Hybrid quantum-classical algorithms and quantum error mitigation

Suguru Endo,1, ∗ Zhenyu Cai,2 Simon C. Benjamin,2 and Xiao Yuan3, †

1NTT Secure Platform Laboratories, NTT Corporation, Musashino 180-8585, Japan2Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom3Stanford Institute for Theoretical Physics, Stanford University, Stanford California 94305, USA

Quantum computers can exploit a Hilbert space whose dimension increases exponentially withthe number of qubits. In experiment, quantum supremacy has recently been achieved by the Googleteam by using a noisy intermediate-scale quantum (NISQ) device with over 50 qubits. However, thequestion of what can be implemented on NISQ devices is still not fully explored, and discoveringuseful tasks for such devices is a topic of considerable interest. Hybrid quantum-classical algorithmsare regarded as well-suited for execution on NISQ devices by combining quantum computers withclassical computers, and are expected to be the first useful applications for quantum computing.Meanwhile, mitigation of errors on quantum processors is also crucial to obtain reliable results. Inthis article, we review the basic results for hybrid quantum-classical algorithms and quantum errormitigation techniques. Since quantum computing with NISQ devices is an actively developing field,we expect this review to be a useful basis for future studies.

CONTENTS

I. Introduction 2

II. Basic variational quantum algorithms 2A. Variational quantum eigensolver 3B. Real and imaginary time evolution quantum

simulator 5

III. Variational quantum optimisation 7A. Quantum approximate optimisation

algorithm 7B. Variational algorithms for machine learning 8

1. Quantum circuit learning 82. Data-driven quantum circuit learning 93. Quantum generative adversarial networks 94. Quantum autoencoder for quantum data

compression 105. Variational quantum state eigensolver 11

C. Variational algorithm for linear algebra 11D. Excited state-search variational algorithms 12

1. Overlap-based method 122. Quantum subspace expansion 133. Contraction VQE methods 134. Calculation of Green’s function 14

E. Variational circuit recompilation 14F. Variational-state quantum metrology 15G. Variational quantum algorithms for quantum

error correction 161. Variational circuit compiler for quantum

error correction 162. Variational quantum error corrector

(QVECTOR) 17H. Dissipative-system variational quantum

eigensolver 17

∗ suguru.endou.uc@hco.ntt.co.jp† xiao.yuan.ph@gmail.com

I. Other applications 18

IV. Variational quantum simulation 18A. Variational quantum simulation algorithm for

density matrix 181. Variational real time simulation for open

quantum system dynamics 182. Variational imaginary time simulation for a

density matrix 18B. Variational quantum simulation algorithms

for general processes 191. Generalised time evolution 192. Matrix multiplication and linear

equations. 193. Open system dynamics 19

C. Gibbs state preparation 20D. Variational quantum simulation algorithm for

Green’s function 20E. Other applications 21

V. Quantum error mitigation 21A. Extrapolation 21

1. Richardson extrapolation 212. Exponential extrapolation 223. Methods to boost physical errors 234. Mitigation of algorithmic errors 24

B. Least square fitting for several noiseparameters 24

C. Quasi-probability method 24D. Quantum subspace expansion 26E. Symmetry verification 27F. Individual error reduction 27G. Measurement error mitigation 28H. Learning-based quantum error mitigation 28

1. Quantum error mitigation via CliffordData Regression 28

2. Learning-based quasi-probability method 29I. Stochastic error mitigation 30J. Combination of error mitigation techniques 30

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1. Symmetry verification with errorextrapolation 31

2. Quasi-probability method with errorextrapolation 31

3. Symmetry verification withquasi-probability method 31

4. Combining quasi-probability, symmetryverification and error extrapolation 32

VI. Conclusion 32

Acknowledgments 32

A. Derivation of Eq. (11) for variational quantumsimulation 33

B. Hadamard test and quantum circuits forvariational quantum simulation 33

C. SWAP test and Destructive SWAP test 34

D. Methodologies for optimisation 341. Local cost function 342. Hamiltonian morphing optimisation 35

E. Subspace expansion 35

References 36

I. INTRODUCTION

As the size of the Hilbert space of a quantum systemincreases exponentially with respect to the system size,general quantum systems are, in principle, hard to sim-ulate on a classical computer. For example, systems ma-nipulating tens to hundreds of qubits have been believedto be classically intractable, and they have been pro-posed for demonstrating quantum advantages over clas-sical supercomputers in the so-called task of ‘quantumsupremacy’ [1]. In October 2019, Google announced thatthey had successfully demonstrated quantum supremacywith a high-fidelity 53 qubit device, named Sycamore [2].The dimension of the computational state-space is aslarge as 253 ≈ 9.0 × 1015. To sample one output of aquantum circuit on the 53 qubits, it is estimated that aclassical supercomputer would need 10000 years, whilstSycamore only took 200 seconds. Although recent ef-forts have significantly reduced the classical simulationcost [3], classical simulation of a general quantum circuitwill certainly become an intractable task as we increasethe gate fidelity, the gate depth, or the number of qubits.

While the tasks considered in quantum supremacy aregenerally mathematically abstract problems, ultimatelythe field must progress to demonstrate true quantum ad-vantage i.e. to solve a problem of practical value with su-perior efficiency using a quantum device. Current quan-tum hardware only incorporates a small number (tens)of qubits with a non-negligible gate error rate, making it

insufficient for implementing conventional quantum algo-rithms such as Shor’s factoring algorithm [4], the phaseestimation algorithm [5], and Hamiltonian simulation al-gorithms [6]. These generally require one to accuratelycontrol millions of qubits when taking account of fault-tolerance [7].

Before realising a universal fault-tolerant quantumcomputer, a more feasible scenario for current andnear-term quantum computing is the so-called noisyintermediate-scale quantum (NISQ) regime [8], where wecontrol tens to thousands of noisy qubits with gate errorsthat may be on the order of 10−3 or lower. AlthoughNISQ computers are not universal, we may exploit themto solve certain computational tasks, such as chemistrysimulation, significantly faster than classical computers,via a combination of quantum and classical computers [9–14]. Intuitively, because a large portion of the computa-tional burden is processed on the classical computer, fullycoherent deep quantum circuits may not be required.As both quantum and classical computers are used, suchsimulation methods are called hybrid quantum-classicalalgorithms. In addition, to compensate computation er-rors, quantum error mitigation techniques can be used bya post-processing of the experiment data. Since quantumerror mitigation does not necessitate encoding of qubitsas full error correction does, it thus contributes to a hugesaving of qubits, which is vital for NISQ simulation.

In this review paper, we aim to summarise the mostbasic ideas of hybrid quantum-classical algorithms andquantum error mitigation techniques. In Sec. II, we in-troduce the basic algorithms — the variational quantumeigensolver and variational quantum simulation — forfinding a ground state or simulating dynamical evolu-tion of a many-body Hamiltonian. In Sec. III, we showhow the variational quantum eigensolver algorithm canbe extended for general optimisation problems includingmachine learning problems, linear algebra problems, ex-cited energy spectra, etc [15–22]. Meanwhile, we showin Sec. IV that the variational quantum simulation algo-rithm may be extended as well for open systems, generalprocesses, thermal states, and calculating Green’s func-tion. Finally, in Sec. V, we show several error mitigationmethods for suppressing errors in NISQ computing. Thisreview does not cover the application of NISQ computersin solving specific physics problems and we refer to McAr-dle et al. [23] and Cao et al. [24] for reviews for its ap-plication in quantum computational chemistry, to Baueret al. [25] in quantum materials, etc.

II. BASIC VARIATIONAL QUANTUMALGORITHMS

Since NISQ devices can only apply a relatively shal-low circuit on a limited number of qubits, conventionalquantum algorithms may not be implemented on NISQdevices. Here we consider hybrid quantum-classical al-gorithms tailored to NISQ computing. Because the algo-

3

rithms generally use parametrised quantum circuits andvariationally update the parameters, they are also calledvariational quantum algorithms (VQAs).

For implementing VQAs [9, 11, 12], we first considerthe parametrised trial wave function as

|ϕ(~θ)〉 = U(~θ) |ϕref 〉 , (1)

where U(~θ) = UN (θN ) . . . Uk(θk) . . . U1(θ1) generally

consists of single or two qubit gates, and ~θ =(θ1, θ2, . . . ., θN )

T is a vector of independent real param-eters, and |ϕref 〉 is the initial state. Typically, whenwe have Nq-qubit quantum processor, we can choose

|ϕref 〉 = |0〉⊗Nq or any initial state from a classical com-putation. Here, we refer to |ϕ(~θ)〉 as the ansatz state, andU(~θ) as the ansatz circuit. The routine of a variationalquantum algorithm typically works by preparing the trialstate, measuring the state, and updating the parametersaccording to a classical algorithm on the measurement re-sults. To circumvent the accumulation of physical errors,

we generally assume that the ansatz U(~θ) is implementedwith a shallow quantum circuit. The schematic figure ofVQAs is shown in Fig. 1.

Although there exist a large number of VQAs, theycan be generally classified into two categories: variationalquantum optimisation (VQO) and variational quantumsimulation (VQS). VQO involves optimising parametersunder a cost function. For example, when we minimisethe energy of the state, i.e. the expectation value of thegiven Hamiltonian as a cost function, the cost functionafter optimisation approximates the ground state energy.The corresponding state also approximates the groundstate. This is the so-called variational quantum eigen-solver [9, 12] and other VQO algorithms can be similarlydesigned by properly changing the cost function to othermetrics. While variational quantum optimisation aims tooptimise a static target cost function, VQS aims to sim-

FIG. 1. Schematic of varational quantum algorithms.

The ansatz state |ψ(~θ)〉 is generated via a short-depthparametrised quantum circuit and measured to extract clas-sical data. The measurement result is fed to a classical com-puter to update the parameters.

ulate a dynamical process, such as the Schrödinger timeevolution of a quantum state [14, 26]. VQS algorithmscan also be applied for optimising a static cost func-tion, such as variational imaginary time simulation, orstudying general many-body physics problems. The dis-tinction between variational quantum optimisation andvariational quantum simulation is not absolute, and al-gorithms for problems in one category may be adaptedfor those in the other category. Before showing how spe-cific VQO or VQS algorithms work for specific tasks, inthis section, we first illustrate the most basic VQO al-gorithm, a variational quantum eigensolver for findingground state energy, and the most basic VQS algorithmsfor simulating real and imaginary time simulation.

A. Variational quantum eigensolver

The variational quantum eigensolver (VQE) is a hybridquantum-classical algorithm for computing the groundstate and the ground state energy of a Hamiltonian Hof interest. In the seminal work by Peruzzo et al. [9],the VQE algorithm was theoretically proposed and ex-perimentally demonstrated for finding the ground stateenergy of the HeH+ molecule using a two-qubit photonicquantum processor. We note that the conventional ap-proach for finding eigenstates of a Hamiltonian with auniversal quantum computer is by adiabatic state prepa-ration and quantum phase estimation (QPE) [27]. Werefer to McArdle et al. [23] and Cao et al. [24] for thereview.

The VQE algorithm relies on the Rayleigh-Ritz varia-tional principle, i.e., for any parameterised quantum state

|ϕ(~θ)〉, we have

min~θ〈ϕ(~θ)|H |ϕ(~θ)〉 ≥ EG, (2)

where EG is the ground state energy of the Hamiltonian

H and the minimisation is over all parameters ~θ [12].As we will explain shortly, we can efficiently calculate

〈ϕ(~θ)|H |ϕ(~θ)〉 with quantum processors. Therefore, byoptimising parameters ~θ via a classical computer, con-

sidering E(~θ) = 〈ϕ(~θ)|H |ϕ(~θ)〉 as a cost function to beminimised, we can approximate the ground state energyand the ground state.

Now, we explain how to measure E(~θ) =

〈ϕ(~θ)|H |ϕ(~θ)〉 for a Hamiltonian H. As Pauli op-erators and products of them {I,X, Y, Z}⊗Nq form thecomplete basis for operators, any Hamiltonian can beexpanded as

H =∑α

fαPα,

Pα ∈ {I,X, Y, Z}⊗Nq ,(3)

where fα are real coefficients and {X,Y, Z} are Paulioperators. While an arbitrary Nq-qubit operator may

4

have exponential terms in the expansion, Hamiltoniansin reality are generally sparse so that the expansion onlyhas the number of terms polynomial to Nq. For example,the Hamiltonian of the 1D Ising model is

H = h∑i

ZiZi+1 + λ∑i

Xi, (4)

where the number of terms is linear to the number ofqubits. This also holds true for other systems. For ex-ample, to simulate the electronic structure of molecules,we consider the fermionic Hamiltonian

Hf =∑ij

tija†iaj +

∑ijkl

uijkla†ia†kalaj , (5)

where a†j (aj) denotes the creation (annihilation) opera-tor for a fermion in the j-th orbital, and tij and uijkl arethe one and two electron interactions, which can be effi-ciently calculated by integrating the basis set wave func-tions [28, 29]. The Hamiltonian consists of a polynomialnumber of terms consisting of products of fermionic op-erators. They can be further mapped to qubit operatorsvia encoding methods such as Jordan-Wigner, parity, andBravi-Kitaev encodings [27, 30, 31]. For example, theJordan-Wigner transformation is defined as

a†i → I⊗i−1 ⊗ σ− ⊗ Z⊗N−i

ai → I⊗i−1 ⊗ σ+ ⊗ Z⊗N−i,(6)

where σ± = (X ∓ iY )/2. We can therefore map thefermion Hamiltonian Hamiltonian to the form of Eq. (3)with a polynomial number of Pauli operators.

By assuming a Pauli decomposition of the Hamiltonian

H =∑α fαPα, the cost function E(

~θ) becomes

E(~θ) =∑α

fα 〈ϕ(~θ)|Pα |ϕ(~θ)〉 , (7)

where each term 〈ϕ(~θ)|Pα |ϕ(~θ)〉 is evaluated by calcu-lating the expectation value of each Pauli operator Pα.Note that the measurement of Pα can be fully parallelisedby using many quantum processors.

After we have obtained E(~θ), we update the parame-ters by using a classical computer to minimise the costfunction. As an example, the gradient descent methodupdates the parameter settings as

~θ(n+1) = ~θ(n) − a∇E(~θ(n)), (8)

where ~θ(n) and ~θ(n+1) denote the parameters at the n-step and the n + 1-step, respectively, a > 0 is a param-

eter determining the step size, and ∇E(~θ(n)) is the gra-dient of the cost function at ~θ(n). The gradient descentmethod deterministically decreases the cost function toa local minimum.The concept of the VQE is illustratedin Fig. 2. In practice, it is important to opt for a fastand accurate optimisation method properly to reach the

global minimum or feasible solution in a reasonable time.Also, the optimisation method should be robust to phys-ical noise and shot noise in the quantum hardware. Inaddition to gradient based methods, another type of opti-misation method is via direct search of the cost function.While the gradient may be more sensitive to physicalnoise – it typically vanishes exponentially in the numberof qubits [32], direct search is believed to be more ro-bust to physical noise, which may necessitate less repeti-tions [33]. We refer to McArdle et al. [23] for the reviewof classical optimisation algorithms.

Whether the VQE algorithm works also depends on thechoice of the ansatz. To have an efficient quantum simu-lation algorithm, we need to use a suitable ansatz for theproblem. If the ansatz state cannot express the solution,e.g., when the solution is a highly entangled state but theansatz can only generate low entangled states, it cannotfind the correct solution. In literature, several differenttypes of ansätze are proposed for different purposes. Forexample, the unitary coupled cluster ansatz is known asa suitable physically inspired ansatz for electronic struc-ture problems in chemistry [9, 34, 35]. However, the uni-tary coupled cluster ansatz generally necessitates a com-plicated form of quantum circuit with gates applied onmultiple number of qubits, where each multi-qubit gatecould be decomposed as a sequence of two-qubit gates,and the number of multi-qubit gates is quadratic to thenumber of qubits and the number of electrons. Sincethe unitary coupled cluster ansatz involves many generaltwo-qubit gates, it may be hard to implement on noisyquantum devices with short coherence time and limitedconnectivity. This problem might be circumvented byleveraging so-called “hardware efficient ansätze”. Thisfamily of ansätze might be more experimentally feasi-ble [9, 10, 13], because they are constructed based onrealisable demands on connectivity and gate operationsthat correspond to real quantum devices. However, ahardware efficient ansatz does not reflect the details of

FIG. 2. Schematic of the variational quantum eigensolver.The expectation values of the Pauli operators Pα are mea-sured for the ansatz state, and the expectation value of theHamiltonain is measured as a cost function on the classicalcomputer. The cost function is sent to a classical optimiserto update the parameters.

5

the simulated quantum system, and it has been shownthat exponentially vanishing gradients (so-called barrenplateaus) are liable to occur for randomly initialised pa-rameters [36]. There are several other methods proposedto circumvent the vanishing gradient problem [37–40].We refer to McArdle et al. [23] for a more detailed dis-cussion for ansatz construction.

The disadvantage of the VQE method is that the cor-rectness of the solution relies on the heuristic choice ofthe ansatz and the optimisation may be caught by a localinstead of global minimum. Furthermore, the total num-ber of measurements is O(�−2) for reaching a precision� due to shot noise [9, 12], which is quadratically worsethan the conventional QPE algorithm that uses universalquantum computers.

The VQE algorithm has been experimentally demon-strated by several groups [9, 10, 41–46]. To date, the thehydrogen chain was simulated with a 12-qubit supercon-ducting system [47].

B. Real and imaginary time evolution quantumsimulator

Now we introduce the basic algorithms for variationalquantum simulation (AQS), in particular, for simulatingreal [14] and imaginary [48] time evolution. The real timeevolution of a quantum system can be described via theSchrödinger equation as

d |ψ(t)〉dt

= −iH |ψ(t)〉 , (9)

where H is the Hamiltonian and |ψ(t)〉 is the time-dependent state. The conventional approach for simu-lating the evolution is to realize the time evolution e−iHt

as a unitary circuit and the state at time t is obtainedby applying the unitary to the initial state [6, 49–51].The circuit depth generally increases polynomially withrespect to the evolution time t. Instead, variational quan-tum simulation algorithms assumes that the quantumstate |ψ(t)〉 is represented by an ansatz quantum cir-cuit, |ϕ(~θ(t))〉 = U(~θ(t)) |ϕref 〉, and the time evolutionof Schrödinger equation of the state |ψ(t)〉 is mapped tothe evolution of the parameters ~θ(t).

Different variational principles can be used to havedifferent evolution equations of the parameters. Thethree most conventional variational principles are — TheDirac and Frenkel variational principle [52, 53], McLach-lan’s variational principle [54], and the time-dependentvariational principle [55, 56]. The Dirac and Frenkelvariational principle is not suitable for variational quan-tum simulation because the equation of the parametersmay involve complex solutions, which contradicts withthe requirement that parameters are real. Although theother two variational principle both gives real solutions,it is shown that the time-dependent variational princi-ple could be more unstable and it cannot be applied for

evolution of density matrices and imaginary time evolu-tion. On the contrary, McLachlan’s principle generallyproduces stable solutions and it is also applicable to allthe other scenarios beyond real time simulation. We re-fer a detailed study of the three variational principlesto Yuan et al. [26].

Now, we focus on McLachlan’s variational principle[54] and show how to derive the evolution of the pa-rameters that effectively simulates the time evolution.McLachlan’s principle requires one to minimise the dis-tance between the ideal evolution and the evolution in-duced of the parametrised trial state as

δ‖(∂/∂t+ iH) |ϕ(~θ(t))〉 ‖ = 0, (10)

where ‖ |ϕ〉 ‖ = 〈ϕ|ϕ〉 is a norm of states |ϕ〉 and δ is thevariation over the derivative of the parameters θ̇j . With

real parameters ~θ, the solution gives the evolution of theparameters ∑

j

Mk,j θ̇j = Vk, (11)

with coefficients

Mk,j = Re

(∂ 〈ϕ(~θ(t))|

∂θk

∂ |ϕ(~θ(t))〉∂θj

)Vk = Im

(〈ϕ(~θ(t))|H∂ |ϕ(

~θ(t))〉∂θk

),

(12)

where Re(·) and Im(·) are the real and imaginary parts,respectively. We refer to Appendix A for the derivation.

Next, we describe the variational imaginary timesimulation algorithm. The normalised Wick-rotatedSchrödinger equation [57] is obtained by replacing t inEq. (9) with = iτ ,

d |ψ(τ)〉dτ

= −(H − 〈H〉) |ψ(τ)〉 , (13)

where 〈H〉 = 〈ψ(τ)|H|ψ(τ)〉 is included for preservingthe norm of the state |ψ(τ)〉. Notably, the imaginarytime evolution can be leveraged for preparing a Gibbsstate and for discovering the ground state of quantumsystems [26, 48]. Following the same procedure for realtime evolution, we first make use of McLachlan’s princi-ple,

δ‖(∂/∂τ +H − 〈H〉) |ϕ(~θ(τ))〉 ‖ = 0, (14)

which then gives the evolution of the parameters:∑j

Mk,j θ̇j = Ck, (15)

with M defined in Eq. (12) and C defined by

Ck = −Re

(〈ϕ(~θ(τ))|H∂ |ϕ(

~θ(τ))〉∂θk

)= −1

2

∂E(~θ)

∂θk,

(16)

6

(|0〉+ eiθ |1〉)/√

2 X • X • H

. . . . . .

|ϕref 〉 U1 Uk−1 σk,i Uk Uj−1 σj,q

(a)

(|0〉+ eiθ |1〉)/√

2 X • X • H

. . . . . .

|ϕref 〉 U1 Uk−1 σk,i Uk UN σj

(b)

FIG. 3. Quantum circuits for computing (a) Re(eiθ 〈ϕref |U†k,iUj,q |ϕref 〉) and (b) Re(eiθ 〈ϕref |U†k,iσjU |ϕref 〉) [14, 26, 48].

with E(~θ) =(〈ϕ(~θ(τ))|H |ϕ(~θ(τ))〉

). Note that the

C vector is related to the gradient of the energy, im-plying that variational imaginary time simulation canbe regarded as a generalisation of the gradient descentmethod. It has been numerically found that the vari-ational imaginary time simulation algorithm might beless sensitive to local minima in contrast to simple gra-dient descent methods [48]. In addition, when imagi-nary time evolution does reach a minimum that is notthe ground state, it tends to be an excited eigenstate ofthe Hamiltonian, which can be thus exploited for findinggeneral eigen-spectra [21]. It has recently been observedthat an equivalent formulation of the variational imag-inary time algorithm can be obtained by exploiting thequantum natural gradient approach [58–60]. Notice thatsince M matrix has to be measured, variational imagi-nary time evolution needs more measurements than theconventional gradient descent method. However, for anincreasing system size and simulation time, the numberof measurements required for M matrix is asymptoticallynegligible [61].

FIG. 4. Schematic of the variational quantum simulation al-gorithm. The elements of M matrix and V (C) vectors aremeasured for the ansatz state. The results are sent to the

classical computer to solve M~̇θ = V (C) to update the pa-rameters, which will be fed to quantum processors.

Given the current parameters ~θ, we now showhow to efficiently measure the M , V , and C

terms with quantum circuits. Suppose |ϕ(~θ(t))〉 =UN (θN ) . . . Uk(θk) . . . U1(θ1) |ϕref 〉 with the derivative ofeach unitary Uk(θk) expressed as

∂Uk(θk)

∂θk=∑i

gk,iUkσk,i, (17)

where σk,i are unitary operators and gk,i are complexcoefficients. For instance, assuming Uk(θk) = e

−iθkX , wehave ∂Uk(θk)/∂θk = −iUkX and hence gk,i = −i andσk,i = X. Thus, the derivative of the ansatz state is

∂ |ϕ(~θ(t))〉∂θk

=∑i

gk,iUk,i |ϕref 〉 , (18)

where we defined

Uk,i = UNUN−1 · · ·Uk+1Ukσk,i · · ·U2U1. (19)

Now each Mk,j can be written as

Mk,j =∑i,q

Re(g∗k,igj,q 〈ϕref |U

†k,iUj,q |ϕref 〉

). (20)

Supposing the Hamiltonian is decomposed as H =∑α fαPα, with real coefficients fα and Pauli operators

Pα, we have Vk and Ck as

Vk = −∑i,α

Re(ig∗k,ifα 〈ϕref |U

†k,iσαU |ϕref 〉

),

Ck = −∑i,α

Re(g∗k,ifα 〈ϕref |U

†k,iσαU |ϕref 〉

),

(21)

Note that each term constituting M , V , and C can bewritten as a sum of terms as

aRe(eiθ 〈ϕref | V |ϕref 〉

), (22)

where a, θ ∈ R depend on the coefficients gk,i and fj ,and V is either U†k,iUj,q or U

†k,iσαU . Then we can cal-

culate M , C, and V with the quantum circuits shown in

7

Fig. 3. Refer to Appendix B for a detailed explanationabout construction of the quantum circuit. Notice thatcomprehensive analysis on the sampling cost for M , Vand C has been given by van Straaten and Koczor [61].We summarise the variational algorithm for simulatingreal (imaginary) time evolution.

Variational quantum simulation algorithmInput Hamiltonian H and initial state|ψ(0)〉;Algorithm Output |ψ(T )〉 under real (imagi-nary) time evolution.

Step 1 Determine the ansatz |ϕ(~θ)〉, the ini-tial parameter ~θ(0) for the initialstate |ψ(0)〉, and set t = 0.

Step 2 Use a quantum computer to computethe M matrix and the V (C) vector.

Step 3 Use a classical computer to solve

Eq. (11) (Eq. (15)) to obtain ~̇θ(t).

Step 4 Set ~θ(t+ δt) = ~θ(t) + δt~̇θ(t) and t = t+ δt.

Step 5 Repeat step 2 to step 4 until t = T.

The schematic figure is also shown in Fig. 4. We canalso simulate time dependent Hamiltonian evolution byusing the time-dependent Hamiltonian at each step. Theaccuracy of the simulation can be computed at each stepby the distance between the evolution of the ansatz stateand that of the ideal evolution [26]. In the case of thereal time simulation, we have

‖(∂/∂t+ iH) |ϕ(~θ(t))〉 ‖2

=∑k,j

Mk,j θ̇kθ̇j − 2∑k

Vkθ̇k + 〈H2〉 , (23)

which is a function of M , V and 〈H2〉. Similar argumentsalso hold for imaginary time evolution. Note that a vari-ational real time simulation algorithm was demonstratedin an experiment using 4 superconducting qubits [62]. Inthis experiment, adiabatic quantum computing was sim-ulated, which was used for discovering eigenstates of anIsing Hamiltonian.

III. VARIATIONAL QUANTUMOPTIMISATION

In this section, we illustrate several examples of vari-ational optimisation algorithms for different problems.The key idea is to construct a Hamiltonian or a cost func-tion such that the solution of the problem corresponds tothe ground state or the minimum of the cost function.

A. Quantum approximate optimisation algorithm

The quantum approximate optimisation algorithm(QAOA) [13] was initially proposed for solving classical

optimisation problems. The algorithm works by map-ping the classical problem to a Hamiltonian HP so thatthe ground state corresponds to the solution. Since wetry to solve a classical optimisation problem, and theHamiltonian HP is diagonal in the computational basisas HP =

∑α fαPα where Pα ∈ {I, Z}⊗Nq .

As an example, we consider the Boolean satisfiabilityproblem, which aims to find solutions of Boolean vari-ables so that all given clauses in a propositional formulaare true. The j-th Boolean variable denoted as xj takesa value either 1 or 0, each of which corresponds to trueand false. The Boolean satisfiability problem consists ofxi∨xj , xi∧xj , and x̄ operations. xi∨xj becomes 1 wheneither of xi or xj is 1, xi ∧ xj is a product of xi and xj ,and x̄ operations flip the value x. An example of Booleansatisfiability problem is (x1 ∨ x2) ∧ (x̄1 ∨ x2) ∧ (x̄1 ∨ x̄2)and the solution that all the clauses are true (with value1) is x1 = 0 and x2 = 1.

The general form of this problem (with clauses involv-ing three or more booleans) is generally NP-hard and haswide range of applications in computer science and cryp-tography [63]. To map the Boolean satisfiability problemto a Hamiltonian, we first get the Hamiltonian with itsground state being the solution of each clause. For ex-ample, the Hamiltonian for the first clause (x1 ∨ x2) is

H1 =1

4(I − Z1)(I − Z2). (24)

After having the Hamiltonian Hk for each clause, theHamiltonian for all clauses is expressed as HP =

∑kHk.

Since the solution corresponds to the ground state ofHP , we can try to solve the problem by using a quantumalgorithm for searching for the ground state. The varia-tional approach for solving such problems is first studiedby Farhi et al. [13] who introduced the ansatz

U(~θ1, ~θ2) =

D∏k=1

e−iθ(k)1 HXe−iθ

(k)2 HP , (25)

with HX =∑Nqj=1Xj , D being the number of repeti-

tions of the ansatz quantum circuit, ~θ1 = (θ(1)1 , θ

(2)1 , . . . ),

and ~θ2 = (θ(1)2 , θ

(2)2 , . . . ). With initial states |−,−, . . .〉

and |−〉 = 1√2(|0〉 − |1〉), we obtain the ansatz state

|ϕ(~θ1, ~θ2)〉 = U(~θ1, ~θ2) |−,−, . . .〉. Directly optimisingthe parameters might be challenging for a fixed Hamilto-nian, so Farhi et al. [13] also suggest to consider graduallychanging a Hamiltonian in each step of optimisation as

H(t) =

(1− t

T

)HX +

t

THP , (26)

withH(0) = HX andH(T ) = HP . Since |−,−, . . .〉 is theground state of H(0) = HX and solution is the groundstate of H(T ) = HP , the optimisation emulates the pro-cess of adiabatic state preparation with D → ∞. Withsufficiently large D, and adaptive optimisations of the

8

parameters for each optimisation step t, the algorithmmay still be able to find the ground state solution.

A recent thorough study of the performance of theQAOA on MaxCut problems can be found in Zhouet al. [64]. Utilising nonadiabatic mechanisms, a heuris-tic strategy was proposed to learn the parameters expo-nentially faster than the conventional approach. Mean-while, the QAOA was implemented with 40 trapped-ionqubits [65].

B. Variational algorithms for machine learning

Now we introduce the application of variational quan-tum algorithms in machine learning. In general, machinelearning provides a universal approach to learn the pat-tern of the given data and to predict or reproduce newdata. For example, in supervised learning, the given dataare described by {(~x1, y1), (~x2, y2), . . . , (~xND , yND )} withND being the number of the data. Then we try to con-struct the model y = f(~x) so that it predicts the outputynew = f(~xnew) for any new data ~xnew. Suppose the

model y = f(~x, ~θ) has parameters ~θ, the training processis to tune the parameters to minimise a cost function,such as

CML(~θ) =∑k

|yk − f(~xk, ~θ)|2. (27)

When the cost function CML(~θ) is minimised to a smallvalue, it indicates that the given data are well-modelled.In practice, there are different ways to choose the model.For example, a linear regression model is,

f(~x, ~θ) = ~w · ~x+ b, (28)

with real parameters ~w and b, and ~θ = (~w, b). In deepleaning, i.e., a multi-layer neural network, the model isdescribed as

f(~x, ~θ) = gNd ◦ uNd · · · ◦ u2 ◦ g1 ◦ u1(~x),

uk(~x) = Wk~x+~bk,(29)

where Nd is the depth of the neural network, gk is

a nonlinear activation function and Wk and ~bk are aparametrised matrix and vector, respectively. To circum-vent overfitting, we can add the norm of the parametersto the cost function to restrict degree of freedom of theparameters, which is called regularisation.

Whether machine learning works or not is highly de-pendent on the choice of the model. While quantumstates could efficiently represent multipartite correlationsthat admit no efficient classical representation, quantummachine learning protocols are proposed involving quan-tum neural networks that consist of variational quantumcircuits. Consequently, we can leverage a quantum neu-ral network to dramatically enhance the representabilityof the model [15, 16]. In addition, as the ansatz circuit is

FIG. 5. Comparison between (a) classical neural networks and(b) quantum neural networks used for supervised learning.The figure (b) is the quantum circuit proposed in quantumcircuit leaning [15].

a unitary operator, the norm of the quantum state is nec-essarily unity, and this constraint may lead to a naturalregularisation of the parameters to avoid overfitting [15].The schematic figures for classical and quantum neuralnetworks are shown in Fig. 5.

Note that there are quantum machine learning algo-rithms [66–69] that are based on universal quantum com-puting without variational quantum circuits. While thesealgorithms are proven to have exponential speedups overclassical algorithms under certain conditions, they gener-ally require a deep quantum circuit and are not suitablefor NISQ devices.

In this section, we illustrate five examples of quantummachine learning algorithms. The first two algorithmsare introduced for leaning classical data for solving a clas-sical problem; the latter three are introduced for learningquantum data.

1. Quantum circuit learning

The quantum circuit learning (QCL) algorithm imple-ments supervised learning with a variational quantumcircuit instead of a classical neural network [15]. In QCL,

a state is prepared as |ϕ(~x, ~θ)〉 = U(~θ)U(~x) |ϕref 〉, andthe output {f(~xk, ~θ)} is generated by measuring the statein a properly chosen basis. The quantum circuit can nat-urally introduce nonlinearality of the model f . Supposethe data is described as {xk, yk} and the initial state en-coded with the information of x is

ρin(x) =1

2Nq

Nq⊗k=1

[I + xXk +

√1− x2Zk

], (30)

9

which can be generated by applying the rotational-Ygate, Ry(sin

−1x) with x ∈ [−1, 1], to each qubit that isinitialised in |0〉. This state involves higher order termsof x up to the Nqth order, where terms such as x

√1− x2

may enhance the learning process. Note that this argu-ment can be naturally generalised to higher dimensionaldata. In addition, nonlinearality can be introduced viathe measurement process and classical post-processing ofthe measurement outcome. As we have discussed above,two potential benefits of QCL are the representability ofmultipartite correlations and the unitarity of the quan-tum circuit for circumventing overfitting. Whether QCLcan outperform classical neural network based machinelearning still needs further study.

2. Data-driven quantum circuit learning

Data-driven quantum circuit learning (DDQCL) im-plements a generative model by using a variational quan-tum circuit [16]. We briefly summarise the classical gen-erative model. Suppose the data are described with{~x1, ~x2, . . . , ~xND}, where ND is the number of data andwe assume ~x has a binary representation, i.e., ~x ={x1, x2 . . . , xNE} with xj ∈ {−1, 1} and NE being thelength of the binary string. We can assume the datais generated from an unknown probability distributionpD(~x), and the task of a generative model is to construct

a model pM (~x|~θ) to approximate the probability distri-bution pD(~x). The joint probability that the data is gen-

erated from pM (~x|~θ) is

L(~θ) =ND∏n=1

pM (~xn|~θ), (31)

which is the so-called likelihood function. By maximising

L(~θ), we can obtain the optimised model probability dis-tribution for the data. Alternatively, we can adopt thecost function

CDD(~θ) = −1

ND

ND∑n=1

log[max(ε, pM (~xn|~θ))], (32)

where a small value ε > 0 is introduced for avoiding sin-gularities of the cost function. For a classical generative

model, an example model pM (~x|~θ) could be generated viaa Boltzman machine on a classical computer.

For DDQCL, the model pM (~x|~θ) is obtained from aquantum computer, for example, by the projection prob-ability

pQ(~x|~θ) = | 〈~x|ϕ(~θ)〉 |2, (33)

where |ϕ(~θ)〉 is an ansatz state generated on the varia-tional quantum circuit, |~x〉 = |x1, x2, . . . xNE 〉 is the com-putational basis, and the probability is defined accord-ing to the Born Rule. The quantum generative model is

FIG. 6. A quantum circuit for sampling pQ(~x|~θ).

also referred to as the Born machine [70]. Since quan-tum states can efficiently represent complex multipar-tite correlations and the model can be efficiently sam-pled by measuring the prepared state, DDQCL may beable to represent generative models that are classicallychallenging. In experiment, DDQCL has been appliedto successfully learn Greenberger-Horne-Zeilinger states(GHZ) states and coherent thermal state by using a 4-qubit trapped ion device [16]. The schematic figure for

sampling pQ(~x|~θ) is shown in Fig. 6.

3. Quantum generative adversarial networks

Quantum generative adversarial networks(QuGANs) [17, 71] are a quantum analogue of generativeadversarial networks (GANs). Conventional GANs arecomposed of three parts — true data, generator, anddiscriminator, as shown in Fig. 7(a). The generatorcompetes with the discriminator, where the formertries to produce fake data and the latter tries to de-termine whether the input data are true or fake. Byoptimising both the generator and the discriminator,the generator can learn the distribution of the truedata until the discriminator cannot tell the differencebetween the true data and the fake data from thegenerator. GANs are generalised to quantum computingby replacing each part with a quantum system [17].By using a quantum circuit as the generator, we canrepresent the Nq-dimensional Hilbert space with log Nqqubits and compute sparse and low-rank matrices withO(poly(logNq)) steps. Suppose the true data are de-scribed with an ensemble of quantum states as a densitymatrix ρtrue, the discriminator implements a quantummeasurement. The schematic figure for quantum GANsis shown in Fig. 7 (b).

Suppose the fake density matrix from the generatoris produced from a parametrised quantum circuit as

ρG(~θG) with parameters ~θG, which aims to learn thetrue density matrix ρtrue. By using ancillary qubits,we can express density matrices with pure states andwe refer to Dallaire-Demers and Killoran [71] for de-tailed ansatz constructions. The discriminator imple-ments a parametrised positive-operator valued measure

{P t(~θD), P f (~θD)}, with parameters ~θD and P t(~θD) +P f (~θD) = I, to distinguish between ρtrue and ρG(~θG).

10

FIG. 7. Schematic diagrams for (a) classical GANs and (b)quantum GANs. In classical GANs, the generator and thediscriminator consist of classical neural network, while quan-tum neural networks are used in quantum GANS.

Assuming that ρtrue and ρG(~θG) are sent to the discrim-inator randomly with equal probability, the probabilitythat the discriminator fails is

Pfail =1

2

(Tr[ρG(~θG)P

t(~θD)] + Tr[ρtruePf (~θD)]

),

=1

2

(CGA(~θD, ~θG) + 1

)(34)

with CGA(~θD, ~θG) = Tr[(ρG(~θG) − ρtrue)P t(~θD)] beingproportional to the trace distance between ρG(~θG) and

ρtrue when optimised over Pt(~θD). With fixed param-

eters ~θG for the generator, we optimise the discrimi-

nator ~θD to minimise the failure probability Pfail or

CGA(~θD, ~θG). With optimised discriminator, we fix ~θDand in turn optimise the generator ~θG to maximise thefailure probability. By repeating this process, the param-

eters ~θD and ~θG arrives at an equilibrium with ρG(~θG) =ρtrue, indicating a successful learning of the data.

4. Quantum autoencoder for quantum data compression

The classical autoencoder is used for compressing clas-sical data as shown in Fig. 8(a). For an input set oftraining data {~x1, ~x2, . . . ., ~xND}, the autoencoder E firstencodes it to {~x′1, ~x′2, . . . ., ~x′ND} with each ~x

′i being a

smaller vector than ~xi. A decoder D is then applied totransform the data to {~x′′1 , ~x′′2 , . . . ., ~x′′ND}, with each ~x

′′i

having the same size as ~xi. The task of an autoencoder

is to compress the input data into smaller size, with therequirement that the input data can be recovered fromthe compressed one with

∑k ‖~x′′k − ~xk‖2 ≤ ε where ε is

a desired accuracy.

Quantum autoencoder implements a similar task forcompressing quantum states [18, 72]. Here, we considerthe scheme proposed in Romero et al. [18]. Consider thecase with input states of n+m qubits being compressedto states of n qubits. Suppose the input states are an en-semble {pk, |φk〉AB} with probabilities pk and unknownstates |φk〉AB . Here the subsystems A and B consists ofn and m qubits, respectively. With a compression circuit

U(~θ) and a post-selection measurement (for example inthe computational basis) on the m-qubits of system B,as shown in Fig. 8(b), each input state |φk〉AB of n+mqubits is mapped to a state ρcompk of n qubits. The in-verse process then decodes the compressed state and mapeach ρcompk to an output state ρ

outk . The average fidelity

between the input and outputs is

C(1)AE(

~θ) =∑k

pkF (|φk〉 , ρoutk (~θ)), (35)

which serves as a cost function and can be computedvia the SWAP test or the destructive SWAP test cir-cuit. Refer to Appendix C for details. When C

(1)AE(

~θ) ismaximised to a desired accuracy, it indicates a successfulcompression of the input state ensemble. Meanwhile, the

successful compression and decoding, i.e., C(1)AE = 1, are

achieved if and only if

U(~θ) |φk〉AB = |φcompk 〉A ⊗ |0̄〉B , ∀k, (36)

where |φcompk 〉A corresponds to the compressed state, and|0̄〉B is some fixed reference state. Thus we can also adopt

FIG. 8. Schematic figures for (a) classical autoencoder and (b)quantum autoencoder. The encoding operation E compressthe data, and the decoding operation D decodes the data tothe original dimension.

11

a simpler cost function

C(2)AE(

~θ) =∑k

pkTr[U(~θ) |φk〉 〈φk|AB U(~θ)†IA ⊗ |0̄〉 〈0̄|B ],

(37)

which can be computed as the probability projected toIA ⊗ |0̄〉 〈0̄|B .

5. Variational quantum state eigensolver

Analysing eigenvalues and eigenvectors of the covari-ance matrix of data is crucial for extracting its importantfeatures. Such a process is called the principal componentanalysis (PCA), which has been widely used in data sci-ence and machine learning. The covariance matrix of thedata could be uploaded onto a quantum computer [73–75]. Here we show how to use the variational quantumstate eigensolver (VQSE) algorithm to diagonalise inputdensity matrices ρ to

ρ =∑j

λj |λj〉 〈λj | , 〈λi|λj〉 = δi,j . (38)

We focus on the VQSE algorithm introduced in Cerezoet al. [38], which only requires a single copy of the stateand other schemes requiring two copies of the state can befound in LaRose et al. [37] and Bravo-Prieto et al. [76].Without loss of generality, we assume the eigenvalues arein an descending order, i.e., λ1 ≥ λ2 ≥ · · · ≥ λf wheref = rank(ρ).

We first map a parametrised quantum circuit U(~θ) to

the input density matrix as ρ(~θ) = U(~θ)ρU†(~θ). Then wedefine a cost function

CDI(~θ) = Tr[ρ(~θ)H], (39)

with H =∑j Ej |~xj〉 〈~xj | being a diagonal Hamiltonian

in the computational basis {|~xj〉} with non-degenerateeigenvalues E1 < E2 < · · · < E2Nq . Then we have

CDI(~θ) = ~E · ~q(~θ), (40)

with ~E = (E1, E2, . . . , E2Nq ), ~q(~θ) = (q1, q2, . . . , q2Nq ),

and qj = Tr[ρ(~θ) |~xj〉 〈~xj |]. Since ~q(~θ) is obtained frommeasuring ρ, the vector ~q(~θ) can be obtained from apply-ing a doubly stochastic matrix on the eigenvalue vector

λ. Hence λ majorises the measurement probabilities ~q(~θ),

i.e., ~λ � ~q(~θ). That is, with ~λ = (λ1, λ2, . . . , λ2Nq ) and~q(θ) = (q1, q2, . . . , q2Nq ) in descending orders, we have∑ki=1 λi ≥

∑ki=1 qi, ∀k = {1, 2, . . . , 2Nq}. Here, we set

λk = 0 for k > f . Since the dot product is a Schur con-

cave function, satisfying f(~a) � f(~b), ∀~b � ~a, we have

CDI(~θ) = ~E · ~q(~θ) ≥ ~E · ~λ. (41)

The equality holds when ~q(~θ) = ~λ, i.e., the diagonalisa-tion is achieved.

Therefore, after minimising the cost function CDI(~θ)

with optimal parameters ~θopt, the state ρ(~θopt) is rotated

to ρ(~θopt) =∑i λj |~xj〉 〈~xj | with λ1 ≥ λ2 ≥ · · · ≥ λf . By

measuring ρ(~θopt) in the computational basis, we thusobtain |~xj〉 with probability determined by the eigen-value λj and the the corresponding eigenvector of ρ is

U(~θopt) |~xj〉.On the other hand, a time dependent cost function

which combines local and global cost functions is usedto make the best of their benefits [38]. See AppendixD for an explanation of local and global cost functions.Also one can find applications of VQSE in quantum errormitigation [14, 77, 78] and entanglement specroscopy [79,80].

C. Variational algorithm for linear algebra

Variational quantum algorithms can be used for solv-ing matrix-vector multiplication and solving linear sys-tems of equations. The task of matrix-vector multipli-cation is to obtain |vM〉 = M|v0〉 /‖M|v0〉 ‖, whereM is a sparse matrix, |v0〉 is a given state vector,and ‖ |ψ〉 ‖ =

√〈ψ|ψ〉. Meanwhile, linear systems of

equation is to solve a linear equation M|vM−1〉 =|v0〉 to have |vM−1〉 = M−1 |v0〉. There exist severalvariational quantum algorithms for implementing thesetasks [19, 20, 81, 82]. Herein, we illustrate the methodsintroduced in Bravo-Prieto et al. [19] and Xu et al. [20].

We first consider matrix-vector multiplication pro-posed by Xu et al. [20], where the solution |vM〉 cor-responds to the ground state of the Hamiltonian,

HM = I −M|v0〉 〈v0|M†

‖M|v0〉 ‖2. (42)

When M is a sparse matrix and the circuit for prepar-ing |v0〉 is known, the expectation value EM =〈ϕ(~θ)|HM |ϕ(~θ)〉 can be efficiently evaluated by using thehadamard test or the swap test circuit. Thus by minimis-ing the expectation value EM, we can find the groundstate |vM〉. Because the ground state has a uniqueground state energy 0, we can further know whether thediscovered state is the ground state or not, unlike con-ventional VQE where the ground state energy is gen-erally unknown. The matrix-vector multiplication al-gorithm can be applied for implementing Hamiltoniansimulation. Suppose we want to simulate a time evo-lution operator U = exp(−iHt) via Trotterisation U ≈∏

exp(−iHδt). By setting M = 1−iHδt, we can approx-imate each exp(−iHδt) and hence the whole evolution asU = MNS + O(t2/NS), where NS = t/δt. Therefore, bysubsequently applying matrix-vector multiplication, wecan simulate the time evolution of quantum systems.

Here we also consider the algorithms for solving linearequations independently proposed by Xu et al. [20] andBravo-Prieto et al. [76]. The solution is mapped to the

12

ground state of the Hamiltonian

HM−1 =M†(I − |v0〉 〈v0|)M, (43)

with the smallest eigenvalue 0 as well. This Hamilto-nian was firstly proposed by Subaş ı et al. [83], who ap-plied adiabatic algorithms to find the ground state witha universal quantum computer. With the variationalmethod, solving the ground state is similar to the one formatrix-vector multiplication. Furthermore, Xu et al. [20]used Hamiltonian morphing optimisation for avoiding lo-cal minima, and Bravo-Prieto et al. [76] used a local costfunction for circumventing a barren plateau issue and runa simulation with up to 50 qubits. Refer to Appendix Dfor these optimisation methods.

D. Excited state-search variational algorithms

Next we show how to find excited states and excitedenergy spectra of a Hamiltonian. Calculating excited en-ergy spectra is important for studying many-body quan-tum physics problem. For example, it can be used tostudy chemical reaction dynamics, important for creatingnew drugs and new methodologies for mass productionof beneficial materials [29, 84]. In addition, evaluationof excited states enables us to calculate the photodisso-ciation rates and absorption bands, which are essentialfor designing solar cells and investigating their dynam-ics [85, 86]. There exist several VQAs for evaluatingexcited states and excited energy spectra [21, 22, 87–94]. These algorithms can be used as a subroutine forother applications, e.g., Green’s function [95, 96], non-adiabatic coupling, and Berry’s phase [97] and simulat-ing real time evolution [98] etc. In this section, we reviewthree VQAs — the overlap-based method, the subspaceexpansion method, and the contraction VQE method —for calculating excited state energy, and the applicationin calculating Green’s function.

1. Overlap-based method

The overlap-based method first uses VQE to find theground state and then sequentially obtains excited statesby penalising the previously obtained eigenstates [21, 22].

Suppose that the ground state |G̃〉 of the given Hamilto-nian H is obtained from either the conventional VQE orvariational imaginary time simulation. Now, suppose wereplace the Hamiltonian H with the a Hamiltonian

H ′ = H + α |G̃〉 〈G̃| . (44)

Here, α is a positive number which is chosen to be suf-ficiently larger compared to the energy gap between theground and the first excited state of the Hamiltonian.Then the first excited state |E1〉 of the original Hamilto-nian H becomes the ground state of the new Hamiltonian

H ′. Therefore, with the new Hamiltonian H ′, we can ob-tain the first excited state |Ẽ1〉 of H with the VQE orvariational imaginary time simulation on H ′.

To realise the VQE or imaginary time evolution of H ′,

we need to measure the energy E′(~θ) = 〈ϕ(~θ)|H ′ |ϕ(~θ)〉of the trial state |ϕ(~θ)〉 as

E′(~θ) = 〈ϕ(~θ)|H |ϕ(~θ)〉+ α 〈ϕ(~θ)|G̃〉 〈G̃|ϕ(~θ)〉 . (45)

The first term can be computed in the same way as theconventional VQE method. The second term is the over-

lap between |ψ(~θ)〉 and |G̃〉, which can be evaluated withthe SWAP test circuit or the destructive SWAP test cir-cuit [99, 100]. These quantum circuit necessitate twocopies of the state. Note that destructive SWAP test cir-cuit only leverages a shallow depth circuit. We leave adetailed explanation about the (destructive) SWAP testcircuit in Appendix C. Alternatively, we can computethe overlap term without using two copies of the statebut using a doubled depth of the quantum circuit [22].

Denoting |G̃〉 = U(~θG) |ϕref 〉, the overlap term can bewritten as

〈ϕ(~θ)|G̃〉 〈G̃|ϕ(~θ)〉 = | 〈ϕref |U†(~θG)U(~θ) |ϕref 〉 |2. (46)

This can be evaluated by applying U(~θ) and U†(~θG) tothe reference state |ϕref 〉, and measuring the probabilityto observe |ϕref 〉.

After finding the first excited state ofH, we can furtherfind the second excited state by replacing the Hamilto-nian H with

H ′′ = H + α(|G̃〉 〈G̃|+ |Ẽ1〉 〈Ẽ1|). (47)

Then the second excited state of H becomes the groundstate of H ′′, which could be similarly solved via the VQEor variational imaginary time simulation. It is not hardto see that this procedure can be repeated to sequentiallydiscover other low energy eigenstates.

In the overlap-based method, an error happens whenthe discovered state |G̃〉 is not the exact ground state,such as the case where it is a superposition of groundstate and excited states. Therefore when the VQEmethod gets trapped to a local minimum, the overlap-based method may fail to work. Note that this problem issevere since if the ground state was calculated incorrectly,all the subsequently calculated excited states are incor-rect. Interestingly, it was numerically observed that thisproblem is less severe for variational imaginary time evo-lution [21]. This is because when variational imaginarytime evolution fails to find the ground state, it may in-stead converge to an eigenstate of the Hamiltonian basedon the definition of imaginary time evolution [21, 48].By penalising the discovered excited state, we can stillconstruct a new Hamiltonian to find another low energystate.

13

2. Quantum subspace expansion

The quantum subspace expansion solves a generalisedeigenvalue problem in terms of the given Hamiltonianin a expanded subspace around an approximated groundstate, and the obtained eigenstates and eigenenergies cor-respond to those of the Hamiltonian [87]. Note that theobtained spectrum is error-mitigated because the excitedstates are approximated as a linear combination of statesin the expanded subspace. Refer to Sec. V D for a de-tailed explanation about error mitigation effect of quan-tum subspace expansion.

Let |G̃〉 be an approximation of the true ground stateobtained from either the VQE or variational imaginarytime simulation. Suppose we approximate an eigenstateof the Hamiltonian as

|ψeig(~c)〉 ≈∑m

cm |ψm〉 , (48)

where |ψm〉 = |G̃〉 with m = 0, |ψm〉 (m ≥ 1) arestates in the expanded subspace, ~c = (c0, c1, c2, . . . )

T ,and 〈ψeig(~c)|ψeig(~c)〉 = 1. For fermionic systems, wecan choose |ψm〉 = a†iaj |G̃〉 , m = (i, j), with a

†j and

aj being the creation and annihilation operators. For

spin systems, we can set |ψm〉 = Pm |G̃〉 with Pm ∈{I,X, Y, Z}⊗Nq . As the number of Pm increases, the sub-space expands, which potentially improves the accuracyof the subspace expansion method. Denote E(~c,~c ∗) =〈ψeig|H |ψeig〉, the state |ψeig(~c)〉 is an eigenstate of Hwhen E(~c) corresponds a local minimum satisfying

δ[E(~c,~c∗)− E 〈ψeig(~c)|ψeig(~c)〉] = 0. (49)

Here δE(~c,~c∗) =∑i δci∂E(~c)/∂ci+c.c. and E is the La-

grangian multiplier for the constraint 〈ψeig(~c)|ψeig(~c)〉 =1. A solution of this equation results in

H̃~c = ES̃~c. (50)

Here H̃ and S̃ are defined by

H̃αβ = 〈ψα|H|ψβ〉 , S̃αβ = 〈ψα|ψβ〉 , (51)

and E corresponds to the energy for eigenstate. BothH̃ and S̃ can be efficiently measured. For example,with |ψm〉 = Pm |G̃〉 we have H̃αβ = 〈G̃|PαHPβ |G̃〉and S̃αβ = 〈G̃|PαPβ |G̃〉, which can be respectively ob-tained by measuring the expectation value of the oper-ators PαHPβ and PαPβ for the approximated ground

state |G̃〉. By solving the generalised eigenvalue problemof Eq. (50), we can obtain the information of eigenspec-tra of the Hamiltonian in the considered subspace. Werefer to Appendix E for the derivation.

3. Contraction VQE methods

The contraction VQE methods firstly discover the low-est energy subspace and then construct eigenstates in

that subspace. Here we introduce two contraction VQEmethods — subspace-search VQE (SSVQE) [89] and mul-tistate contracted variant of VQE (MC-VQE) [90].

The procedure of SSVQE is as follows. We first prepare

a set of orthogonal states {|φi〉ki=0} (〈φj |φi〉 = δi,j), suchas states in the computational basis. Then we minimisethe cost function

C(1)CO(

~θ) =

k∑j=0

〈φj |U†(~θ)HU(~θ) |φj〉 (52)

to constrain the subspace {U(~θ∗) |φi〉}ki=0 to the lowestk+ 1 eigenstates, where ~θ∗ is the parameter set after theoptimisation and we define 0 th excited state |E0〉 as theground state. At this stage, U(~θ∗) |φi〉 is generally a su-perposition of the eigenstate {|Ei〉}ki=0 of H. To projectU(~θ∗) |φs〉 (s ∈ {0, . . . , k}) to the k th excited state |Ek〉,we maximise

C(2)CO(

~φ) = 〈φs|V †(~φ)U†(~θ∗)HU(~θ∗)V (~φ) |φs〉 . (53)

By finding the optimal parameters ~φ∗, we can approxi-

mate the kth excited state |Ek〉 by U(~θ∗)V (~φ∗) |φs〉. Inpractice, we can start with k = 0 to find the ground stateand then increase the value of k to find excited states.

For the MC-VQE method, it also first projects the sub-space to the lowest energy subspace in the same way asSSVQE. Different from SSVQE, MC-VQE assumes theexcited states can be expanded in the lowest energy sub-space as

|E〉 =k∑i=0

ciU(~θ∗) |φi〉 , (54)

and the goal is to find the matrix ~c = (c0, c1, . . . , ck).This task is similar to subspace expansion method men-tioned above. Now, ~c can be computed by solving thefollowing eigenvalue problem

H̃~c = E~c, (55)

where the matrix H̃ is

H̃α,β = 〈φα|U†(~θ∗)HU(~θ∗) |φβ〉 , (56)

and E is the corresponding excited state energy. Wecan regard Eq. (55) as the special case of Eq. (50) withS = I, because the states in the subspace are mutu-ally orthogonal. The diagonal term of the matrix H̃can be evaluated as the same procedure to the con-ventional VQE. Non-diagonal terms of H̃ can be ob-tained by calculating expectation values of H with states|±α,β〉 = (|φβ〉 ± |φα〉)/

√2 and linearly combining them

as

2H̃α,β = 〈+α,β |U†(~θ)HU(~θ) |+α,β〉

− 〈−α,β |U†(~θ)HU(~θ) |−α,β〉 .(57)

14

The potential problem of the contraction VQE meth-

ods is that the energy landscape of C(1)CO(

~θ) may bemore complicated than the conventional VQE. This isbecause the conventional VQE only aims to find the cor-rect ground state, while the contraction VQE methodsneed to find the correct unitary for all the k orthogonalstates. The benefit of this method is that the opitimiza-

tion of U(~θ) is averaged over multiple states, therefore theexcited states can be calculated with an equal accuracy.

4. Calculation of Green’s function

Now, we show the application of the VQE algorithms inthe calculation of Green’s function [95, 96], which playsa crucial role in investigating many-body physics suchas high Tc superconductivity [101], topological insula-tors [102] and magnetic materials [103]. The definitionof the retarded Green’s function at zero temperature is

G(R)αβ (t) = −iΘ(t) 〈G| aα(t)a

†β(0) + a

†β(0)aα(t) |G〉 . (58)

Here, Θ(t) is a Heaviside step function, a(†)α(β) is the an-

nihilation (creation) fermionic operator for the fermionicmode α(β), aα(t) = e

iHtaαe−iHt, and |G〉 is a ground

state. Here, we show how to calculate Green’s functionwith the algorithms for finding excited states. Note thatanother approach is to use the variational quantum sim-ulation algorithm in section IV.

For simplicity, we consider Green’s function in themomentum space for identical spins. We thus have

α = β = (k, ↑) and denote Green’s function as G(R)k (t).Green’s function has another expression called Lehmannrepresentation as

Gk(t) = −i∑m

ei(EG−Em)t| 〈Em| a†k |G〉 |2 (59)

where |Em〉 and Em are eigenstates and eigenenergiesof the Hamiltonian. Thus, if the transition amplitudes

| 〈Em| a†k |G〉 |2 can be calculated, we can obtain Green’sfunction.

In literature, Endo et al. [95] used the contraction VQEmethod and Rungger et al. [96] employed the overlapmethod to calculate the transition amplitudes and fur-ther Green’s function. The algorithm using the over-lap method [96] was originally employed for computingGreen’s function in the specific Jordan Wigner encod-ing, whose generalisation was further studied in Ibe et al.[104] for general operators. Here we review the algorithmbased on the MC-VQE method [95]. By using the expres-sion for mth excited state in Eq. (54), we have

|Em〉 ≈k∑i=0

c(m)i U(

~θ∗) |φi〉 , (60)

and hence

〈Em| a†k |En〉 =∑ij

c(m)∗i c

(n)j 〈φi|U

†(~θ∗)a†kU(~θ∗) |φj〉 .

(61)

Denoting a†k = Ak + iBk, |φ±ij〉 = U(~θ∗)(|φi〉 ± |φj〉)/

√2

and |φi±ij 〉 = U(~θ∗)(|φi〉 ± i |φj〉)/√

2, where Ak and Bkare Hermitian operators, we have

Re[〈φi|U†(~θ∗)AkU(~θ∗) |φj〉] = 〈φ+ij |Ak |φ+ij〉

− 〈φ−ij |Ak |φ−ij〉

Im[〈φi|U†(~θ∗)AkU(~θ∗) |φj〉] = 〈φi+ij |Ak |φi+ij 〉

− 〈φi−ij |Ak |φi−ij 〉 ,

(62)

and similarly for Bk. Thus we can calculate

〈φi|U†(~θ∗)a†kU(~θ∗) |φj〉 and hence 〈Em| a†k |En〉 by mea-

suring the expectation values of Ak and Bk for states|φ±ij〉 and |φ

i±ij 〉.

Note that based on calculation of the Green’s function,Rungger et al. [96] implemented dynamical mean-fieldtheory (DMFT) calculation on two-sites DMFT model byusing a superconducting system and a trapped ion sys-tem. Also, in the work by Ibe et al. [104], the accuracy ofthe calculation of transition amplitudes are compared forthe overlap method and the contraction VQE methods indetail.

E. Variational circuit recompilation

Circuit recompilation aims to approximate a givenquantum circuit with ones that are compatible with apractical experiment hardware. Concerning noisy gatesor restricted set of realisable gates, the compiler runs toreduce the circuit noise or the implementation cost, asshown in Fig. 9. For example, an arbitrary two-qubitunitary is generally not directly supported on a practicalquantum hardware and we need to compile the unitaryinto a sequence of realisable gates. While a naive decom-position of the unitary may induce too many unnecessarygates for hardware with specific topological structures,finding efficient and simple decomposition of quantumcircuits is vital for near-term quantum computing.

FIG. 9. Schematic figure for circuit recompilation algorithms.A variational quantum circuit with hardware constraints triesto approximate the target quantum circuit.

15

Denote the target unitary as UT and the variational

circuit as U(~θ), which consists of gates compatible withthe hardware. Circuit recompilation is to tune the pa-

rameters ~θ so that UT ≈ U(~θ). We can mathematicallydefine a metric of the distance between UT and U(~θ) viathe average gate infidelity (AGI) [105, 106]

CI(~θ) = 1−∫dϕ| 〈ϕ|U†TU(~θ) |ϕ〉 |

2 (63)

where pure states ϕ are randomly chosen according to theHaar measure. By construction, AGI indicates how dif-ferent two unitary gates are on average for randomly sam-pled pure states, and it vanishes when the circuit is per-fectly re-compiled. The AGI method has been applied forcompiling high fidelity CNOT gate with cross-resonancegate suffering from crosstalk and single qubit operations.Furthermore, a high-fidelity four-qubit syndrome extrac-tion circuit was recompiled to be achievable with simul-taneous cross resonance drives under crosstalk [106].

Another related cost function [105] is

CHS(~θ) = 1−1

d|Tr(U†TU(~θ))|

2, (64)

which can be efficiently calculated with the ‘Hilbert-Schmidt test’ circuit [105] by using two copies of states.In particular, we have

1

d2Tr(U†TU

~θ)|2 = | 〈Ψ+|UT ⊗ U∗(~θ) |Ψ+〉 |2, (65)

where |Ψ+〉 = d−1∑i |i〉⊗|i〉 is the maximally entangled

state with d being the dimension of the system. Then,

Tr(U†TU~θ)|2/d2 can be evaluated by applying UT ⊗U∗(~θ)

to |Ψ+〉 and measure the probability to observe |Ψ+〉.Note that CHS(~θ) and AGI is related [107, 108],

CHS(~θ) =d+ 1

dCI(~θ). (66)

Thus CHS(~θ) can be used as an alternative cost function

for circuit recompilation. Note that CI(~θ) and CHS(~θ)generally exponentially vanish for an increasing systemsize. This problem is solved by using the weighted av-erage of this global cost function and the correspondinglocal cost function (see Appendix D for an explanationof global and local costs). Then simple 9-qubit unitarieswere successfully recompiled for noiseless simulator andfor the Rigetti and IBM experimental processor.

Meanwhile, circuit recompilation can be implementedfor specific input states, which may reduce the optimisa-tion complexity [105, 109]. The goal is to find the recom-

piled circuit U(~θ) so that UT |ψin〉 ≈ U(~θ) |ψin〉 holdsfor the state |ψin〉 and the target unitary UT . When|ψin〉 is a product state, we can easily define the Hamil-tonian Hin whose ground state is |ψin〉. For example, for|ψin〉 = |00 . . . 0〉, we can set Hin = −

∑j Zj , where Zj is

the Pauli Z operator acting on the j th qubit. Then, by

minimising the expectation value of Hin for the trial state

|ϕ(~θ)〉 = U†(~θ)UT |ψin〉, we can obtain U(~θ) such thatU†(~θ)UT |ψin〉 ≈ |ψin〉 and hence UT |ψin〉 ≈ U(~θ) |ψin〉.We can leverage the conventional VQE method or varia-tional imaginary time simulation for the optimisation andthe fidelity of the re-compiled state is lower-bounded as

FR ≥ 1−δ

E1 − E0, (67)

where δ = 〈ϕ(~θ)|Hin|ϕ(~θ)〉 − E0 is the deviation of thecost function from the ground state energy, and E0 andE1 are the ground state and first excited state energy.Note that E0 = −Nq and E1 = 2 − Nq when the uni-tary acts on Nq qubits. This algorithm successfully re-compiled a unitary operator involving 7 qubit systeminto another quantum circuit with different topology anddramatically reduced the number of two-qubit gates to72 from 144, while the number of single-qubit gates in-creased to 77 from 42 [109].

F. Variational-state quantum metrology

Quantum metrology aims to discover the optimal setupfor probing a parameter with the minimal statistical shotnoise [110–112]. The basic setup of quantum metrology isas follows — firstly, we prepare the initial probe state |ψp〉and evolve the state under the Hamiltonian H(ω) withω being the target parameter. After time t, the probestate can be described as |ψp(ω, t)〉, which is measuredand analysed to extract the information of ω. Typically,ω can be a magnetic field, for example, with Hamiltonian

H(ω) = ω∑j σ

(j)z with σ

(j)z being the Pauli Z operator,

and the measurement is a Ramsey type measurement.When separable states are used as probes, the statisti-cal error of the parameter behaves as δω ∝ 1/

√Nq with

Nq being the number of qubits. This scaling is called thestandard quantum limit (SQL) [112]. Notably the scalingcan be improved by using entangled states, such as GHZstates, symmetric Dicke states, and squeezed states. Inthe absence of noise or for specific types of noise in theevolution under the Hamiltonian, the optimal strategyhas been revealed. For example, with no environmen-tal noise, the optimal probe state has been proved tobe the GHZ state which achieves the Heisenberg limitδω ∝ 1/Nq [111–113]. However, in the presence of gen-eral types of noise, analytical arguments about the opti-mal strategy are usually very hard.

Variational-state quantum metrology is to use a quan-tum computer to find the optimal quantum state forquantum metrology with noisy hardware. Differentproposals have been studied with either general vari-ational quantum circuits [114] or specific experimen-tal setups [115], i.e., optical tweezer arrays of neutralatoms [116, 117]. In general, suppose the initial probe

state is created on a quantum device, as |ϕp(~θ)〉. Byevolving the state under the Hamiltonian H(ω) followed

16

with a proper measurement, we can define a cost func-tion to reflect the metrology performance. In particu-lar, we can either use the quantum Fisher information(QFI) [114] or the spin squeezing parameter [115, 118].Here, we focus on the QFI, which characterises the min-imum uncertainty of the estimated parameter as

δω ≥ 1√NsFQ[ρP (ω, t)]

, (68)

where FQ[ρP (ω, t)] is the QFI for the noisy output stateρP (ω, t) after the evolution time t and Ns is the numberof samples. Denote T = Nst to be the effective total timeof all Ns samples, we can define a cost function as

CME(~θ, t) =T

tFQ[ρP (ω, t, ~θ)]. (69)

For fixed T , we aims to optimise ~θ and t to maximise

CME(~θ, t). We show the schematic of this method inFig. 10. Although QFI of mixed states cannot be directlyevaluated in an efficient way, classical Fisher information(CFI) defined for a fixed measurement basis lower boundsQFI and it can be computed efficiently. We note that CFIis equivalent to QFI when the measurement basis is op-timal, so QFI can in principle be obtained by optimisingthe measurement.

With variational-state quantum metrology, a highlyasymmetric state has been discovered for a 9-qubit sys-tem that outperforms previous results [114]. Interest-ingly, even though the Hamiltonian and the noise modelis symmetric under the permutation of qubits, there is asymmetry breaking for the optimal solution. Note thatunlike conventional analytical approaches employed inquantum metrology, we do not have to know the noisemodel of the quantum device to obtain the optimisedstate. Recently, this algorithm was generalised to multi-parameter estimation [119].

FIG. 10. Schematic figures for variational quantum-statemetrology. After the probe state is prepared by the vari-ational quantum circuit, it evolves under the HamiltonianH(ω) with environmental noise, and is measured to evaluatethe cost function.

G. Variational quantum algorithms for quantumerror correction

Quantum error correction (QEC) makes use of largernumber of physical qubits to encode logical qubits toprotect them against physical errors. In general, con-ventional formulation of QEC does not take into accountthe experimental implementation of the code. However,in the NISQ era, for example, the set of possible op-erations and the qubit topology are restricted for eachphysical hardware. Therefore, hardware-friendly imple-mentation of QEC, tailored to actual experiments, is cru-cial for near-term quantum computers [120–122]. Here,we illustrate two examples [121, 122] for realising QECon NISQ computers.

1. Variational circuit compiler for quantum error correction

This variational circuit compiler is for automaticallydiscovering the optimal quantum circuit satisfying user-specified requirements for a given QEC code [121]. Typ-ically, such requirements tend to come from hardwareproperties, such as available gate sets, limited topology,and achievable error rate. More concrete example maybe two-qubit gate implementation, e.g., superconductingqubits employing CNOT gate, and ion trap systems us-ing Møren-Sørensen gate. We prepare an ansatz unitary

U(~θ), or a process E(~θ), that takes account of gate noiseand reflects the requirements. The compiler is to opti-

mise the parameters ~θ so that the ansatz emulates theencoded target state |ψ0〉L for a given QEC code.

The essential point of the compiler is to design theHamiltonian whose ground state is the target state. Thenthe target state can be obtained via conventional VQEor variational imaginary time simulation algorithm. Gen-erally, the code space is defined by a set of commutingPauli operators, so- called stabiliser generators. For ex-ample, when we consider the three qubit code, the logicalstate |ψ〉three = α |000〉+β |111〉 is an eigenstate of Z1Z2and Z2Z3 with eigenvalue 1. Suppose the target logicalstate |ψ0〉L is determined by the stabiliser generator set{Gk} with Gk |ψ0〉L = |ψ0〉L. For determining a particu-lar logical qubit state, we can choose an additional logicaloperator, ML = |ψ0〉L 〈ψ0|L−|ψ⊥0 〉L 〈ψ⊥0 |L, which can bedecomposed as a linear combination of the logical I, X,Y , Z operators. Then the logical state |ψ0〉 is the uniqueground state of the Hamiltonian

HL = −∑k

akGk − a0ML (70)

with energy E0 = −(∑k ak + a0), where ak, a0 > 0.

When using the variational algorithm to minimise theaverage energy, an approximation of the encoding circuitis found. Suppose the energy of the optimally discoveredstate is Edis, the fidelity between the discovered state

17

and the target logical state is lower bounded as

f ≥ 1− (Edis − E0)/a, (71)

where a = min{ak, a0}. This inequality ensures that ifEdis is sufficiently close to E0, the discovered encodingcircuit approximates the target QEC code well. The algo-rithm has been numerically tested for the five and sevenqubit codes with different available gate sets and undernoise free and noisy circuits [121].

2. Variational quantum error corrector (QVECTOR)

Variational quantum error corrector (QVECTOR)) isfor discovering a device-tailored quantum error correctingcode [122]. Different from the compiler for preparing atarget logical state [121], QVECTOR aims to discover theoptimal encode circuit that preserves the quantum stateunder noise. As shown in Fig. 11, the circuit US is used

for preparing the to-be-encoded k-qubit state |ϕ〉, V (~θ1)and V †(~θ1) are the noisy encoding and decoding circuits

on n ≥ k qubits, and W (~θ2) is noisy gates for state re-covery, which operates on n + r qubits. This quantumcircuit corresponds to creating the [n, k] quantum codes

with r additional qubits. The parameters ~θ1 and ~θ2 areoptimised to maximise the average code fidelity,

CQV (~θ1, ~θ2) =

∫ψ∈C〈ψ| ER(~θ1, ~θ2)

(|ψ〉 〈ψ|

)|ψ〉 dψ,

(72)

where ER(~θ1, ~θ2) is the encoding, recovery, and decodingoperations V (~θ1), W (~θ2), and V

†(~θ1), and the integra-tion is calculated following the Haar distribution with

|ψ〉 = US |0〉⊗k. In practice, we can set US to be theunitary 2-design for efficiently calculating the averagefidelity [123]. With numerical simulation, QVECTORlearned the three qubit code [124] under the phase damp-ing noise, resulting in a six times longer T2 than conven-tional methods. In the presence of amplitude and phasedamping noise, QVECTOR learned the quantum codeoutperforming the five qubit stabilizer code.

FIG. 11. Schematic figure of the variational circuit for QVEC-TOR.

H. Dissipative-system variational quantumeigensolver

Dissipative-system variational quantum eigensolver(dVQE) is for obtaining the non-equilibrium steady state(NESS) in an open quantum system. In practice, quan-tum systems inevitably interact with their environments,and quantum states decohere owing to the noise. There-fore, the ability to simulate open quantum systems is in-dispensable for studying practical quantum phenomena.Notably, investigating NESS of open quantum systems isvery important, e.g., in revealing the transport mech-anism in nano-scale devices such as single-atom junc-tions [125].

The time independent Markovian open quantum dy-namics can be described by the Lindblad master equa-tion,

d

dtρ = −i[H, ρ] + L(ρ), (73)

where ρ is the system state, H is the Hamiltonian, L(ρ) =∑k(2LkρL

†k − L

†kLkρ − ρL

†kLk) and Lk is a Lindblad

operator. Denoting L′(ρ) = −i[H, ρ] + L(ρ), the non-equilibrium steady state corresponds to the state withL′(ρ) = 0. To find the steady state, we first vectorise thestate ρ =

∑nm ρnm |n〉 〈m| to

|ρ〉〉 = 1N∑nm

ρnm |n〉 |m〉 , (74)

where N is a normalisation factor. With vectorised L′,the steady state satisfies L′|ρness〉〉 = 0 and hence wehave

〈〈ρness|L′†L′|ρness〉〉 = 0, (75)

where

L′ =(− i(H ⊗ I − I ⊗HT ) +Dv

),

Dv =∑k

(Lk ⊗ L∗k −

1

2L†kLk ⊗ I −

1

2I ⊗ LTk L∗k

).

(76)

Here A∗ denotes the complex conjugate of operator A.

As |ρ(~θ)〉〉 is a vector, we can express this vector rep-resentation of the quantum state through using a puretrial state on a variational quantum circuit by imposingproper constraints so that the corresponding density ma-

trix ρ(~θ) will be physical, i.e., positive semi-definitenessand hermiteness have to be satisfied. The trace of thestate needs to be unity but this condition will be consid-ered when the expectation value of a given observable ismeasured. We can prepare the vectorised physical stateon a variational quantum circuit as

|ρ(~θ)〉〉 = U(~θ1)⊗ U∗(~θ1)|ρd(~θ2)〉〉,

|ρd(~θ2)〉〉 =1

N∑j

αj(~θ2) |j〉s ⊗ |j〉a ,(77)

18

where ~θ ≡ {~θ1, ~θ2}, αj(~θv) > 0 and∑j α(

~θ2) = 1, |j〉 isin the computational basis, and the corresponding den-

sity matrix is ρ(~θ) =∑j αjU(

~θ1) |j〉s 〈j|s U†(~θ1). Here,N =

√∑j α

2j ensures the normalisation. The subscripts

a and s denote system and ancilla, respectively. We referto Yoshioka et al. [126] for the detailed ansatz construc-

tion. By preparing a trial state |ρ(~θ)〉〉, and minimisingthe cost function CNE(~θ) = 〈〈ρ(~θ|L′†L′|ρ(~θ)〉〉, we canobtain an approximation of NESS. After the optimal pa-

rameters ~θ(op)1 and

~θ(op)2 are found, we can measure the

expectation value of any given observable O for ρ(~θ) by

randomly generating the state U(~θ(op)1 ) |j〉 with probabil-

ity αj(~θ(op)2 ), and averaging the measurement outcome.

This method was demonstrated for 8-qubit dissipativeIsing model with a 16-qubit classical simulation.

I. Other applications

There are other applications, such as VQAs for non-linear problems [127], fidelity estimation [128], factor-ing [129], singular value decomposition [76, 130], quan-tum foundations [131], circuit QED simulation [132], andGibbs state preparation [130, 133, 134].

IV. VARIATIONAL QUANTUM SIMULATION

In this section, we review the variational quantum sim-ulation algorithms for simulating dynamical evolution ofquantum systems and the application in linear algebratasks, Gibbs state prepration, and evaluating Green’sfunction.

A. Variational quantum simulation algorithm fordensity matrix

We first show how to generalise the simulation algo-rithm for real and imaginary time evolution from purestates to mixed states [26]. The main idea is again to con-sider a parametrised representation of mixed states andmap the dynamics to the evolution of the parameters.As we are considering mixed states, only McLachlan’svariational principle applies, which leads to evolution ofparameters with information determined by the densitymatrix. Although conventional quantum computers op-erate on pure states, we can also represent mixed stateswith their purified pure states by using ancilla qubits.

1. Variational real time simulation for open quantumsystem dynamics

In practice, a quantum system interacts with its envi-ronment, so open quantum system simulation algorithms

are useful for investigating practical quantum phenom-ena. Here we aim to simulate real time evolution of openquantum systems described by the Lindblad master equa-tion dρ/dt = L(ρ) with L(ρ) being a super-operator onthe state as in Eq. (73). By parametrising the state as

ρ(~θ(t)) with real parameters, and applying McLachlan’s

variational principle δ‖dρ(~θ(t))/dt − L(ρ)‖ = 0, we canobtain a similar equation of the parameters as the onefor closed systems in Eq. (11) as∑

j

Mk,j θ̇j = Vk, (78)

where

Mk,j = Tr

[(∂ρ(~θ(t))

∂θk

)†∂ρ(~θ(t))

∂θj

]Vk = Tr

[(∂ρ(~θ(t))

∂θk

)†L(ρ)

].

(79)

Note that the evaluation of M and V can be reduced tothe computation of terms like c · Re(Tr[eiϕρ1ρ2]), withρ1 and ρ2 being two quantum states, and c, ϕ ∈ R [26].By encoding the mixed state via a purified pure state,this term can be evaluated via the SWAP test circuit.Note that, in order to simulate open system of Nq qubits,we first need 2Nq qubits for representing its purification.We also need two copies of the purification for evaluat-ing M and V , so we need in total 4Nq qubits to simulateopen quantum system dynamics on Nq qubits. We reviewshortly that an alternative algorithm that simulates thestochastic Schördinger equation which enables the simu-lation of Nq-qubit open systems on an Nq-qubit quantumhardware.

2. Variational imaginary time simulation for a densitymatrix

The variational quantum simulation algorithm can beapplied for simulating imaginary time evolution of den-sity matrices as well [26], which is defined as

ρ(τ) =e−Hτρ0e

−Hτ

Tr[e−Hτρ0e−Hτ ], (80)

where ρ0 is the initial state. The time derivative equationfor ρ(τ) is

dρ(τ)

dτ= −{H, ρ(τ)}+ 2 〈H〉 ρ(τ), (81)

where {A,B} = AB + BA. By applying McLach-lan’s variational principle as δ‖dρ/dτ + {H, ρ(τ)} −2 〈H〉 ρ(τ)‖ = 0, we have the evolution of the parame-ters as ∑

j

Mk,j θ̇j = Wk, (82)

19

where

Wk = −Tr[(

∂ρ(~θ(t))

∂θk

){H, ρ(~θ(τ))}

]. (83)

We note that Wk can be computed similarly to Vk.

B. Variational quantum simulation algorithms forgeneral processes

In this section, we review the variational quantum sim-ulation algorithms for general processes, including thegeneralised time evolution, its application in solving lin-ear algebra tasks, and simulating open system dynamics.

1. Generalised time evolution

Apart from real and imaginary time evolution, we con-sider the generalised time evolution defined as

A(t)d

dt|u(t)〉 = |du(t)〉 , (84)

where |du(t)〉 =∑k Bk(t) |u′k(t)〉, and A(t) and Bk(t) are

general (possibly non-Hermitian) sparse matrices thatmay be efficiently decomposed as sums of Pauli opera-tors, and each |u′k(t)〉 could be either |u(t)〉 or any fixedstates. The quantum states |u(t)〉 and |u′k(t)〉 can beunnormalised and we assume a parametrisation of the

states as |u(~θ)〉 = α(~θ1) |ψ(~θ2)〉, where ~θ = {~θ1, ~θ2}, α(~θ1)is a classical coefficient, and |ψ(~θ2)〉 is a quantum stategenerated on quantum computer. By using MchLaclan’svariational principle

δ‖A(t) ddt|u(~θ(t))〉 −

∑k

Bk(t) |u′k(t)〉 ‖ = 0, (85)

we obtain the evolution of the parameters similar toEq. (11) as ∑

j

M̄k,j θ̇j = V̄k. (86)

Each matrix element M̄k,j or V̄k could be expanded asa sum of terms that can be efficiently measured via aquantum circuit. We refer to Endo et al. [135] for details.

The real time evolution corresponds to A(t) = 1and |du(t)〉 = −iH |u(t)〉 and the imaginary time evo-lution corresponds to A(t) = 1 and |du(t)〉 = −(H −〈u(t)|H|u(t)〉) |u(t)〉 for Hamiltonian H. Therefore, thegeneralised time evolution unifies real and imaginarytime evolution. In addition, the generalised time evolu-tion describes a general first-order differential equationswith non-Hermitian Hamiltonians, which may have ap-plications in non-Hermitian quantum mechanics. In thefollowing, we show its application in solving linear alge-bra tasks and simulating the stochastic Schördinger equa-tion.

2. Matrix multiplication and linear equations.

Now, we explain how we can apply the algo-rithm for generalised time evolution to realise matrix-multiplication and to solve linear systems of equa-tions [135]. This is an alternative method for linear alge-bra discussed in Sec. III C [20, 76]. For sparse matrixMand a (unnormalised) state vector |u0〉, we aim to obtain

|uM〉 =M|u0〉 , |u−1M 〉 =M−1 |u0〉 , (87)

for matrix-multiplication and solving linear systems ofequations. In terms of matrix-multiplication, by setting|uM(t)〉 = C(t) |u0〉 with C(t) = tTM+(1−

tT )I, we have

|uM(0)〉 = |u0〉 being the given vector and |uM(T )〉 =M|u0〉 = |uM〉 being the solution. The time dependentstate |uM(t)〉 follows a generalised time evolution

d

dt|uM(t)〉 = D |u0〉 (88)

with D = (M−I)/T , which corresponds to the case withA(t) = I and |du(t)〉 = |u0〉 in Eq. (84). For solving linearsystems of equations, by considering C(t) |uM−1(t)〉 =|u0〉 with C(t) = tTM+ (1−

tT )I, we have |uM−1(0)〉 =

|u0〉 being the given vector and |uM−1(T )〉 =M−1 |u0〉 =|uM−1〉 being the solution. The state |uM−1(t)〉 followsthe evolution equation as

C(t)d

dt|uM−1(t)〉 = −D |uM−1(t)〉 (89)

which corresponds to the generalised time evolution ofEq. (84) with A(t) = C(t) and |du(t)〉 = −D |u(t)〉.Therefore, the ability of efficiently simulating generalisedtime evolution enables us to solve these two linear algebraproblems.

3. Open system dynamics

Now we show how to simulate open quantum sys-tem dynamics with the variational algorithm of gener-alised time evolution [135]. Instead of directly simulat-ing the Lindblad master equation defined in Eq. (73), weconsider its alternative representation via the stochas-tic Schrödinger equation, where the whole evolution is amixture of pure state trajectories [136]

d |ψc(t)〉 =

(−iH − 1

2

∑k

(L†kLk − 〈L†kLk〉)

)|ψc(t)〉 dt

+∑k

[(Lk |ψc(t)〉||Lk |ψc(t)〉 ||

− |ψc(t)〉)dNk

].

(90)Here |ψc(t)〉 is the state of each trajectory, d |ψc(t)〉 =|ψc(t+ dt)〉−|ψc(t)〉, and dNk is a random variable whichtakes either 0 or 1 and satisfies dNkdNk′ = δkk′dNkand E[dNk] = 〈ψc(t)|L†kLk |ψc〉 dt. This implies that

20

the state |ψc〉 jumps to Lk |ψc〉 /‖Lk |ψc〉 ‖ with a prob-ability E[dNk] when the state evolves from time t tot + dt. Each trajectory can be regarded as a contin-uous evolution of the state continuously measured by

{O0 = I −∑k L†kLkdt, Ok = L

†kLkdt}. When the mea-

surement corresponds to O0, the state evolves under thegeneralised time evolution of Eq. (84) with A = I and

|du(t)〉 =

[−iH − 1

2

∑k

(L†kLk − 〈L†kLk〉)

]|u(t)〉 . (91)

This process can