Variational quantum algorithm, a powerful tool for practical applications of quantum computing

 

Applications such as simulating large quantum systems or solving large-scale linear algebra problems are extremely challenging for classical computers because they have extremely high computational losses. Quantum computers promise to open up these applications, although fault-tolerant quantum computers may not be available for several years. Currently available quantum devices have severe limitations, including a limited number of quantum bits and noisy processes that limit the depth of the circuit. Variational quantum algorithms (VQA), which use classical optimizers to train parametric quantum circuits, have emerged as the main strategy to address these constraints. vQA has now been proposed for almost all applications that researchers envision for quantum computers, with the most promising quantum advantages. Therefore, this paper presents in detail the principles, application scenarios, challenges and ways to cope with the challenges of the Variational Quantum Algorithm.

 

01Steps and basic concepts of Variational Quantum Algorithms

 

VQA, including VQE, VQS, QAOA, QCBM, QNN and their increasingly sophisticated improvements, have become increasingly mature as the most suitable body of relevant technology for achieving quantum dominance using gated NISQ computers.VQA utilizes classical optimization tools to run parametric quantum circuits on quantum computers.

 

Figure 1 Schematic diagram of the Variational Quantum Algorithm (VQA). the inputs to the VQA are: the loss function C(θ) that encodes the solution to the problem, the ansatz that trains the parameters to minimize the loss, and (possibly) a set of training data to be used in the optimization process. In each iteration of the loop, we use a quantum computer to efficiently estimate the loss (or its gradient). This information is fed to a classical computer that uses the capabilities of the optimizer to solve the optimization problem. Once the termination condition is satisfied, the VQA outputs an estimate of the problem solution. The form of the output depends on the specific problem. Some of the most common types of outputs are shown in the figure.

 

In order for hybrid classical-quantum computing to succeed, two challenges need to be overcome. First, we need to find parametric quantum circuits that have the expressive power to produce a good enough approximation to the optimal solution of the optimization problem in question. Second, the classical optimization of quantum circuit parameters needs to be solved fast and accurately enough. Figure 2 illustrates the process of variational quantum algorithm optimization.

 

Figure 2 Flow chart of VQA optimization. This work addresses the complexity of the classical optimization part (red).

 

One of the main advantages of Variational Quantum Algorithms (VQA) is that they provide a general framework that can be used to solve a wide variety of problems. For an optimization problem, the loss function is defined by the expectation of an observable state generated by a parameterized quantum circuit; the parameter optimization is then outsourced to a classical optimizer, and the classical computer trains the quantum circuit by optimizing the expectation of the circuit parameters. As shown in Figure 1, the steps of the Variational Quantum Algorithm (VQA) are shown as follows.

 

1) Define the loss function C, which encodes the solution of the problem.

 

2) propose ansatz, i.e., quantum operations that depend on a set of continuous or discrete parameters, θ, that can be optimized.

 

3) train this ansatz in a hybrid quantum classical loop using data from the training set to solve the optimization task

 

 

It is somewhat like a natural analog of machine learning in quantum computing. For a classical machine learning algorithm, the model is usually a neural network running on a classical computer; for a variational quantum algorithm, the model is a quantum circuit running on a quantum computer [3]. A schematic representation of both is as follows.

 

Figure 3 Comparison of machine learning algorithm and VQA algorithm

 

02Introduction of terms in the variable component sub algorithm

 

2.1 Loss function

 

An important aspect of VQA is the encoding of the problem as a loss function. Similar to classical machine learning, the loss function maps the values of the trainable parameters θ to real numbers and the optimization task is to find the global minimum of the loss function.

 

 

where U(θ) is the parameterized You, θ consists of discrete and continuous parameters, {ρk } is the input state of the training set, {Ok } is a set of observables, and fk is a function of the coding task. For a given problem, there can be different choices of fk.

 

The loss function should satisfy the following ideal criteria: (1) the loss must be plausible because the minimum value of C(θ) corresponds to the solution of the problem, and (2) measurements and possibly classical post-processing are performed on a quantum computer to come to an efficient estimate of C(θ). The implicit assumption here is that the loss cannot be efficiently computed with a classical computer, as this would imply that VQA cannot achieve quantum dominance; (3) C(θ) is operationally meaningful, with smaller values of loss representing better solutions; and (4) the loss must be trainable in order to efficiently optimize the parameters.

 

2.2 Ansatz

 

Figure 4 Schematic diagram of ansatz. You U (θ) can be expressed as a product of L You Ul (θl ), which act sequentially on the input states. ul (θl ) can be decomposed sequentially into a series of parametric and nonparametric gates.

 

The form of ansatz determines the parameters θ and they need to be trained to minimize the losses. On the one hand, the specific structure of ansatz usually depends on the specific problem, since in many cases one can use information about the problem to solve it. On the other hand, some ansatz architectures are generic and problem-independent, which means that one can use them even if no relevant information is available. For (2), the parameters can be encoded in You U(θ), applying You U(θ) to the input states of the quantum circuit. the representation of U(θ) is shown below.

 

 

where Wm is a nonparametric You and Hm is an Ermey operator.

 

2.3 Optimizer

 

For any variational quantum algorithm, the success of VQA depends on the efficiency and reliability of the optimization method employed. Here, we will discuss some optimizers designed or promoted for variational quantum algorithms. For convenience, they are divided into two categories according to whether they implement gradient descent or not.

 

Gradient descent methods. One of the most common optimization methods is to iterate in the direction indicated by the gradient. Given that only statistical estimates are available for these gradients, these strategies fall under the category of stochastic gradient descent (SGD). One SGD method introduced from the field of machine learning is Adam, which adjusts the size of the steps taken in the optimization process to allow for more efficient and accurate solutions than those obtained through basic SGD. Another approach inspired by the machine learning literature adjusts the precision (i.e., the number of executions per estimate) in each iteration, instead of the step size, in order to save the quantum resources used.

 

A different gradient-based approach is based on assuming a hypothetical time evolution, or equivalently by using a quantum natural gradient decay method based on the concept of information geometry. While standard gradient descent takes steps in the steepest direction of descent in the l2 (Euclidean) geometry of the parameter space, natural gradient descent works on a space with a metric tensor that encodes the sensitivity of the quantum state to parameter changes. Using this metric typically accelerates the convergence of the gradient update step, allowing a given level of accuracy to be achieved with fewer iterations. This approach has also been extended to include the effect of noise.

 

Other methods. Among the optimization methods for VQA that do not directly utilize gradients, the one that is probably most closely related to SGD is the simultaneous perturbative stochastic approximation (SPSA) method. sPSA can be thought of as an approximation to gradient descent, where the gradient is approximated by a single partial derivative computed from finite differences along a randomly chosen direction. Therefore, SPSA has been proposed as an effective method for VQA because it avoids computing many gradient components in each iteration. Moreover, it is shown that for a finite set of problems, the theoretical convergence of SPSA is faster than finite-difference SGD.

 

Finally, another noteworthy gradient-free method was developed specifically for VQA for problems where the objective function is a linear function of the expectation of the operator, so that C(θ) can be expressed as a sum of trigonometric functions. We can match the function dependence with a few parameters (keeping the remaining parameters unchanged) so that local parameter updates can be performed. Performing this local update sequentially over all parameters or a subset of parameters and iterating over all parameters gives a gradient-free and hyperparameter-independent optimization method. In addition, a method to accelerate convergence using Anderson acceleration is proposed.

 

03Applications

 

One of the main advantages of the VQA paradigm is that it allows task-oriented programming. That is, VQA provides a framework that can be used to handle a wide variety of tasks, and the field of VQA has become a thriving and rapidly growing area with major applications as shown below.

 

3.1 Finding ground and excited states

 

The most typical application of VQA is to estimate the lower eigenstates and the corresponding eigenvalues of a given Hamiltonian. Previous quantum algorithms for finding the ground state of a given Hamiltonian quantity were based on adiabatic state preparation and quantum phase estimation subroutines, both of which required circuit depths beyond the level of the NISQ era. Therefore, the first proposed VQA is a variational quantum instanton solver (VQE) that was developed to estimate the ground and low-excited states of a given Hamiltonian. Here, the original VQE architecture and some more advanced methods for finding ground and excited states are presented.

 

Fig. 5 Variable component quantum solver (VQE) implementation. the VQE algorithm can be used to estimate the ground state energy of a molecule EG. the interactions of the system are encoded in terms of the Hamiltonian H, which is usually represented as a linear combination of simple operator pictures. Using H as input, VQE outputs an estimated picture of the ground-state energy. In the figure, we show the results of a VQE implementation of the electronic structure problem for a picture molecule whose exact energy is indicated by the dashed line. The experimental results were obtained using two quantum bits in a superconducting quantum processor from IBM (the inset shows the quantum bit connection). Due to the presence of hardware noise, there is a gap between the estimated energy picture and the real energy. In fact, increasing the noise intensity (i.e., increasing the number c) deteriorates the quality of the solution.

 

Variational quantum eigensolver (VQE). As shown in Figure 5, the purpose of the VQE is to find a picture of the ground state energy of a Hamiltonian quantity H. The loss function is defined as the picture. For a certain ansatz U(θ) and the initial state picture, the expectation value of minimizing H over the test state picture is sought. According to the Rayleigh-Ritz variational principle, the loss is meaningful and reliable when the picture, where the equation holds when the picture is the ground state picture of H. In fact, the Hamiltonian quantity H is usually expressed as a linear combination of the product of the pictures of the bubble operator, i.e., pictures, and thus the loss function C(θ) is obtained from a linear combination of the expected values of the pictures of the bubble operator. Since actual physical systems are usually described by sparse Hamiltonian quantities, the loss function can be efficiently estimated on a quantum computer, whose computational loss usually grows polynomially at most with the system size.

 

Orthogonality constrained VQE (orthogonality constrained VQE). Once an approximate picture of the ground state is obtained, it can be used to find the excited state of H. If a is a positive constant much larger than the energy gap between the ground state and the first excited state, then the picture is a Hamiltonian quantity whose ground state is the first excited state of H. By updating the loss using VQE for the picture.

 

 

The first excited state of h can be found. The first term in Eq. is estimated by the VQE algorithm, and the second term is obtained by calculating the picture and picture superposition states. This procedure can be further extended to approximate higher excited states. In addition, it is shown that the introduction of imaginary time evolution can improve the robustness of the calculation.

 

Subspace VQE (subspace expansion method). The main idea behind subspace VQE is to train a least positive preparatory state in the lowest energy subspace of H. The subspace VQE is called weighted subspace VQE and unweighted subspace VQE. for the weighted subspace VQE, the loss function picture and the easy-to-prepare picture of the mutually orthogonal states are considered for the weighted subspace VQE. By minimizing the loss function, the subspace of the lowest eigenstate can be approximated as pictures. Since the weights are decreasing, each state picture corresponds to an eigenstate of increasing energy Hamiltonian quantity. On the other hand, the unweighted subspace VQE gives again the lowest eigenstate subspace using loss function pictures. When each state picture is a superposition of eigenstates, a further optimization of the second loss picture is required in order to rotate each state picture to an eigenstate.

 

Multistate contracted VQE (multistate contracted VQE). multistate contracted VQE can be seen as the midpoint between subspace expansion and subspace VQE. It first obtains the same lowest energy subspace picture as the unweighted subspace VQE by optimizing the picture. Instead of optimizing an extra You value, Multistate contracted VQE approximates each eigenstate as a picture whose coefficients are obtained by solving a generalized eigenvalue problem for the subspace expansion approximating S = I.

 

Adiabatically assisted VQE (Adiabatically assisted VQE). Quantum adiabatic optimization is the search for a solution to an optimization problem by slowly transforming the ground state of a simple problem into the ground state of a complex problem. These methods are closely related to classical homotopy schemes (classical homotopy schemes) for solving classical problems in optimization. According to this connection, the adiabatic auxiliary VQE uses loss function pictures, where pictures. Here the picture is the problem Hamiltonian to be concerned with, and the picture is a simple Hamiltonian whose known ground state is set as the initial state picture. During parameter optimization, s is changed from 0 to 1. The idea of Hamiltonian transformation has been used as an ansatz to obtain more challenging solutions near the endpoints.

 

Accelerated VQE (accelerated VQE.). Although Quantum Phase Estimation (QPE) provides a way to estimate the eigenenergy in the fault-tolerant era, it is not achievable in the short term. However, a nice feature of the algorithm is that the accuracy ε can be obtained through a series of measurements on a picture-based scale. this is in contrast to VQE, which requires pictures to obtain the same accuracy. This motivates the accelerated VQE algorithm, which interpolates between the VQE and QPE algorithms, using an adjustable version of the replacement metric process called α-QPE, allowing the measurement loss to be interpolated between VQE and QPE.

 

VQE is a series of techniques that offer the possibility to perform calculations in areas such as computational chemistry. In order to achieve the level of accuracy normally required, repeated output measurements must be performed for each test circuit. The large number of repetitions required can seem like a serious bottleneck that can push run times beyond availability. Although this problem first appeared in VQE, it has the potential to affect other variational algorithms as we better understand their need for accuracy.

 

In real-world research, Zapata, Cambridge Quantum, Phasecraft, QC Ware, Rahko, QunaSys, and other startups have leveraged academic talent to drive innovation and provide variants and improvements to VQE.

 

The startup Zapata has completed work on runtime estimation for a range of short-chain hydrocarbons (from methane to propargyl). For the traditional VQE approach, they estimated a runtime of 2 to 71 days per required optimization iteration, requiring multiple iterations, and potentially much longer for other molecules of interest. To close this gap, a robust amplitude estimation technique was introduced to help us use this technique on noisy quantum processors; a "classically enhanced" VQE technique was developed that allows some orbitals to be calculated using traditional approximations, while only those orbitals with strong quantum correlations are processed on quantum processors.

 

Cambridge Quantum and Roche have combined traditional computational chemistry techniques from density matrix embedding theory with VQE to detect protein-ligand binding energies. In such demonstrations, the calculations are performed on IBM and Honeywell processors.

 

The Dutch startup Qu&Co collaborated with Schrödinger, a specialist in classical computational chemistry software. Their work benefits from a deep understanding of the potential of quantum computing and the strengths and weaknesses of current classical methods. Their work questions whether VQE implementations in the NISQ era hold promise for chemical precision (typically considered as 1 kcal/mol or 1.6 mH). A common observation is that many quantum demonstrations to date have confused accuracy and precision: they have calculated accurate results, but simulated too few orbits to achieve true accuracy.

 

3.2 Dynamical quantum simulation (DQS)

 

In addition to the static eigenstate problem, variational quantum algorithms can be used to simulate the dynamical evolution of quantum systems. Traditional quantum Hamiltonian simulation algorithms, such as the Trotter-Suzuki product formulation, usually discretize time into small time steps and simulate each time evolution with a quantum circuit. As a result, the circuit depth generally grows polynomially with the system size and simulation time. The hardware errors accumulated by such deep quantum circuits can be prohibitive given the inherent noise of NISQ devices. In contrast, the variational quantum algorithm for time-containing quantum simulations uses only a shallow circuit, significantly reducing the impact of hardware noise.

 

Iterativeapproach (Iterativeapproach). Instead of directly using the positive evolution of the youngest described by the picture of the Schrödinger equation, the iterative variational algorithm considers the picture of the experimental state and maps the evolution of the state to the evolution of the parameter θ. By iteratively updating the parameters, the quantum states are effectively updated and evolved. Specifically, the linear equations of the parameters at picture time are obtained by solving the minimum picture using variational principles, such as the McLach-lan principle. By means of a modified Hadamard test circuit, each element of M and V can be measured efficiently. By solving the linear equation, the parameters from θ to the picture can be updated iteratively with a small time step ∆t. A similar variational algorithm can be used to model the Wick-rotated Schrödinger imaginary time evolution equation and general first-order differential equations with non-Ermy Hamiltonian quantities.

 

Subspaceapproach (Subspaceapproach). The weighted subspace VQE provides an alternative method for modeling dynamics in low-energy eigenstate subspaces. Using the weighted subspace VQE, it maps the computational ground state picture to the low energy eigenstate picture, i.e. the picture, which is an unknown phase. Considering the low-energy subspace, the time-evolution operator can be approximated as a picture, where the picture, the implementation steps are shown as follows: (1) rotate the state to the computational basis with the picture; (2) evolve the state with T(T), and (3) rotate the basis with the picture. Thus, for any state picture, i.e., a superposition of low-energy eigenstates, its time evolution can be simulated as a picture. Since the time evolution is directly implemented by T(t), no iterative parameter updates are involved and the circuit depth is independent of the simulation time

 

3.3 Optimization

 

So far, we have discussed the use of variational quantum algorithms for the intrinsically inherent quantum task of finding ground states and simulating the evolution of quantum states. Here, we discuss a different possibility with its use of VQA to solve classical optimization problems.

 

The best-known variational quantum algorithm for quantum-enhanced optimization is the Quantum Approximate Optimization Algorithm (QAOA), originally introduced to approximate combinatorial problems such as the constraint satisfaction (SAT) and maximum cut problems.The goal of QAOA is to prepare a quantum state and obtain a binary string that maximizes the value of the objective function with high probability at measurement time. The combinatorial optimization problem is defined as a picture whose task is to maximize a given classical objective function L(s).

 

Fig. 6 Quantum approximate optimization algorithm (QAOA). (a) Schematic representation of the adiabatic transition in ansatz. For each t ∈ [0,1], the algorithm only loosely follows the evolution of the ground state of the picture as the final state approaches the ground state. (b) Picture of the Hamiltonian volume problem for the max-cut task with the picture of the diagram. Each node (circle) in the picture represents a spin. The pictures in the vertex representation connecting the two node pictures interact with each other, and the pictures represent the Bubbleley z-calculus on spin k. The solution is encoded in the picture basis state, where green indicates spin up and down (blue).

 

QAOA prepares the ground state of the picture by lifting each classical variable to a picture of the bubbleley spin 1/2 operator, encoding L(s) in the quantum Hamiltonian picture. Inspired by the quantum adiabatic algorithm, QAOA replaces the adiabatic evolution with a p-round alternating time propagation between the problem Hamiltonian picture and a suitably chosen mixer Hamiltonian picture, see Fig. 4. As discussed in part II of the ansatz, the evolution time interval is treated as a variational parameter and is optimized classically. Thus, defining the picture, the loss function is :

 

 

Workflow of QAOA [5].

 

Step 1: preparation of the initial state of the line.

 

Step 2: initialization of the parameters β and γ to be optimized, which are mainly used to determine the rotation angles of the RZ and RX gates (usually the parameters are initialized to 0).

 

Step 3: generation of the quantum line according to the parameters.

 

Step 4: measurement of the quantum states to calculate the expectation of each subterm.

 

Step 5: calculation of the total expectation corresponding to the current parameter.

 

Step 6: pass the current parameters and their corresponding expectation values to the classical optimizer for optimization to obtain a new set of parameters.

 

Step 7: Repeat steps 3-6 until the pre-defined end conditions are met.

 

The quantum approximate optimization algorithm (QAOA) is one of the most promising hybrid quantum-classical algorithms that can be used to solve combinatorial optimization problems, maximum partitioning problems, and other difficult problems. However, the high error rate and limited fidelity of quantum gates in recent quantum devices are major obstacles to the successful implementation of QAOA algorithms.

 

With support from the Quantum Computing User Program (QCUP) at Oak Ridge National Laboratory, Ruslan Shaydulin of the Argonne National Laboratory Research Institute and Alexey Galda, research assistant professor at the University of Chicago and chief scientist at Menten AI, a quantum algorithm company, collaborated to reduce errors in QAOA evolution by exploiting the symmetry present in classical objective functions. Specifically, QAOA states are projected into symmetry-constrained subspaces, with the projection performed at the end of the circuit or throughout the evolution. They verified this approach using five quantum bits in an IBM quantum processor. When global bit-flipping symmetry is utilized, quantum state fidelity is improved by an average of 23% - one of the most successful improvements to quantum problems to date. The paper has been presented at the IEEE International Conference on Quantum Computing and Engineering in 2021 [6].

 

Wei-Feng Zhuang et al. (2021) proposed a new classical algorithm based on graph decomposition, which has a running time linearly proportional to the number of quantum bits in shallow QAOA circuits for most optimization problems except complete graphs. Numerical experiments on maximum cut, GraphColor and Sherrington-Kirkpatrick model problems show performance improvements of several orders of magnitude compared to state-of-the-art methods. The algorithm and the associated software Qcover can be used in conjunction with the NISQ processor to accelerate the demonstration of application-level quantum advantages [7].

 

3.4 Mathematical applications

 

Several variational quantum algorithms have been proposed to deal with related mathematical problems, such as solving systems of linear equations or integer decomposition. Since quantum algorithms for these tasks exist in many cases for the fault-tolerant era, the goal of VQA is to have a heuristic scale comparable to the provable scale of these non-short-term algorithms while keeping the algorithmic requirements compatible with the NISQ era.

 

Linear systems. The solution of systems of linear equations has a wide range of applications in science and engineering. Quantum computers offer the possibility of exponential acceleration for this task. Specifically, for the N × N linear system Ax = b, we consider the quantum linear system problem (QLSP), where the task is to prepare a normalized state picture such that the picture, where the picture is also in the normalized state. The complexity of the classical algorithm for this task scales polynomially in dimension N, while the complexity of the now well-known HHL quantum algorithm scales logarithmically in N. Some extension improvements are proposed. However, these pioneering quantum algorithms will be difficult to implement in the short term due to the huge requirements on circuit depth.

 

Factor decomposition. Although Shor's factorization algorithm is well known, it is still a long time away from its large-scale implementation. Therefore, variational quantum algorithms are introduced as a potential near-term alternative. This proposal relies on the fact that factorization can be formulated as an optimization problem, in particular a classical Ising model of the ground state, using QAOA for variational search of the ground state, and their numerical trial method shows that the linear layers in ansatz (p ∈ O(n)) lead to a large overlap with the ground state.

 

Principal component analysis. An important concept in data science is dimensionality reduction with principal component analysis (PCA). This involves diagonalizing the covariance matrix of the data set and selecting the eigenvector with the largest eigenvalue as the key feature of the data. Because the covariance matrix is semi-positive definite, it can be stored in the density matrix, i.e., the quantum state, and then any quantum state diagonalization method can be used for PCA, and a quantum algorithm for PCA is proposed. However, quantum phase estimation and density matrix power finding are subroutines in this algorithm, which makes it unrealizable in the NISQ era. To bring this application closer to reality, a variational quantum state diagonalization algorithm is proposed where the loss function pictures quantize the Hilbert-Schmidt distance between pictures, where Z is the decoherence channel.

 

3.5 Compilation

 

One task that can be accelerated by NISQ devices is compiling quantum programs. In quantum compilation, the goal is to transform a given YouV into a sequence of native gates U(θ) with optimal short-circuit depth. Quantum compilation plays an important role in error mitigation when the error increases with the depth of the circuit. Quantum compilation is an extremely challenging problem for classical computers. Due to the exponential complexity of classical analog quantum dynamics, several variational quantum algorithms have been introduced that can be used to speed up this task. These algorithms can be categorized as Full Mississippi Matrix Compilation (FUMC) or Fixed Input State Compilation (FISC), compiling the target Mississippi V on all input states or on a specific input state, respectively. for FISC, a loss function closely related to the entanglement fidelity is used to quantize the distance between V and U(θ). For FUMC, an alternative approach to quantify the loss is adopted, using an average gate fidelity that averages over many input and output states. It is worth noting that both FUMC and FISC exhibit resilience to hardware noise, since the loss global minimum is independent of various types of noise, and this noise recovery property is essential for error reduction using variable fractional subcompilation.

 

3.6 Error Correction

 

Quantum error correction (QEC) protects quantum bits from hardware noise. Due to the large number of sub-bits required for QEC schemes, their implementation is beyond the capability of NISQ devices. Nevertheless, QEC can still benefit NISQ hardware by reducing errors to some extent and combining it with other error mitigation methods. Specifically, traditional generic methods used to implement QEC codes typically involve unnecessarily long circuits that do not take into account hardware structure or noise type. Therefore, variable fraction quantum algorithms are introduced to address these issues in order to automatically discover or compile small quantum error correction codes for any quantum hardware and any noise.

 

The Variational Quantum Error Corrector (QVEC-TOR) was originally proposed to discover device-tailored quantum error correction codes for quantum memory. For an arbitrary k-quantum bit input state picture, prepared from the You picture acting on the reference state picture, QVEC-TOR considers two parameterized circuit pictures (n ≥ k quantum bits) and pictures (n + r quantum bits), encodes the input logic state into n quantum bits and n - k auxiliary quantum bits, respectively, and implements the recovery operation for r auxiliary quantum bits. By sequentially encoding, recovering and decoding the input state, obtaining the output, projecting the n - k auxiliary quantum bits back to the picture and discarding the last r auxiliary quantum bits, a quantum channel picture is found on the input state of ψ. The goal of QVEC-TOR is to maximize the fidelity of the picture between the output ρ and the input ψ. The solution will allow the quantum circuit to maximize the protection of the input state. Numerical simulations show that QVEC-TOR is able to find quantum codes with better performance than existing quantum codes.

 

3.7 Machine Learning and Data Science

 

Quantum machine learning (QML) typically refers to the task of learning patterns with the goal of making accurate predictions about unknown and invisible data. Specifically learning a parametric quantum circuit to solve a given task, this connection between VQA and (typical) QML applications suggests that lessons learned in one domain can be of great use in the other, thus providing a close link between the two domains.

 

Classifiers. Data classification is a prevalent task in machine learning. Given training data of the form {x(i), y(i)}, where x(i) is the input and y(i) is the label, the goal is to train a classifier to accurately predict the label of each input. Since a key aspect of the success of classical neural networks is their nonlinearity, it can be expected that this property will also be present in quantum classifiers. Parameterized quantum circuits can support linear transformations, and the tensor product structure of quanta can exploit nonlinearities. More precisely, define an input data that depends on You V(x), and then the tensor product picture or the multiplicative picture generates a nonlinear function of the input data x. In this sense, You V(x) can be used as a quantum nonlinear characteristic mapping that can exploit the eigenspace of Hilbert space; after embedding the input data x into the quantum state, a linear transformation is performed using a parametric quantum circuit picture; the loss function is defined as the error between the true value and the expected value of the measurable observable A. This approach has been used for generalization and classification tasks as well as for experimental demonstrations of variational classification.Edwin Stoudenmire and David J Schwab (2016) found that the tensor network structure of quantum mechanics even inspired classical machine learning methods.

 

Autoencoders. Autoencoders for data compression are an important foundation for machine learning. The idea here is to make information pass through bottlenecks while still keeping the data recoverable. As a quantum simulation, a variational quantum algorithm for quantum self-coding is introduced with the goal of compressing quantum data. The input to the algorithm is a picture of pure quantum states on a system composed of two parts of AB. The goal is to train an ansatz U(θ) to compress the system into subsystem A, so that each state picture can be recovered from subsystem A with high fidelity, discarding subsystem B, which is considered "garbage". Given the close connection between data compression and decoupling, the loss function is based on the overlap between the output states on B and the fixed pure states. a local version of this loss function was proposed by M. Cerezo et al. (2020), which can be well trained for large-scale problems. alex Pepper et al. (2019) experimentally implemented quantum self-encoders on quantum hardware, and is likely to be an important foundation in quantum machine learning.

 

Generating models. The idea of implementing training parametric quantum circuits for QML can also be applied to generative models, an unsupervised statistical learning task where the goal is to learn to generate probability distributions for a given data set. Let the picture be a dataset sampled from a probability distribution q(x). q(x) is the picture of the parametric probability distribution, obtained by applying to the input state and measuring the computational basis, corresponding to the quantum circuit Born machine. In principle, one would like to minimize the difference between the two distributions. However, since q(x) is not available, the loss function is defined by a negative log-likelihood picture. Brian Coyle et al. (2020) showed that quantum circuit Born machines can simulate restricted Boltzmann machines and perform sampling tasks that are difficult for classical computers to perform.

 

Quantum neural network architectures. Several quantum neural network architectures have been proposed that can be used for some of the previously mentioned tasks. In perceptron based quantum neural network architectures, each node in the neural network represents a quantum bit and their connections are given by parameterized yous acting on the input states.Iris Cong et al. (2019) and Lukas Franken et al. (2020) used quantum convolutional neural networks (Quantum Convolutional Neural Networks (QCNN) for error correction, image recognition, and to distinguish quantum states belonging to different topological phases.

 

3.8 New Fields

 

In this section, we will discuss an exciting application of the VQA framework that exploits the fact that variational quantum algorithms deal with quantum mechanical systems. That is, many variational quantum algorithms are proposed to understand and develop the mathematical and physical structure of quantum states, as well as the generalized quantum theory.

 

Quantum Foundations. NISQ computers will likely play an important role in understanding the foundations of quantum mechanics. In a sense, these devices provide the experimental platform to test fundamental theories ranging from quantum gravity to quantum Darwinism. For example, the emergence of classicality in quantum systems will soon become a computationally tractable area of research due to the increased size of NISQ computers. Along these lines, A. Arrasmith et al. (2019) proposed the Variational Consistent Histories (VCH) algorithm. Variational Consistent Histories is a formal approach to quantum mechanics that has proven useful in the study of the quantum-to-classical transition as well as quantum cosmology; however, it requires the calculation of decoherence functions between all pairs of history records. Since the number of histories grows exponentially in terms of both system size and number of dynamics, classical numerical methods are difficult to handle for this form. the VCH algorithm aims to store the decoherence functions in a quantum density matrix and then use quantum devices to efficiently compute the loss functions.

 

Quantum information theory. Another area that is likely to be of renewed interest due to NISQ computers is quantum information theory or quantum Shannon theory. For example, quantum auto-coding algorithms could potentially be used to learn the achievable rates of coding and quantum channel transmission. Another area of research is the use of NISQ computers to compute key quantities in quantum information theory, such as the von Neumann entropy. Although these problems are known to be difficult for general quantum states, M. Cerezo et al. (2020) introduced VQA to estimate the quantum fidelity between an arbitrary state σ and a low-rank state ρ. Moreover, GuillaumeVerdon et al. (2019) introduced VQA to learn modular Hamiltonian quantities, which provide an upper bound on the von Neumann entropy of quantum states.

 

Entanglement spectroscopy. Describing entanglement is essential for understanding condensed matter systems, entanglement spectroscopy has proven useful in studying topological order, and several variational quantum algorithms have been introduced to extract the entanglement spectra of quantum states. Since the entanglement spectrum can be considered as the main component of the simplified density matrix, PCA algorithms can be used for this purpose. In addition, variational algorithms for quantum singular value decomposition can be used to describe the entanglement in the ground state prepared by VQE (e.g., topological order), and different variational quantum algorithms can be used together in a complementary way.

 

Quantum metrology. Quantum metrology is a field of study in which one seeks the optimal setup to find the parameter of interest with minimal scattered particle noise. The parameter being probed is usually the magnetic field, which gives an analytical solution for the best probe state in the absence of noise during the probe; however, when general physical noise is present, analytical arguments are difficult and variational quantum metrology searches for the best probe state in a variational way.

 

04 Challenges and possible solutions

 

Despite the tremendous developments in the field of VQA, there are still many challenges that need to be addressed. Understanding the limitations of VQA is crucial for developing strategies that can be used to construct better algorithms with guaranteed performance, and even to make better quantum hardware. In this section, we review some of the known challenges of VQA and how to address them, namely trainability of loss functions, efficiency and accuracy of loss estimation.

 

4.1 Trainability

 

The Barren Plateau (BP) phenomenon in loss functions has recently received considerable attention as one of the major bottlenecks in VQA. When a given loss function C(θ) exhibits a BP, the magnitude of its partial derivatives will vanish exponentially with the size of the system on average, as shown in the figure below. Therefore, in BP, an exponential accuracy is required to resolve the finite sampling noise and determine the direction of minimum loss. The exponential accuracy destroys the hope of achieving quantum advantage using VQA because its computational complexity will be no better than that of classical algorithms.

 

Fig. 7 Barren plateau (BP) phenomenon. a) Variance of the partial derivatives of the loss function with respect to the number of quantum bits. Results are obtained from a VQE implementation with deep unstructured ansatz, the y-axis is on a logarithmic scale, and the variance vanishes exponentially with system size as the number of quantum bits increases. b) Visualization of the landscape of the global loss function, which shows the BP used for the quantum compiled implementation, the orange (blue) landscape is obtained for n = 24 (n = 4) quantum bits. The landscape becomes flatter as the number of quantum bits increases. In addition, the loss exhibits a narrow valley phenomenon, in which the number of parameters leading to small loss values decreases exponentially with n.

 

When randomly initialized, the deeply unstructured parametric quantum circuit exhibits BP. the proof of this result relies on the fact that the unstructured ansatz becomes 2-design as the depth of the ansatz grows polynomially with the number of quantum bits n. One can view this phenomenon as a leap, since the ansatz is problem agnostic and needs to be explored an exponentially large space to find a solution. Thus, the probability of finding a solution when randomly initializing ansatz is exponentially small.

 

The BP phenomenon relies on the loss function: the global loss function exhibits BP when comparing operators or states existing in an exponentially large Hilbert space, while the local loss function exhibits a gradient when comparing objects at the level of single quantum bits, which vanishes polynomially in n as long as the circuit depth is at most logarithmic in n.

 

When training a perceptron based quantum neural network with random initialization, the BP phenomenon arises from massive entanglement, which is generated by the perceptron connecting a large number of quantum bits in the visible and hidden layers. When tracking the quantum bits in the hidden layer, the state of the visible quantum bits exponentially approaches the maximum mixed state. Obviously, it is difficult to extract information from such a state.

 

There are two main approaches to avoid or mitigate the effect of BP.

 

1) Parameter initialization. Selection of suitable parameters at the beginning of the optimization.

 

2) ansatz selection. Use a structured ansatz that limits the space for ansatz exploration during the optimization process.

 

4.2 Efficiency

 

Another requirement that the variational quantum algorithm must satisfy in order to achieve quantum dominance is to have an efficient method for estimating the expectation values of the loss function. the presence of BP can exponentially increase the accuracy required for the optimization part of VQA, but even in the absence of BP, the estimation of these expectation values is not guaranteed to be efficient. Indeed, early estimates of resource requirements suggest that the number of measurements required for interesting quantum chemical VQE problems would be astronomical, which makes solving this problem critical for quantum dominance. For Hamiltonian quantities, the decomposition into bubble operator pictures, this decomposition contains many terms, and the measurement difficulty increases.

 

There are the following methods to improve the efficiency in estimating expectations.

 

1) Swap the set of operators. To reduce the number of measurements needed to estimate the expectation of an operator, one can divide the set of Bubbleley strings into subsets that can be swapped and measured. The simplest way to do this division is to look for subsets with qubit-wise commuting, i.e. each quantum bit swaps bubble operators.

 

2) Optimal sampling. Giving each bubblelee operator a number of measurement opportunities proportional to the picture, where the picture is the variance.

 

Currently, the startup Zapata has developed an open source tool orqviz to help solve optimization challenges by helping visualize optimization scenarios.

 

4.3 Accuracy

 

One of the main goals of the Variational Quantile algorithm is to provide practical applications for recent noisy devices. In this sense, VQAs provide a strategy to deal with hardware noise as they minimize the depth of the quantum circuit. Hardware noise can affect the accuracy of VQA in several ways: hardware noise may slow down the training process so that the global optimum with noise no longer corresponds to the global optimum without noise, which in turn affects the final optimal loss value.

 

Quantum error mitigation (QEM) typically reduces physical errors in the expected value of observations by classical post-processing of measurement results, such as extrapolation.

 

Fig. 8 Study of quantum bit trajectories in Bloch spheres using the zero-noise extrapolation (ZNE) technique. The accuracy of noisy quantum computers can be improved with the ZNE error mitigation method. a) Here, one repeats a given calculation with different noise levels. The green curve corresponds to a rotation in the Bloch sphere that has a higher noise level than the rotation of the red curve. b) From the data obtained from the red and green curves, ZNE can be used to estimate the blue trajectory in the absence of noise.

 

05Summing up

 

The analysis of Variational Quantum Algorithms (VQA) will become increasingly important in the pursuit of quantum dominance. In this paper, we first detailed the principle of VQA and the steps to implement it. VQA runs parametric quantum circuits on a quantum computer and then outsources the parametric optimization to a classical optimizer. Then eight applications of the Variational Quantum Algorithm are presented, such as finding ground and excited states, time-containing quantum simulations, and optimization, showing that the field of Variational Quantum Algorithm has become a thriving and rapidly growing one.

 

The real promise of quantum computers, i.e., the acceleration of practical applications, often referred to as quantum advantage, has not yet been realized. Moreover, the emergence of fault-tolerant quantum computers seems to be many years, if not decades, away. Therefore, the key technical question is how to best leverage today's NISQ devices to achieve quantum advantage. In order to construct better algorithms as well as quantum hardware, this paper reviews some known challenges of VQA, namely trainability of loss functions, efficiency and accuracy of loss estimation, and how to address them.

 

Reference：

[1] M. Cerezo,et al.Variational Quantum Algorithms.arXiv:2012.09265

[2] Bittel L,Kliesch M.Training variational quantum algorithms is NP-hard -- even for logarithmically many qubits and free fermionic systems[J].2021.

[3] https://zhuanlan.zhihu.com/p/358552587

[4] https://www.factbasedinsight.com/quantum-algorithms-outlook-2022/

[5] 郭国平，陈昭昀，郭光灿.量子计算编程与入门[M]科学出版社，2020.

[6] https://ieeexplore.ieee.org/document/9605320/authors#authors

[7] Efficient Classical Computation of Quantum Mean Values for Shallow QAOA Circuits，arXiv: 2112.11151