optimal scaling ratio for particular rotational quantum gates by exploiting the additional resources such as ancillary qubits, special states, and classical feedback. The second category contains the optimization-based quantum compilers, which have good universal properties but lack a theoretical guarantee. ... ... Optimality in theory [32] --O C log 1.000 (1/ε) Brute force [35] Fibonacci anyons Single-qubit O 2.24 log 1. 43 (1/ε) Inverse-closed S-K theorem [35] Fibonacci anyons Single-qubit O log 5. 18 (1/ε) Inverse-closed S-K theorem [36] -Single-qubit O log 3 (1/ε) Inverse-free S-K theorem [34] -Single-qubit O log 8. 62 (1/ε) Inverse-free algorithm [37] Inverse-free diffusive set Single-qubit O log 1.585 (1/ε) Our RL-based compiler Inverse-free diffusive set Single-qubit O 2.683 log 0.9735 (1/ε) Other RL-based compiler [35] Fibonacci anyons Single-qubit O 1. 55 including discrete parameters (e.g., quantum circuit layouts) and continuous parameters (e.g., the angles in rotational quantum gates). However, these optimizationbased compilers fail to show any prominent advantage in the inference time compared with those deterministic quantum compilers. ... Preprint Full-text available Efficient quantum compiling tactics greatly enhance the capability of quantum computers to execute complicated quantum algorithms. Due to its fundamental importance, a plethora of quantum compilers has been designed in past years. However, there are several caveats to current protocols, which are low optimality, high inference time, limited scalability, and lack of universality. To compensate for these defects, here we devise an efficient and practical quantum compiler assisted by advanced deep reinforcement learning (RL) techniques, i.e., data generation, deep Q-learning, and AQ* search. In this way, our protocol is compatible with various quantum machines and can be used to compile multi-qubit operators. We systematically evaluate the performance of our proposal in compiling quantum operators with both inverse-closed and inverse-free universal basis sets. In the task of single-qubit operator compiling, our proposal outperforms other RL-based quantum compilers in the measure of compiling sequence length and inference time. Meanwhile, the output solution is near-optimal, guaranteed by the Solovay-Kitaev theorem. Notably, for the inverse-free universal basis set, the achieved sequence length complexity is comparable with the inverse-based setting and dramatically advances previous methods. These empirical results contribute to improving the inverse-free Solovay-Kitaev theorem. In addition, for the first time, we demonstrate how to leverage RL-based quantum compilers to accomplish two-qubit operator compiling. The achieved results open an avenue for integrating RL with quantum compiling to unify efficiency and practicality and thus facilitate the exploration of quantum advantages. ... Quantum computing has shown immense promise for accelerating chemistry simulation [15], prime factorization [44], database search [12], and machine learning [8]. ... Preprint Despite rapid advances in quantum computing technologies, the qubit connectivity limitation remains to be a critical challenge. Both near-term NISQ quantum computers and relatively long-term scalable quantum architectures do not offer full connectivity. As a result, quantum circuits may not be directly executed on quantum hardware, and a quantum compiler needs to perform qubit routing to make the circuit compatible with the device layout. During the qubit routing step, the compiler inserts SWAP gates and performs circuit transformations. Given the connectivity topology of the target hardware, there are typically multiple qubit routing candidates. The state-of-the-art compilers use a cost function to evaluate the number of SWAP gates for different routes and then select the one with the minimum number of SWAP gates. After qubit routing, the quantum compiler performs gate optimizations upon the circuit with the newly inserted SWAP gates. In this paper, we observe that the aforementioned qubit routing is not optimal, and qubit routing should \textit{not} be independent on subsequent gate optimizations. We find that with the consideration of gate optimizations, not all of the SWAP gates have the same basis-gate cost. These insights lead to the development of our qubit routing algorithm, NASSC (Not All Swaps have the Same Cost). NASSC is the first algorithm that considers the subsequent optimizations during the routing step. Our optimization-aware qubit routing leads to better routing decisions and benefits subsequent optimizations. We also propose a new optimization-aware decomposition for the inserted SWAP gates. Our experiments show that the routing overhead compiled with our routing algorithm is reduced by up to69.30\%$($21.30\%$on average) in the number of CNOT gates and up to$43.50\%$($7.61\%$on average) in the circuit depth compared with the state-of-the-art scheme, SABRE. ... Fault-tolerant quantum computers promise to solve some computational problems much faster than classical machines, such as quantum chemistry simulation [1], prime factorization [2], solving linear systems of equations [3]. However, quantum information carried by current noisy intermediate-scale quantum (NISQ) systems is highly fragile and can be easily altered by the environment. ... Preprint Quantum error-correcting codes (QECCs) are believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various methods of QECC constructions have been developed, leading to many good families of codes. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g.,$((n,2^{n-6},3))_2$for$n$from 7 to 14. We also found new$((6,2,3))_2$and$((7,2,3))_2$codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a$((7,3,3))_2\$ code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.