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Quantum computers can exploit a Hilbert space whose dimension increases exponentially with the number of qubits. In experiment, quantum supremacy has recently been achieved by the Google team by using a noisy intermediate-scale quantum (NISQ) device with over 50 qubits. However, the question of what can be implemented on NISQ devices is still not fully explored, and discovering useful tasks for such devices is a topic of considerable interest. Hybrid quantum-classical algorithms are regarded as well-suited for execution on NISQ devices by combining quantum computers with classical 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. In this article, we review the basic results for hybrid quantum-classical algorithms and quantum error mitigation 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**

1. Introduction 2

2. Basic Variational Quantum Algorithms 2

2.1 Variational quantum eigensolver 3

2.2 Real and imaginary time evolution quantum simulator 4

3. Variational Quantum Optimisation 5

3.1 Quantum approximate optimisation algorithms 6

3.2 Variational algorithms for machine learning 6

3.2.1 Quantum circuit learning 7

3.2.2 Data-driven quantum circuit learning 7

3.2.3 Quantum generative adversarial networks 8

3.2.4 Quantum autoencoder for quantum data compression 8

3.2.5 Variational quantum state eigensolver 9

3.3 Variational algorithms for linear algebra 9

3.4 Excited state-search variational algorithms 10

3.4.1 Overlap-based method 10

3.4.2 Quantum subspace expansion 11

3.4.3 Contraction VQE methods 11

3.4.4 Calculation of the Green's function 11

3.5 Variational circuit recompilation 12

3.6 Variational-state quantum metrology 13

3.7 Variational quantum algorithms for quantum error correction 13

3.7.1 Variational circuit compiler for quantum error correction 14

3.7.2 Variational quantum error corrector (QVECTOR) 14

3.8 Dissipative-system variational quantum eigensolver 14

3.9 Other applications 15

4. Variational Quantum Simulation 15

4.1 Variational quantum simulation algorithms for density matrix 15

4.1.1 Variational real time simulation for open quantum system dynamics 15

4.1.2 Variational imaginary time simulation for a density matrix 15

4.2 Variational quantum simulation algorithms for general processes 16

4.2.1 Generalised time evolution 16

4.2.2 Matrix multiplication and linear equations 16

4.2.3 Open system dynamics 16

4.3 Gibbs state preparation 17

4.4 Variational quantum simulation algorithms for estimating the Green's function 17

4.5 Other applications 17

5. Quantum Error Mitigation 17

5.1 Extrapolation 18

5.1.1 Richardson extrapolation 18

5.1.2 Exponential extrapolation 19

5.1.3 Methods to boost physical errors 19

5.1.4 Mitigation of algorithmic errors 20

5.2 Least square fitting for several noise parameters 20

5.3 Quasi-probability method 21

5.4 Quantum subspace expansion 22

5.5 Symmetry verification 22

5.6 Individual error reduction 23

5.7 Measurement error mitigation 23

5.8 Learning-based quantum error mitigation 24

5.8.1 Quantum error mitigation via Clifford data regression 24

5.8.2 Learning-based quasi-probability method 24

5.9 Stochastic error mitigation 25

5.10 Combination of error mitigation techniques 25

5.10.1 Symmetry verification with error extrapolation 25

5.10.2 Quasi-probability method with error extrapolation 26

5.10.3 Symmetry verification with quasi-probability method 26

5.10.4 Combining quasi-probability, symmetry verification and error extrapolation 27

6. Conclusion 27

Acknowledgments 27

A. Derivation of Eq. (11) for Variational Quantum Simulation 27

B. Hadamard Test and Quantum Circuits for Variational Quantum Simulation 28

C. SWAP Test and Destructive SWAP Test 28

D. Methodologies for Optimisation 29

D·1 Local cost function 29

D·2 Hamiltonian morphing optimisation 30

E. Subspace Expansion 30

References 30

As the dimension of the Hilbert space of a quantum system increases exponentially with respect to the system size, general quantum systems are, in principle, hard to simulate on a classical computer. For example, systems manipulating tens to hundreds of qubits have been believed to be classically intractable, and they have been proposed for demonstrating quantum advantages over classical supercomputers in the so-called task of “quantum supremacy”.^{1}^{)} In October 2019, Google announced that they had successfully demonstrated quantum supremacy with a 53 qubit device, named Sycamore.^{2}^{)} The dimension of the computational state-space is as large as ^{3}^{)} classical simulation of a general quantum circuit will certainly become an intractable task as we increase the gate fidelity, the gate depth, or the number of qubits.

While the tasks considered in quantum supremacy are generally mathematically abstract problems, ultimately the field must progress to demonstrate true quantum advantage, i.e., to solve a problem of practical value with superior efficiency using a quantum device. Current quantum hardware only incorporates a small number (tens) of qubits with a non-negligible gate error rate, making it insufficient for implementing conventional quantum algorithms such as Shor's factoring algorithm,^{4}^{)} the phase estimation algorithm,^{5}^{)} and Hamiltonian simulation algorithms.^{6}^{)} These generally require one to accurately control millions of qubits when taking account of fault-tolerance.^{7}^{)}

Before realising a universal fault-tolerant quantum computer, a more feasible scenario for current and near-term quantum computing is the so-called noisy intermediate-scale quantum (NISQ) regime,^{8}^{)} where we control tens to thousands of noisy qubits with gate errors that may be on the order of ^{9}^{–}^{15}^{)} Intuitively, because a large portion of the computational burden is processed on the classical computer, fully coherent deep quantum circuits may not be required. As both quantum and classical computers are used, such simulation methods are called hybrid quantum-classical algorithms. In addition, to compensate computation errors, quantum error mitigation techniques can be used by a post-processing of the experiment data.^{14}^{,}^{16}^{,}^{17}^{)} Since quantum error mitigation does not necessitate encoding of qubits as full error correction does, it thus contributes to a huge saving of qubits, which is vital for NISQ simulation.

In this review paper, we aim to summarise the most basic ideas of hybrid quantum-classical algorithms and quantum error mitigation techniques. In Sect. 2, we introduce the basic algorithms — the variational quantum eigensolver and variational quantum simulation — for finding a ground state or simulating dynamical evolution of a many-body Hamiltonian.^{9}^{,}^{14}^{,}^{18}^{,}^{19}^{)} In Sect. 3, we show how the variational quantum eigensolver algorithm can be extended for general optimisation problems including machine learning problems, linear algebra problems, excited energy spectra, etc.^{20}^{–}^{33}^{)} Meanwhile, we show in Sect. 4 that the variational quantum simulation algorithm may be extended as well for open systems,^{19}^{,}^{34}^{)} general processes,^{34}^{)} thermal states,^{19}^{)} and calculating the Green's function.^{35}^{)} Finally, in Sect. 5, we show several error mitigation methods for suppressing errors in NISQ computing.^{14}^{,}^{16}^{,}^{17}^{,}^{36}^{–}^{47}^{)} This review does not cover the application of NISQ computers in solving specific physics problems and we refer to McArdle et al.^{48}^{)} and Cao et al.^{49}^{)} for reviews for its application in quantum computational chemistry, to Bauer et al.^{50}^{)} in quantum materials, etc.

Since NISQ devices can only apply a relatively shallow circuit on a limited number of qubits, conventional quantum algorithms may not be implemented on NISQ devices. Here we consider hybrid quantum-classical algorithms tailored to NISQ computing. Because the algorithms generally use parametrised quantum circuits and variationally update the parameters, they are also called variational quantum algorithms (VQAs).

For implementing VQAs,^{9}^{,}^{11}^{,}^{12}^{)} we first consider the parametrised trial wave function as

Figure 1. Schematic of variational quantum algorithms. The ansatz state

Although there exist a large number of VQAs, they can be generally classified into two categories: variational quantum optimisation (VQO) and variational quantum simulation (VQS). VQO involves optimising parameters under a cost function. For example, when we minimise the energy of the state, i.e., the expectation value of the given Hamiltonian as a cost function, the cost function after optimisation approximates the ground state energy. The corresponding state also approximates the ground state. This is the so-called variational quantum eigensolver^{9}^{,}^{12}^{)} and other VQO algorithms can be similarly designed by properly changing the cost function to other metrics. While variational quantum optimisation aims to optimise a static target cost function, VQS aims to simulate a dynamical process, such as the Schrödinger time evolution of a quantum state.^{14}^{,}^{19}^{)} VQS algorithms can also be applied for optimising a static cost function, such as variational imaginary time simulation, or studying general many-body physics problems. The distinction between variational quantum optimisation and variational quantum simulation is not absolute, and algorithms for problems in one category may be adapted for those in the other category. Before showing how specific VQO or VQS algorithms work for specific tasks, in this section, we first illustrate the most basic VQO algorithm, a variational quantum eigensolver for finding ground state energy, and the most basic VQS algorithms for simulating real and imaginary time simulation.

The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for computing the ground state and the ground state energy of a Hamiltonian *H* of interest. In the seminal work by Peruzzo et al.,^{9}^{)} the VQE algorithm was theoretically proposed and experimentally demonstrated for finding the ground state energy of the HeH^{+} molecule using a two-qubit photonic quantum processor. We note that the conventional approach for finding eigenstates of a Hamiltonian with a universal quantum computer is by adiabatic state preparation and quantum phase estimation (QPE).^{51}^{)} We refer to McArdle et al.^{48}^{)} and Cao et al.^{49}^{)} for the review.

The VQE algorithm relies on the Rayleigh–Ritz variational principle, i.e., for any parameterised quantum state *H* and the minimisation is over all parameters ^{12}^{)} As we will explain shortly, we can efficiently calculate

Now, we explain how to measure *H*. As Pauli operators and products of them *j*-th orbital, and ^{52}^{,}^{53}^{)} The Hamiltonian consists of a polynomial number of terms consisting of products of fermionic operators. They can be further mapped to qubit operators via encoding methods such as Jordan–Wigner, parity, and Bravi–Kitaev encodings.^{51}^{,}^{54}^{,}^{55}^{)} For example, the Jordan–Wigner transformation is defined as

By assuming a Pauli decomposition of the Hamiltonian

After we have obtained *n*-step and the ^{56}^{)} direct search is believed to be more robust to physical noise, which may necessitate less repetitions.^{57}^{)} We refer to McArdle et al.^{48}^{)} for the review of classical optimisation algorithms.

Figure 2. Schematic of the variational quantum eigensolver. The expectation values of the Pauli operators

Whether the VQE algorithm works also depends on the choice of the ansatz. To have an efficient quantum simulation algorithm, we need to use a suitable ansatz for the problem. If the ansatz state cannot express the solution, e.g., when the solution is a highly entangled state but the ansatz can only generate low entangled states, it cannot find the correct solution. In literature, several different types of ansätze are proposed for different purposes. For example, the unitary coupled cluster ansatz is known as a suitable physically inspired ansatz for electronic structure problems in chemistry.^{9}^{,}^{58}^{,}^{59}^{)} However, the unitary coupled cluster ansatz generally necessitates a complicated form of quantum circuit with gates applied on multiple number of qubits, where each multi-qubit gate could be decomposed as a sequence of two-qubit gates, and the number of multi-qubit gates is quadratic to the number of qubits and the number of electrons. Since the unitary coupled cluster ansatz involves many general two-qubit gates, it may be hard to implement on noisy quantum devices with short coherence time and limited connectivity. This problem might be circumvented by leveraging so-called “hardware efficient ansätze”. This family of ansätze might be more experimentally feasible,^{9}^{,}^{10}^{,}^{13}^{)} because they are constructed based on realisable demands on connectivity and gate operations that correspond to real quantum devices. However, a hardware efficient ansatz does not reflect the details of the simulated quantum system, and it has been shown that exponentially vanishing gradients (so-called barren plateaus) are liable to occur for randomly initialised parameters.^{60}^{)} There are several other methods proposed to circumvent the vanishing gradient problem.^{61}^{–}^{64}^{)} We refer to McArdle et al.^{48}^{)} for a more detailed discussion for ansatz construction.

The disadvantage of the VQE method is that the correctness of the solution relies on the heuristic choice of the ansatz and the optimisation may be caught by a local instead of global minimum. Furthermore, the total number of measurements is *ϵ* due to shot noise,^{9}^{,}^{12}^{)} which is quadratically worse than the conventional QPE algorithm that uses universal quantum computers.

The VQE algorithm has been experimentally demonstrated by several groups.^{9}^{,}^{10}^{,}^{65}^{–}^{70}^{)} To date, the hydrogen chain was simulated with a 12-qubit superconducting system.^{71}^{)}

Now we introduce the basic algorithms for variational quantum simulation (AQS), in particular, for simulating real^{14}^{)} and imaginary^{18}^{)} time evolution. The real time evolution of a quantum system can be described via the Schrödinger equation as *H* is the Hamiltonian and *t* is obtained by applying the unitary to the state.^{6}^{,}^{72}^{–}^{74}^{)} The circuit depth generally increases polynomially with respect to the evolution time *t*. Instead, variational quantum simulation algorithms assume that the quantum state

Different variational principles can be used to have different evolution equations of the parameters. The three most conventional variational principles are — The Dirac and Frenkel variational principle,^{75}^{,}^{76}^{)} McLachlan's variational principle,^{77}^{)} and the time-dependent variational principle.^{78}^{,}^{79}^{)} The Dirac and Frenkel variational principle is not suitable for variational quantum simulation because the equation of the parameters may involve complex solutions, which contradicts with the requirement that parameters are real. Although the other two variational principles both give real solutions, it is shown that the time-dependent variational principle could be more unstable and it cannot be applied for evolution of density matrices and imaginary time evolution. On the contrary, McLachlan's principle generally produces stable solutions and it is also applicable to all the other scenarios beyond real time simulation. We refer a detailed study of the three variational principles to Yuan et al.^{19}^{)}

Now, we focus on McLachlan's variational principle^{77}^{)} and show how to derive the evolution of the parameters that effectively simulates the time evolution. McLachlan's principle requires one to minimise the distance between the ideal evolution and the evolution of the parametrised trial state as *δ* is the variation over the derivative of the parameters

Next, we describe the variational imaginary time simulation algorithm.^{18}^{)} The normalised Wick-rotated Schrödinger equation^{80}^{)} is obtained by replacing *t* in Eq. (9) with = ^{18}^{,}^{19}^{)} Following the same procedure for real time evolution, we first make use of McLachlan's principle, *M* defined in Eq. (12) and *C* defined by *C* vector is related to the gradient of the energy, implying that variational imaginary time simulation can be regarded as a generalisation of the gradient descent method. It has been numerically found that the variational imaginary time simulation algorithm might be less sensitive to local minima in contrast to simple gradient descent methods.^{18}^{)} In addition, when imaginary time evolution does reach a minimum that is not the ground state, it tends to be an excited eigenstate of the Hamiltonian, which can be thus exploited for finding general eigen-spectra.^{32}^{)} It has recently been observed that an equivalent formulation of the variational imaginary time algorithm can be obtained by exploiting the quantum natural gradient approach.^{81}^{–}^{83}^{)} Notice that since the *M* matrix has to be measured, variational imaginary time simulation needs more measurements than the conventional gradient descent method. However, for an increasing system size and simulation time, the number of measurements required for *M* matrix is asymptotically negligible.^{84}^{)}

Given the current parameters *M*, *V*, and *C* terms with quantum circuits. Suppose *M*, *V*, and *C* can be written as a sum of terms as *M*, *C*, and *V* with the quantum circuits shown in Fig. 3. Refer to Appendix B for a detailed explanation about construction of the quantum circuit. Notice that comprehensive analysis on the sampling cost for *M*, *V*, and *C* has been given by van Straaten and Koczor.^{84}^{)} We summarise the variational algorithm for simulating real (imaginary) time evolution.

Figure 3. Quantum circuits for computing (a) ^{14}^{,}^{18}^{,}^{19}^{)}

The schematic figure is also shown in Fig. 4. We can also simulate time-dependent Hamiltonian evolution by using the time-dependent Hamiltonian at each step. The accuracy of the simulation can be computed at each step by the distance between the evolution of the ansatz state and that of the ideal evolution.^{19}^{)} In the case of the real time simulation, we have *M*, *V*, and ^{85}^{,}^{86}^{)} Similar arguments also hold for imaginary time evolution. Note that a variational real time simulation algorithm was demonstrated in an experiment using 4 superconducting qubits.^{87}^{)} In this experiment, adiabatic quantum computing was simulated, which was used for discovering eigenstates of an Ising Hamiltonian.

Figure 4. Schematic of the variational quantum simulation algorithm. The elements of *M* matrix and *V* (*C*) vectors are measured for the ansatz state. The results are sent to the classical computer to solve *C*) to update the parameters, which will be fed to quantum processors.

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

The quantum approximate optimisation algorithm (QAOA)^{13}^{)} was initially proposed for solving classical optimisation problems. The algorithm works by mapping the classical problem to a Hamiltonian

As an example, we consider the Boolean satisfiability problem, which aims to find solutions of Boolean variables so that all given clauses in a propositional formula are true. The *j*-th Boolean variable denoted as *x*. An example of Boolean satisfiability problem is

The general form of this problem (with clauses involving three or more booleans) is generally NP-hard and has a wide range of applications in computer science and cryptography.^{88}^{)} To map the Boolean satisfiability problem to a Hamiltonian, we first get the Hamiltonian with its ground state being the solution of each clause. For example, the Hamiltonian for the first clause

Since the solution corresponds to the ground state of ^{13}^{)} who introduced the ansatz *D* being the number of repetitions of the ansatz quantum circuit, ^{13}^{)} suggested to consider gradually changing a Hamiltonian in each step of optimisation as *D*, and adaptive optimisations of the parameters for each optimisation step *t*, the algorithm may still be able to find the ground state solution.

A recent thorough study of the performance of the QAOA on MaxCut problems can be found in Zhou et al.^{89}^{)} Utilising nonadiabatic mechanisms, a heuristic strategy was proposed to learn the parameters exponentially faster than the conventional approach. Meanwhile, the QAOA was implemented with 40 trapped-ion qubits.^{90}^{)}

Now we introduce the application of variational quantum algorithms in machine learning. In general, machine learning provides a universal approach to learn the pattern of the given data and to predict or reproduce new data. For example, in supervised learning, the given data are described by *b*, and

Whether machine learning works or not is highly dependent on the choice of the model. Since quantum states could efficiently represent multipartite correlations that admit no efficient classical representation, quantum machine learning protocols are proposed involving quantum neural networks that consist of variational quantum circuits. Consequently, we can leverage a quantum neural network to dramatically enhance the representability of the model.^{20}^{,}^{21}^{)} In addition, as the ansatz circuit is a unitary operator, the norm of the quantum state is necessarily unity, and this constraint may lead to a natural regularisation of the parameters to avoid overfitting.^{20}^{)} The schematic figures for classical and quantum neural networks are shown in Fig. 5.

Figure 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 quantum circuit leaning.^{20}^{)}

Note that there are quantum machine learning algorithms^{91}^{–}^{94}^{)} that are based on universal quantum computing without variational quantum circuits. While these algorithms are proven to have exponential speedups over classical algorithms under certain conditions, they generally require a deep quantum circuit and are not suitable for NISQ devices.

In this section, we illustrate five examples of quantum machine learning algorithms. The first two algorithms are introduced for leaning classical data for solving a classical problem; the latter three are introduced for learning quantum data.

The quantum circuit learning (QCL) algorithm implements supervised learning with a variational quantum circuit instead of a classical neural network.^{20}^{)} In QCL, a state is prepared as *f*. Suppose the data is described as *x* is *Y* gate, *x* up to the

Data-driven quantum circuit learning (DDQCL) implements a generative model by using a variational quantum circuit.^{21}^{)} We briefly summarise the classical generative model. Suppose the data are described with

For DDQCL, the model ^{95}^{)} Since quantum states can efficiently represent complex multipartite correlations and the model can be efficiently sampled by measuring the prepared state, DDQCL may be able to represent generative models that are classically challenging. In experiment, DDQCL has been applied to successfully learn Greenberger–Horne–Zeilinger states (GHZ) states and coherent thermal states by using a 4-qubit trapped ion device.^{21}^{)} The schematic figure for sampling

Figure 6. A quantum circuit for sampling

Quantum generative adversarial networks (QuGANs)^{22}^{,}^{96}^{)} are a quantum analogue of generative adversarial networks (GANs). Conventional GANs are composed of three parts — true data, generator, and discriminator, as shown in Fig. 7(a). The generator competes with the discriminator, where the former tries to produce fake data and the latter tries to determine whether the input data are true or fake. By optimising both the generator and the discriminator, the generator can learn the distribution of the true data until the discriminator cannot tell the difference between the true data and the fake data from the generator. GANs are generalised to quantum computing by replacing each part with a quantum system.^{22}^{)} By using a quantum circuit as the generator, we can represent the

Figure 7. (Color online) Schematic diagrams for (a) classical GANs and (b) quantum GANs. In classical GANs, the generator and the discriminator consist of classical neural networks, while quantum neural networks are used in quantum GANS.

Suppose the fake density matrix from the generator is produced from a parametrised quantum circuit as ^{96}^{)} for detailed ansatz constructions. The discriminator implements a parametrised positive-operator valued measure

The classical autoencoder is used for compressing classical data as shown in Fig. 8(a). For an input set of training data *ε* is a desired accuracy.

Figure 8. Schematic figures for (a) classical autoencoder and (b) quantum autoencoder. The encoding operation

Quantum autoencoder implements a similar task for compressing quantum states.^{23}^{,}^{97}^{)} Here, we consider the scheme proposed in Romero et al.^{23}^{)} Consider the case with input states of *n* qubits. Suppose the input states are an ensemble *A* and *B* consists of *n* and *m* qubits, respectively. With a compression circuit *m*-qubits of system *B*, as shown in Fig. 8(b), each input state *n* qubits. The inverse process then decodes the compressed state and map each

Analysing eigenvalues and eigenvectors of the covariance matrix of data is crucial for extracting its important features. Such a process is called the principal component analysis (PCA), which has been widely used in data science and machine learning. The covariance matrix of the data could be uploaded onto a quantum computer.^{98}^{–}^{100}^{)} Here we show how to use the variational quantum state eigensolver (VQSE) algorithm to diagonalise input density matrices *ρ* to ^{62}^{)} which only requires a single copy of the state and other schemes requiring two copies of the state can be found in LaRose et al.^{61}^{)} and Bravo-Prieto et al.^{101}^{)} Without loss of generality, we assume the eigenvalues are in an descending order, i.e.,

We first map a parametrised quantum circuit *ρ*, the vector

Therefore, after minimising the cost function *ρ* is

On the other hand, a time dependent cost function which combines local and global cost functions is used to make the best of their benefits.^{62}^{)} See Appendix D for an explanation of local and global cost functions. Also one can find applications of VQSE in quantum error mitigation^{14}^{,}^{16}^{,}^{17}^{)} and entanglement specroscopy.^{102}^{,}^{103}^{)}

Variational quantum algorithms can be used for solving matrix-vector multiplication and solving linear systems of equations. The task of matrix-vector multiplication is to obtain ^{30}^{,}^{31}^{,}^{104}^{,}^{105}^{)} Herein, we illustrate the methods introduced in Bravo-Prieto et al.^{30}^{)} and Xu et al.^{31}^{)}

We first consider matrix-vector multiplication proposed by Xu et al.,^{31}^{)} where the solution

Here we also consider the algorithms for solving linear equations independently proposed by Xu et al.^{31}^{)} and Bravo-Prieto et al.^{101}^{)} The solution is mapped to the ground state of the Hamiltonian ^{106}^{)} who applied adiabatic algorithms to find the ground state with a universal quantum computer. With the variational method, solving the ground state is similar to the one for matrix-vector multiplication. Furthermore, Xu et al.^{31}^{)} used Hamiltonian morphing optimisation for avoiding local minima, and Bravo-Prieto et al.^{101}^{)} used a local cost function for circumventing a barren plateau issue and run a simulation with up to 50 qubits. Refer to Appendix D for these optimisation methods.

Next we show how to find excited states and excited energy spectra of a Hamiltonian. Calculating excited energy spectra is important for studying many-body quantum physics problems. For example, it can be used to study chemical reaction dynamics, important for creating new drugs and new methodologies for mass production of beneficial materials.^{53}^{,}^{107}^{)} In addition, evaluation of excited states enables us to calculate the photodissociation rates and absorption bands, which are essential for designing solar cells and investigating their dynamics.^{108}^{,}^{109}^{)} There exist several VQAs for evaluating excited states and excited energy spectra.^{32}^{,}^{33}^{,}^{36}^{,}^{110}^{–}^{117}^{)} These algorithms can be used as a subroutine for other applications, e.g., the Green's function,^{35}^{,}^{118}^{)} non-adiabatic coupling, and Berry's phase^{119}^{)} and simulating real time evolution^{120}^{)} etc. In this section, we review three VQAs — the overlap-based method, the subspace expansion method, and the contraction VQE method — for calculating excited state energy, and the application in calculating the Green's function.

The overlap-based method first uses VQE to find the ground state and then sequentially obtains excited states by penalising the previously obtained eigenstates.^{32}^{,}^{33}^{)} Suppose that the ground state *H* is obtained from either the conventional VQE or variational imaginary time simulation. Now, suppose we replace the Hamiltonian *H* with the Hamiltonian *α* is a positive number which is chosen to be sufficiently larger compared to the energy gap between the ground and the first excited state of the Hamiltonian. Then the first excited state *H* becomes the ground state of the new Hamiltonian *H* with the VQE or variational imaginary time simulation on

To realise the VQE or imaginary time evolution of ^{121}^{,}^{122}^{)} These quantum circuit necessitate two copies of the state. Note that destructive SWAP test circuit only leverages a shallow depth circuit. We leave a detailed explanation about the (destructive) SWAP test circuit in Appendix C. Alternatively, we can compute the overlap term without using two copies of the state but using a doubled depth of the quantum circuit.^{33}^{)} Denoting

After finding the first excited state of *H*, we can further find the second excited state by replacing the Hamiltonian *H* with *H* becomes the ground state of

In the overlap-based method, an error happens when the discovered state ^{32}^{)} This is because when variational imaginary time evolution fails to find the ground state, it may instead converge to an eigenstate of the Hamiltonian based on the nature of imaginary time evolution.^{18}^{,}^{32}^{)} By penalising the discovered excited state, we can still construct a new Hamiltonian to find another low energy state.

The quantum subspace expansion solves a generalised eigenvalue problem in terms of the given Hamiltonian in an expanded subspace around an approximated ground state, and the obtained eigenstates and eigenenergies correspond to those of the Hamiltonian.^{36}^{)} Note that the obtained spectrum is error-mitigated because the excited states are approximated as a linear combination of states in the expanded subspace. Refer to Sect. 5.4 for a detailed explanation about error mitigation effect of quantum subspace expansion.

Let *H* when *E* is the Lagrangian multiplier for the constraint *E* corresponds to the energy for the eigenstate. Both

The contraction VQE methods firstly discover the lowest energy subspace and then construct eigenstates in that subspace. Here we introduce two contraction VQE methods — subspace-search VQE (SSVQE)^{111}^{)} and multistate contracted variant of VQE (MC-VQE).^{112}^{)}

The procedure of SSVQE is as follows. We first prepare a set of orthogonal states *H*. To project *k*th excited state *k*th excited state *k* to find excited states.

For the MC-VQE method, it also first projects the subspace to the lowest energy subspace in the same way as SSVQE. Different from SSVQE, MC-VQE assumes the excited states can be expanded in the lowest energy subspace as *E* is the corresponding excited state energy. We can regard Eq. (55) as the special case of Eq. (50) with *H* with states

The potential problem of the contraction VQE methods is that the energy landscape of

Now, we show the application of the VQE algorithms in the calculation of the Green's function,^{35}^{,}^{118}^{)} which plays a crucial role in investigating many-body physics such as high ^{123}^{)} topological insulators,^{124}^{)} and magnetic materials.^{125}^{)} The definition of the retarded the Green's function at zero temperature is

For simplicity, we consider the Green's function in the momentum space for identical spins. We thus have

In literature, Endo et al.^{35}^{)} used the contraction VQE method and Rungger et al.^{118}^{)} employed the overlap method to calculate the transition amplitudes and further the Green's function. The algorithm using the overlap method^{118}^{)} was originally employed for computing the Green's function in the specific Jordan Wigner encoding, whose generalisation was further studied in Ibe et al.^{126}^{)} for general operators. Here we review the algorithm based on the MC-VQE method.^{35}^{)} By using the expression for *m*th excited state in Eq. (54), we have

Note that based on calculation of the Green's function, Rungger et al.^{118}^{)} implemented dynamical mean-field theory (DMFT) calculation on two-site DMFT model by using a superconducting system and a trapped ion system. Also, in the work by Ibe et al.,^{126}^{)} the accuracy of the calculation of transition amplitudes are compared for the overlap method and the contraction VQE methods in detail.

Circuit recompilation aims to approximate a given quantum circuit with ones that are compatible with a practical experiment hardware. Concerning noisy gates or restricted set of realisable gates, the compiler runs to reduce the circuit noise or the implementation cost, as shown in Fig. 9. For example, an arbitrary two-qubit unitary is generally not directly supported on a practical quantum hardware and we need to compile the unitary into a sequence of realisable gates. While a naive decomposition of the unitary may induce too many unnecessary gates for hardware with specific topological structures, finding efficient and simple decomposition of quantum circuits is vital for near-term quantum computing.

Figure 9. Schematic figure for circuit recompilation algorithms. A variational quantum circuit with hardware constraints tries to approximate the target quantum circuit.

Denote the target unitary as ^{127}^{,}^{128}^{)} *φ* are randomly chosen according to the Haar measure. By construction, AGI indicates how different two unitary gates are on average for randomly sampled pure states, and it vanishes when the circuit is perfectly re-compiled. The AGI method has been applied for compiling high fidelity CNOT gate with cross-resonance gate suffering from crosstalk and single qubit operations. Furthermore, a high-fidelity four-qubit syndrome extraction circuit was recompiled to be achievable with simultaneous cross resonance drives under crosstalk.^{128}^{)}

Another related cost function^{127}^{)} is ^{127}^{)} by using two copies of states. In particular, we have *d* being the dimension of the system. Then, ^{129}^{,}^{130}^{)} ^{127}^{)}

Meanwhile, circuit recompilation can be implemented for specific input states, which may reduce the optimisation complexity.^{127}^{,}^{131}^{)} The goal is to find the recompiled circuit *Z* operator acting on the *j*th qubit. Then, by minimising the expectation value of ^{131}^{)}

Quantum metrology aims to discover the optimal setup for probing a parameter with the minimal statistical shot noise.^{132}^{–}^{134}^{)} The basic setup of quantum metrology is as follows — firstly, we prepare the initial probe state *ω* being the target parameter. After time *t*, the probe state can be described as *ω*. Typically, *ω* can be a magnetic field, for example, with Hamiltonian *Z* operator, and the measurement is a Ramsey type measurement. When separable states are used as probes, the statistical error of the parameter behaves as ^{134}^{)} Notably the scaling can be improved by using entangled states, such as GHZ states, symmetric Dicke states, and squeezed states. In the absence of noise or for specific types of noise in the evolution under the Hamiltonian, the optimal strategy has been revealed. For example, with no environmental noise, the optimal probe state has been proved to be the GHZ state which achieves the Heisenberg limit ^{133}^{–}^{135}^{)} However, in the presence of general types of noise, analytical arguments about the optimal strategy are usually very hard.

Variational-state quantum metrology is to use a quantum computer to find the optimal quantum state for quantum metrology with noisy hardware. Different proposals have been studied with either general variational quantum circuits^{136}^{)} or specific experimental setups,^{137}^{)} i.e., optical tweezer arrays of neutral atoms.^{138}^{,}^{139}^{)} In general, suppose the initial probe state is created on a quantum device, as ^{136}^{)} or the spin squeezing parameter.^{137}^{,}^{140}^{)} Here, we focus on the QFI, which characterises the minimum uncertainty of the estimated parameter as *t* and *T*, we aim to optimise *t* to maximise

Figure 10. Schematic figure for variational quantum-state metrology. After the probe state is prepared by the variational quantum circuit, it evolves under the Hamiltonian

With variational-state quantum metrology, a highly asymmetric state has been discovered for a 9-qubit system that outperforms previous results.^{136}^{)} Interestingly, even though the Hamiltonian and the noise model is symmetric under the permutation of qubits, there is a symmetry breaking for the optimal solution. Note that unlike conventional analytical approaches employed in quantum metrology, we do not have to know the noise model of the quantum device to obtain the optimised state. Recently, this algorithm was generalised to multi-parameter estimation.^{141}^{)}

Quantum error correction (QEC) makes use of a larger number of physical qubits to encode logical qubits to protect them against physical errors. In general, conventional formulation of QEC does not take into account the experimental implementation of the code. However, in the NISQ era, for example, the set of possible operations and the qubit topology are restricted for each physical hardware. Therefore, hardware-friendly implementation of QEC, tailored to actual experiments, is crucial for near-term quantum computers.^{142}^{–}^{144}^{)} Here, we illustrate two examples^{143}^{,}^{144}^{)} for realising QEC on NISQ computers.

This variational circuit compiler is for automatically discovering the optimal quantum circuit satisfying user-specified requirements for a given QEC code.^{143}^{)} Typically, such requirements tend to come from hardware properties, such as available gate sets, limited topology, and achievable error rate. More concrete example may be two-qubit gate implementation, e.g., superconducting qubits employing CNOT gate, and ion trap systems using Møren-Sørensen gate. We prepare an ansatz unitary

The essential point of the compiler is to design the Hamiltonian whose ground state is the target state. Then the target state can be obtained via the conventional VQE or variational imaginary time simulation algorithm. Generally, the code space is defined by a set of commuting Pauli operators, so-called stabiliser generators. For example, when we consider the three qubit code, the logical state

When using the variational algorithm to minimise the average energy, an approximation of the encoding circuit is found. Supposing the energy of the optimally discovered state is ^{143}^{)}

Variational quantum error corrector (QVECTOR) is for discovering a device-tailored quantum error correcting code.^{144}^{)} Different from the compiler for preparing a target logical state,^{143}^{)} QVECTOR aims to discover the optimal encode circuit that preserves the quantum state under noise. As shown in Fig. 11, the circuit *k*-qubit state *r* additional qubits. The parameters ^{145}^{)} With numerical simulation, QVECTOR learned the three qubit code^{146}^{)} under the phase damping noise, resulting in a six times longer

Figure 11. Schematic figure of the variational circuit for QVECTOR.

Dissipative-system variational quantum eigensolver (dVQE) is for obtaining the non-equilibrium steady state (NESS) in an open quantum system.^{147}^{)} In practice, quantum systems inevitably interact with their environments, and quantum states decohere owing to the noise. Therefore, the ability to simulate open quantum systems is indispensable for studying practical quantum phenomena. Notably, investigating NESS of open quantum systems is very important, e.g., in revealing the transport mechanism in nano-scale devices such as single-atom junctions.^{148}^{)}

The time independent Markovian open quantum dynamics can be described by the Lindblad master equation, *ρ* is the system state, *H* is the Hamiltonian, *A*.

As *s* and *a* denote system and ancilla, respectively. We refer to Yoshioka et al.^{147}^{)} for the detailed ansatz construction. By preparing a trial state *O* for

There are other applications, such as VQAs for nonlinear problems,^{149}^{)} fidelity estimation,^{150}^{)} factoring,^{151}^{)} singular value decomposition,^{101}^{,}^{152}^{)} quantum foundations,^{153}^{)} circuit QED simulation,^{154}^{)} and Gibbs state preparation.^{152}^{,}^{155}^{,}^{156}^{)}

In this section, we review the variational quantum simulation algorithms for simulating the dynamical evolution of quantum systems^{19}^{,}^{34}^{)} and the application in simulation of open quantum systems,^{19}^{,}^{34}^{)} linear algebra tasks,^{34}^{)} Gibbs state preparation,^{19}^{)} and evaluating the Green's function.^{35}^{)}

We first show how to generalise the simulation algorithm for real and imaginary time evolution from pure states to mixed states.^{19}^{)} The main idea is again to consider a parametrised representation of mixed states and map the dynamics to the evolution of the parameters. As we are considering mixed states, only McLachlan's variational principle applies, which leads to the evolution of parameters with information determined by the density matrix.^{19}^{)} Although conventional quantum computers operate on pure states, we can also represent mixed states with their purifications by using ancilla qubits.

In practice, a quantum system interacts with its environment, so open quantum system simulation algorithms are useful for investigating practical quantum phenomena. Here we aim to simulate real time evolution of open quantum systems described by the Lindblad master equation *M* and *V* can be reduced to the computation of terms like ^{19}^{)} By encoding the mixed state via its purification, this term can be evaluated via the SWAP test circuit. Note that, in order to simulate open system of *M* and *V*, so we need in total

The variational quantum simulation algorithm can be applied for simulating imaginary time evolution of density matrices as well,^{19}^{)} which is defined as

In this section, we review the variational quantum simulation algorithms for general processes, including the generalised time evolution, its application in solving linear algebra tasks, and simulating open system dynamics.^{34}^{)}

Apart from real and imaginary time evolution, we consider the generalised time evolution defined as ^{34}^{)} for details.

The real time evolution corresponds to *H*. Therefore, the generalised time evolution unifies real and imaginary time evolution. In addition, the generalised time evolution describes a general first-order differential equations with non-Hermitian Hamiltonians, which may have applications in non-Hermitian quantum mechanics.^{157}^{)} In the following, we show its application in solving linear algebra tasks and simulating the stochastic Schrödinger equation.

Now, we explain how we can apply the algorithm for generalised time evolution to realise matrix-multiplication and to solve linear systems of equations.^{34}^{)} This is an alternative method for linear algebra discussed in Sect. 3.3.^{31}^{,}^{101}^{)} For a sparse matrix

Now we show how to simulate open quantum system dynamics with the variational algorithm of generalised time evolution.^{34}^{)} Instead of directly simulating the Lindblad master equation defined in Eq. (73), we consider its alternative representation via the stochastic Schrödinger equation, where the whole evolution is a mixture of pure state trajectories^{158}^{)} *t* to *U* and *V*, and diagonal matrix *D*. Since all these matrices only act on a few qubits, the decomposition is efficient and we can further have

Note that this algorithm only needs to control a single copy of the state, thus only uses one-fourth of the number of qubits compared to the algorithm presented in the previous section. The variational quantum simulation algorithm for the stochastic Schrödinger equation is numerically implemented for the 2D Ising Hamiltonian with 6 qubits.^{34}^{)} The simulation result witnesses a dissipation induced phase transition,^{159}^{)} indicating its potential in probing general interesting physics phenomena of many-body open systems with intermediate scale quantum hardware.

The variational quantum simulation algorithm for imaginary time evolution can be applied for preparing a Gibbs state.^{19}^{)} Starting at the maximally mixed state *d*, imaginary time evolution of the state with Hamiltonian *H* and time *τ* prepares the state *s* and *a* denote the target system and the ancilla system for purification. It is easy to verify that *s*, i.e., ^{160}^{)}

Figure 12. The ansatz quantum circuit for obtaining the Gibbs state for

Now we discuss the application of the variational quantum simulation algorithms for calculating the Green's function and the spectral function.^{35}^{)} We refer to Sect. 3.4.4 for the review of the Green's function. The definition of the retarded the Green's function at zero temperature is ^{51}^{,}^{54}^{,}^{55}^{)} as ^{35}^{)}

Furthermore, it is worth noting that, by combining this method with the Gibbs state preparation algorithm in the last section,^{19}^{)} we can also compute the Green's function for finite temperatures.

There are several other applications of variational quantum simulation algorithms. The first one is for financial problems.^{161}^{)} The partial differential equation for pricing is equivalent to imaginary time evolution, so we can apply variational imaginary time simulation algorithm to simulate it. Secondly, variational imaginary time simulation can be used to simulate non-hermitian transcorrelated Hamiltonian for reducing the resource overhead and improving simulation accuracy for electronic structure calculations.^{162}^{)} Finally, it can be used for preparing a Boltzmann distribution with only a single copy of a quantum state.^{163}^{)}

In this section, we review the error mitigation techniques for suppressing errors in noisy quantum devices. We will use *U* to denote the ideal quantum gate with the corresponding channel representation being

With an input state

When the noise is below a certain threshold, we can make use of quantum error correction (QEC) to recover the ideal state. However, implementing QEC requires a huge overhead of the number of qubits. For example, Fowler et al.^{164}^{)} have shown that the surface code, one of the most popular codes for practical fault-tolerant quantum computing, requires around a thousand qubits per logical qubit to perform Shor's algorithm with a reasonable success probability. With the limited number of qubits available in near-term quantum computers, realising QEC on them might not be practical.

Alternative schemes under the name of quantum error mitigation (QEM) are developed instead for error suppression on NISQ devices, which rely on the cleverer post-processing of measurement results. Instead of recovering the ideal output state, most QEM methods aim to recover the ideal measurement statistics. Supposing we measure an observable *M* on the output state, QEM methods target at recovering

Figure 13. (Color) Schematic of QEM. In the presence of noises, the probability distribution of the expectation value of an observable is shifted from the noise-free one. Using QEM we can make the probability distribution centred around the correct expectation value; however the variance is amplified and we need more samples to achieve the same accuracy as the one before QEM.

We first introduce the most simple yet very powerful error mitigation technique based on error extrapolation. The basic idea is to run the circuit with different error rates and use them to extrapolate for the error-free result, which is illustrated in Fig. 14.

Figure 14. (Color) Schematic of quantum error extrapolation. We boost the error rate and obtain other data points. Then we extrapolate the original result and those at boosted error rates to estimate the error-free result. Fitting curves should be chosen depending on the situation. Here, we show the case of two-point (green) and three-point (blue) Richardson extrapolation as examples.

We first review the Richardson extrapolation error mitigation method introduced by Li and Benjamin^{14}^{)} and Temme et al.^{16}^{)} We will look at a stochastic noise process *M*, we can expand the expectation value *ε* according to Taylor expansion, *ε*.

In order to estimate the ideal measurement result *n*th order and use Richardson extrapolation to approximate *ε* to *ε*.

The reader might think that the error can be arbitrarily suppressed by increasing the number of points *n*. This is not possible because the estimation of Eq. (101) also introduces a big shot noise from measuring the expectation values. In particular, the variance of the approximate *k*. This implies that we have to take *n*,^{165}^{)} we can only opt for a small constant value of *n*.

Notably, error mitigation via Richardson extrapolation has been experimentally implemented for finding the ground state energy of H_{2} and LiH with the variational quantum eigensolver method.^{166}^{)} The error mitigation method dramatically reduces the calculation error for several orders, leading to results close to the chemical accuracy.

The Richardson extrapolation assumes a valid Taylor expansion with a polynomial function of the error rates and negligible higher order terms. However the expansion based on polynomial function might be inaccurate for the practical scenario with small error rate *ε* and large number of gates *k* errors. In the third line, we denote the binomial distribution as *ε* satisfying

Therefore, instead of a polynomial expansion of the expectation value, Endo et al.^{17}^{)} considered the exponential decay. In particular, approximating the sum of Eq. (107) to the first order, and considering the two noisy measurements *ε* and ^{17}^{)} and Giurgica-Tiron et al.^{167}^{)} Exponential extrapolation was used on IBM's superconducting qubit device for implementing dynamical mean-field theory simulation via Hamiltonian simulation algorithm.^{168}^{)} Recently, exponential extrapolation was further generalised to multi-exponential extrapolation in the form of ^{47}^{)}

Since the extrapolation methods use measurement results with different error rates, here we review three different methods for effectively increasing the error rates. Firstly, by increasing the number of noisy gates, the amount of physical noise can be effectively increased. For example, letting *ε* to ^{169}^{)} To obtain fine-grained resolution for boosted error rates, unitary folding method was introduced.^{167}^{)} Meanwhile, it is shown that by setting *n* to a random variable, the boosted error rate can take a continuous value.^{170}^{)} Let *n*. Then the error rate can be boosted by a factor of

The second method is via the re-scaling of the Hamiltonian in realising quantum gates.^{16}^{,}^{166}^{)} Suppose a gate operation in a quantum circuit is described as follows *λ* is the dissipation strength. Now we re-scale the Hamiltonian to *c* times greater, we can boost the noise strength from *λ* to *c* may become different from the expected one, leading to estimation errors in error mitigation. Hamiltonian re-scaling method was demonstrated in experiment using superconducting qubits^{166}^{)} and the IBM OpenPulse framework.^{171}^{)}

The third method is via Pauli twirling.^{14}^{)} Suppose the dominant error source are from two-qubit gates so that single-qubit gate error rate is sufficiently low. By randomly applying Pauli gates before and after a Clifford entangling two-qubit gates, we can first convert an arbitrary error into stochastic Pauli errors.^{172}^{)} Then, by randomly generating additional Pauli gates after the twirled two qubit gates, we can boost the physical error rate to any desired amount. This method assumes small single qubit error rate and exact knowledge of the error form. It may not work when the single-qubit error rate is high, or knowing the exact error channel is experimentally challenging.

In addition to mitigating physical errors of the circuit, the extrapolation error mitigation method can also be applied to mitigating algorithmic errors in Hamiltonian simulation via Trotterisation.^{173}^{)} Suppose the Hamiltonian of interest is decomposed as

By properly choosing the optimal number of Trotter steps *M* on the output state. Then similar to the scenario of mitigating physical errors, we can expand the measurement result as a function of the algorithmic error

Now, consider quantum error mitigation via the least square fitting.^{40}^{)} Although the assumption in literature is good characterisation of error rates and amplification of noise for extrapolation is not used, it may be employed to prepare results corresponding to several error rates. Consider well-characterised different error rates *n*th order corresponds to a linear equation, *n*. A solution of the linear equation can be obtained from minimising

In practice, there may exist multiple noise parameters from different noisy effects, e.g., *ν*, and we expand the expectation value ^{40}^{)}

The quasi-probability method is another error mitigation technique, which was first introduced by Temme et al.^{16}^{)} for special channels and then generalised to practical Markovian noise by Endo et al.^{17}^{)} The main idea is for any noise process, its effect can be cancelled out by probabilistically implementing its inverse process. Considering a single noisy gate as *ρ* and measure the output with an observable *M*, the ideal measurement statistic

As an example, we consider error mitigation of the depolarising noise *I*, *X*, *Y*, and *Z* gates as *X*, *Y*, *Z* gates with probability

The quasi-probability method can be applied to a general quantum circuit, as illustrated in Fig. 15. Let us assume that the ideal process of the entire quantum circuit is described as *b* and error probability *ε*. For stochastic noise, *b* is generally less than 2.^{17}^{)} We can see that the cost increases exponentially to

Figure 15. (Color online) Schematic of the quasi-probability method. We apply the inverse channel to cancel the noise process

In contrast to the extrapolation method, we also need to exactly identify the noise model ^{17}^{)} where gate set tomography (GST) was applied so that the quasi-probability decomposition obtained from GST is free from SPAM errors. The same paper also showed how to use experimentally feasible operations to suppress general Markovian errors. The application of quasi-probability in the presence of non-Markovian errors, such as temporally and spatially correlated errors, was further studied by Huo and Li.^{174}^{)} Detailed theoretical analysis of the cost using resource theory approach was given by Takagi.^{175}^{)} The quasi-probability method has been demonstrated using superconducting^{176}^{)} and trapped ion systems^{177}^{)} for single- and two-qubit gates.

The quantum subspace expansion (QSE) can not only be used for evaluating excited states, but also for quantum error mitigation in variational quantum eigensolver (VQE).^{36}^{)} Suppose that we already have the approximation of the ground state *ρ* of the Hamiltonian *H* via either the VQE or variational imaginary time evolution. The approximation *ρ* may not be the exact ground state of *H* due to gate errors or imperfect ansatz and optimisation. To mitigate such errors, we consider a small subset of operators *E* is the eigenvalue, *S*. Once *M* as

The QSE method is more effective for coherent errors such as over-rotation of quantum gates, and it may not mitigate all local stochastic errors. Specifically, the subspace expansion method is equivalent to optimising states from the subspace *ρ* to be a mixture of the ideal pure state *σ* as *S* and hence the QSE method is less effective for stochastic errors on a general quantum state. Nevertheless, as we shortly discussion in the next section, when the target state preserves certain symmetries either inherently or by encoding via an error correcting code, the QSE method can indeed effectively suppress stochastic errors by efficiently projecting to the subspace with the correct symmetry. It can thus mitigate a large portion of stochastic errors and significantly improve the calculation accuracy when further combined with other QEM techniques.^{37}^{)}

In practice, the set *S* can be chosen to be creation and annihilation operators for fermionic Hamiltonians. The QSE method was experimentally demonstrated for mitigating errors in quantum chemistry calculation of ground and excited state energies of the H_{2} molecule.^{65}^{)}

When the physical system we try to simulate exhibits certain symmetries,^{178}^{,}^{179}^{)} they can be exploited for QEM using symmetry verification.^{37}^{,}^{38}^{)} We can design ansätze with unitary components that conserve the particle number and the spin symmetries, such as the unitary coupled cluster ansatz^{69}^{)} or certain hardware efficient ansätze.^{180}^{)} We will focus on symmetries that can be mapped to Pauli operators since they can be efficiently measured. For example, the parity operators for the particle number and for the spins,

While the ideal quantum state preserves the symmetries, states prepared by the noisy circuit may break them. Since symmetries are broken only if an error happens, symmetry verification works by discarding the cases that break the symmetries.^{37}^{,}^{38}^{)} We can implement the measurement of ^{181}^{)} If the symmetry outcome is not the expected one, the circuit run is discarded. More errors can be detected by considering more general symmetries, such as the one that preserves the particle number. Symmetry verification can also be combined with other QEM methods as will be discussed in Sect. 5.10.

Figure 16. Quantum circuit used for symmetry verification. This quantum circuit is for four register qubits. The ancilla qubit is measured in 0 (1) when the the total particle number is even (odd). Thus, by reading out the ancilla qubit, we can detect errors in the register qubits.

Alternatively, we can effectively implement the parity check circuit via a post-processing approach.^{38}^{)} Here we consider the conventional VQE scenario and the case where *ρ* that may have components with both positive and negative eigenvalues of *H*, which commutes with *m*, optimising *m* would be equivalent to the QSE procedure. It indeed has been shown that the solution of QSE coincides with the state of Eq. (133) given the symmetry of the target ground state.^{38}^{)}

In contrast to the parity-check circuit based approach, the current one does not need the additional ancilla and the parity-check circuit even in the most general cases. However it does need more samples since the error is mitigated via post-processing. For example, suppose the probability that the noisy state is in the correct subspace is *ε*, the parity-check circuit method requires ^{182}^{)} The post-processing approach was experimentally demonstrated by using a two-qubit superconducting processor.^{183}^{)}

When considering the logical states of a stabiliser quantum error correcting code, it has code symmetries described by the stabilisers. The conventional approach for realising error detection and error correction is based on the parity check circuit with additional ancillary qubits. In McClean et al.,^{184}^{)} the post-processing approach was extended for realising the error detection and error correction without the parity check circuit. Since an error correcting code has multiple symmetries, we can probabilistically apply each symmetry projection to force the state back into the code space. The post-processing error decoder for the five qubit quantum error correcting code was numerically implemented with an effective threshold of 50% and applied for chemistry simulation.^{184}^{)}

The individual error reduction method makes use of quantum error correction (QEC) on a single qubit and post-processing to mitigate errors.^{39}^{)} Assume that physical noise after each quantum gate is described using the Lindblad master equation *τ*. Note that *k*th qubit. Suppose that the *l*th qubit is encoded as a logical qubit and the noise on it can be reduced by a factor of *l*th qubit described by Eq. (136). Then it can be shown that *M* by measuring the expectation value

Measurement error mitigation is designed for suppressing errors during the measurement process.^{41}^{–}^{43}^{)} Suppose that the ideal measurement is described by a set of positive-operator valued measure (POVM) operators *k* is *ρ*. Denote the ideal error-free probability distribution as *ε* and *η* denoting the error probabilities that the outcome flip from 0 to 1 and vice versa.

In practice, estimation of the matrix N^{185}^{)} so that N^{41}^{)} and Chen et al.^{42}^{)} Recently, measurement error mitigation technique was proposed to further suppress cross-talk errors between qubits during the readout process.^{186}^{)} This method was demonstrated on the IBM 20-qubit superconducting processor.

While most error mitigation methods rely on exact or partial information of the noise model, the learning-based QEM method aims to suppress errors via an automatical learning process.^{44}^{,}^{45}^{,}^{187}^{)} Herein, we illustrate two examples by Czarnik et al.^{45}^{)} and Strikis et al.^{44}^{)}

Assume that we are given test data ^{44}^{)} Here, ^{44}^{)} and illustrated in Fig. 17. Clifford Data Regression was demonstrated by using the 16-qubit IBMQ quantum computer, and a 64-qubit classical simulator.^{44}^{)} Since the noise effect are naturally incorporated in the learning process, this method may work for both local Markovian noise and correlated noise.

Figure 17. (Color online) Schematic figures for Clifford data regression.^{44}^{)} The relationship between noisy results from a quantum computer and ideal results from a classical computer is learned via regression. The training data are sampled from Clifford circuits. Then we use the relationship to predict the error mitigated result for an arbitrary non-Clifford quantum circuit.

Instead of directly learning the ideal measurement outcome from noisy ones, Strikis et al.^{45}^{)} introduced a learning-based method for realising quasi-probability error mitigation without depending on process tomography. Focusing on quantum circuits that consist of arbitrary single-qubit gates and Clifford two-qubit gates, which are sufficient for universal quantum computing. Suppose the dominant error source comes from the two-qubit gates, which are called frame gates. Denote the sequence of single-qubit gates as

Figure 18. (Color) An example of quantum circuits for learning-based quasi-probability method.^{45}^{)} The gates denoted as *Q* are for mitigating errors for frame gates *U* is for sampling a training set *Q* and *U* between every layer of gates.

The learning based approach provides an alternative solution without the tomography of the error channel.^{45}^{)} Denoting ^{44}^{)} the training set

Note that the space ^{45}^{)} For the truncation method, we convert general errors to Pauli errors via Pauli twirling and truncate the space to a subset whose dimension only increases polynomially with the circuit size. Alternatively, we can use the Monte-Carlo method to evaluate the cost function and variationally search the parameters to minimise the cost function. In practice, we can set the starting point of the optimisation to the one specified with the results of imperfect gate set tomography.

The previous error mitigation methods mostly focus on the scenario of digital quantum simulation with discrete noisy gates. Now we review the stochastic error mitigation method for mitigating errors in continuous processes, which works for analog quantum simulation and digital quantum computing.^{46}^{)}

Suppose the dynamics of the system of interest is described by the Lindblad master equation as *H*, noise Lindblad operator *t* to *T*, we can continuously apply the recovery

Figure 19. (Color) Schematic figures for applying quasi-probability operations continuously.^{46}^{)} Note that we can obtain the result corresponding to

However, continuously applying the recovery operation could be challenging. Note that each recovery operation for a small ^{46}^{)}

Here we illustrate ways to combine different error mitigation techniques. More specifically we focus on possible combinations of error extrapolation, quasi-probability, and symmetry verification studied by Cai.^{47}^{)}

Symmetry verification cannot detect errors that stand for transformations within the same symmetry subspace. When acting on the eigenstate of the symmetry, such errors are the errors that commute with the symmetry operator, and we refer to these errors as commuting errors. Similarly we also have anti-commuting errors, and individual occurrence of them would be detectable by symmetry verification. However, when anti-commuting errors occur an even number of times, they will commute with the symmetry and thus cannot be detected. The noisy expectation value of an observable *M* at the error rate *ε* after applying symmetry verification will be denoted as ^{37}^{)} and its application to Hubbard VQE has been discussed by Cai.^{181}^{)}

Figure 20. (Color) Diagrams showing how different error components are suppressed using different combinations of quantum error mitigation techniques: (a) using only symmetry verification, (b) combining quasi-probability and symmetry verification, and (c) combining quasi-probability, symmetry verification and error extrapolation.

It is also possible to combine quasi-probability method with error extrapolation.^{46}^{,}^{47}^{)} Rather than using quasi-probability method to completely suppress all the errors, we can use it to reduce the error rate instead without changing the form of the error channel. In this way, we can obtain the noisy expectation values at several reduced error rates, and apply error extrapolation using these data points. Compared to naive error extrapolation, it does not require the ability to adjust the error rate and can achieve lower estimation errors due to lower effective error rates, albeit at a higher sampling cost due to the use of quasi-probability.^{47}^{)} The combination of these two QEM methods has also been discussed for suppressing both physical and model estimation errors of a continuous process.^{46}^{)}

Besides reducing the effective error rates, quasi-probability can also be used for transforming the form of error channels.^{47}^{)} In particular, when combined with symmetry verification, we would naturally want to use quasi-probability to remove errors that cannot be detected by symmetry verification, i.e., the commuting errors. Note that the additional gates we apply for quasi-probability might anti-commute with the symmetry and in that case we need to flip our target symmetry outcome accordingly. Additional quasi-probability can be applied to suppress the noise level of the anti-commuting errors. For the remaining anti-commuting errors, the erroneous circuit runs with an odd number of them can be detected using symmetry verification, while the erroneous circuit runs with an even number of them will still remain since they cannot be detected using symmetry verification. This is illustrated in Fig. 20(b).

Letting *μ* to be the expected number of errors in each circuit run after applying quasi-probability, then using Eq. (107) the expectation value of the observable can be described as *k* errors occurring in the circuit run.

For the exponential extrapolation that we discussed in Sect. 5.1.2, in which the observable follows a exponential decay curve: ^{47}^{)}

As mentioned, after we apply symmetry verification, only the cases with an even number of errors will remain, giving a resultant expectation value of: *μ* is large.

The remaining errors in the last section after applying quasi-probability and symmetry verification can be further suppressed by using hyperbolic extrapolation.^{47}^{)} We saw that the circuit runs with the right symmetry are those with an even number of commuting errors and their expectation value is of the form *hyperbolic extrapolation*. This is illustrated in Fig. 20(c).

Hence, by using the expectation values obtained from the two different symmetry outcomes, we can estimate the noise-free expectation value using hyperbolic extrapolation, without needing to probe at multiple error rates like in the conventional error extrapolation. This combined method can achieve a much better balance between the sampling cost and the estimation errors by utilising the strengths of different error mitigation techniques to fight different parts of the errors. Its effectiveness has been numerically demonstrated using 8-qubit Fermi-Hubbard model simulation under Pauli errors.^{47}^{)}

In this review article, we illustrated two types of hybrid quantum-classical algorithms and different quantum error mitigation methods. The variational algorithms only employ shallow depth quantum circuits and are hence tailored for noisy intermediate-scale quantum (NISQ) devices. We have classified variational algorithms into variational quantum optimisation and variational quantum simulation. Variational quantum optimisation algorithms aim to optimise a cost function tailored to a specific problem, and have wide applications for Hamiltonian spectra, machine-learning, liner algebra, etc. On the other hand, variational quantum simulation algorithms are used for simulating dynamics of quantum systems, and have applications in open quantum system simulation, linear algebra, Gibbs state preparation, etc. Meanwhile, quantum error mitigation aims to suppress errors in NISQ devices so that variational quantum algorithms can be implemented to achieve a desired calculation accuracy. Since quantum error mitigation does not generally use encoding but rely on classical post-processing, they are applicable to NISQ devices with restricted number of qubits. We have reviewed several quantum error mitigation methods and effective combination of them. Since quantum computing with NISQ devices is yet a relatively young field, whether and how these variational algorithms and error mitigation methods can be applied to solve any practically meaningful problem is still under active research investigation. This article only aims to review the most basic results in NISQ computing, and we hope it will serve as a useful reference for future researches along this direction.

## Acknowledgements

We are grateful for useful discussions with Yuuki Tokunaga, Yasunari Suzuki, Tyson Jones, Bálint Koczor, Sam McArdle, Armands Strikis, Jinzhao Sun, Nobuyuki Yoshioka, Kosuke Mitarai, Yuya Nakagawa, Yuichiro Matsuzkai, Hideaki Hakoshima, Shunsuke Kamimura, and Akira Sone. S.E. thanks Tomoyuki Uemiya and Atsushi Furukawa for insightful discussions. X.Y. acknowledges support from the Simons Foundation. Z.C. acknowledges support from St John's College, Oxford. This work is supported by MEXT Q-LEAP (Grant Nos. JPMXS0120319794 and JPMXS0118068682), JST ERATO (Grant No. JPMJER1601), and EPSRC Hub EP/T001062/1.

Applying McLachlan's variational principle,^{77}^{)} we can map the real time evolution of

The Hadamard test circuit is for calculating the expectation value of a unitary operator *U* for a given state *a* and *s* denote the ancilla and the system, respectively. By applying the controlled unitary operation *X* operator can be implemented by applying Hadamard gate and subsequently measuring the state in the computational basis. From Eq. (B·2), we can compute the real and imaginary part of *ϕ*. When the input state is described by a mixed state

Figure B·1. Quantum circuit for computing

We can also compute *U* and *V* by preparing the state *X* operator of the ancilla qubit. The quantum circuit for this task is shown in Fig. B·2.

Figure B·2. Quantum circuit for computing

Now, we will explain that coefficients in variational quantum simulation algorithms can be evaluated based on the Hadamard test circuit. Here, we consider the *M* matrix but almost the same argument holds for the *V* and *C* vectors. Note that each term constituting the *M* matrix in the variational simulation algorithms can be represented as a sum of terms as *V* and *U* share a large portion of quantum gates, the controlled operation need to be applied only to

The SWAP test and Destructive SWAP test circuits are for evaluating the overlap *ρ* and *σ*. Let

The SWAP test circuit is the quantum circuit where a unitary operator *U* is replaced with ^{188}^{)} The quantum circuit is shown in Fig. C·1. Swap test circuit needs a relatively deep quantum circuit, e.g., the number of gates scales as *ρ* and *σ*.

Figure C·1. SWAP test circuit for computing the overlap of two states *ρ* and *σ*. The number of required qubits is

Meanwhile, Destructive SWAP test circuit requires a much shallower quantum circuit. *j*th qubit of the state *j*th qubit of the state *ρ*. Thus, the measurement of

Figure C·2. Quantum circuit for computing the expectation value of *j*th qubit state of *ρ* and *σ*. When 00, 01, and 10 appear, the measurement result is +1, while it is −1 for 11. Unlike SWAP test, Destructive SWAP test does not require an ancilla qubit and its circuit depth is much shallower.

Given an ^{63}^{)} rigorously showed that a local cost function can be used for avoiding the barren plateau issue and demonstrating the trainability of the target quantum circuit.

Here we show a simple example given by Cerezo et al.^{63}^{)} Suppose we try to approximate a known target state *k*th qubit. Note that each qubit is individually measured to have

Now, we assume ^{101}^{)}

Application of the above example is to design local cost functions for linear algebra problems. The local cost function corresponding to Eq. (43) is defined via the Hamiltonian ^{101}^{)}

When naively employing conventional VQE method for parameter updates, the trial state may be trapped in local minima, and those algorithms must be repeated with random initial parameters until the energy become sufficiently close to zero. Alternatively, Hamiltonian morphing optimisation can be used to circumvent this problem.^{143}^{)} This method is analogous to adiabatic state preparation, which uses the fact that when the Hamiltonian is adiabatically changed from

Here, we show the proof of Eq. (50). The expectation value of energy in the subspace is *E* is the Lagrangian multiplier. Thus we have

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## Author Biographies

**Suguru Endo** was born in Kanagawa, Japan. He received his bachelor's and master's degrees from Keio University in 2014 and 2016 and his Ph.D. degree from University of Oxford in 2019. He has been working as a researcher in NTT secure platform laboratories since 2020. His research interests focus on hybrid quantum-classical algorithms, quantum error mitigation, and circuit QED.

**Zhenyu Cai** was born in Guangdong, China. He obtained his Ph.D. degree from University of Oxford in 2020. Since then he has been a Junior Research Fellow in Physics at St John's College, Oxford. His research interests centre around quantum error correction, quantum error mitigation and the practical applications of near-term noisy quantum computers.

**Simon C. Benjamin** is the Professor of Quantum Technologies at Materials Department in the University of Oxford. From 2014 through 2020 he was an Associate Director of the £40M Oxford-led UK National Hub on Networked Quantum Information Technologies (NQIT) and he is part of the extended leadership team for the successor Hub in Quantum Computation and Simulation (HQCS). In Oxford, Simon leads a team of 12 applied theorists who look at various aspects of quantum computing, including architectures, fault tolerance, and algorithms that are robust against imperfections in the computer. In 2017 Simon co-founded the company Quantum Motion Technologies where he is now Chief Scientific Officer leading the theory and design effort.

**Xiao Yuan** received his Bachelor in theoretical physics from Peking University in 2012 and got his Ph.D. in physics from Tsinghua University in 2016. Then he worked as a postdoc at University of Science and Technology China in 2017, at Oxford University from 2017 to 2019, and at Stanford University from 2019 to 2020. He is now an assistant professor at Center on Frontiers of Computing Studies, Peking University. Xiao Yuan's research interests focus on three aspects of quantum information science, including near-term and universal quantum computing, quantum foundation, and quantum cryptography.