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arXiv:2209.10615 · Quantum

Iteration Complexity of Variational Quantum Algorithms

Vyacheslav Kungurtsev, Georgios Korpas, Jakub Mareček, Elton Yechao Zhu

Published Oct 10, 2024• 9 citations
arXivHTMLPDF

Abstract

There has been much recent interest in near-term applications of quantum computers, i.e., using quantum circuits that have short decoherence times due to hardware limitations. Variational quantum algorithms (VQA), wherein an optimization algorithm implemented on a classical computer evaluates a parametrized quantum circuit as an objective function, are a leading framework in this space. An enormous breadth of algorithms in this framework have been proposed for solving a range of problems in machine learning, forecasting, applied physics, and combinatorial optimization, among others.In this paper, we analyze the iteration complexity of VQA, that is, the number of steps that VQA requires until its iterates satisfy a surrogate measure of optimality. We argue that although VQA procedures incorporate algorithms that can, in the idealized case, be modeled as classic procedures in the optimization literature, the particular nature of noise in near-term devices invalidates the claim of applicability of off-the-shelf analyses of these algorithms. Specifically, noise makes the evaluations of the objective function via quantum circuits biased . Commonly used optimization procedures, such as SPSA and the parameter shift rule, can thus be seen as derivative-free optimization algorithms with biased function evaluations, for which there are currently no iteration complexity guarantees in the literature. We derive the missing guarantees and find that the rate of convergence is unaffected. However, the level of bias contributes unfavorably to both the constant therein, and the asymptotic distance to stationarity, i.e., the more bias, the farther one is guaranteed, at best, to reach a stationary point of the VQA objective.

DOI
10.22331/q-2024-10-10-1495

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There has been much recent interest in near-term applications of quantum computers, i.e., using quantum circuits that have short decoherence times due to hardware limitations. Variational quantum algorithms (VQA), wherein an optimization algorithm implemented on a classical computer evaluates a parametrized quantum circuit as an objective function, are a leading framework in this space. An enormous breadth of algorithms in this framework have been proposed for solving a range of problems in machine learning, forecasting, applied physics, and combinatorial optimization, among others.In this paper, we analyze the iteration complexity of VQA, that is, the number of steps that VQA requires until its iterates satisfy a surrogate measure of optimality. We argue that although VQA procedures incorporate algorithms that can, in the idealized case, be modeled as classic procedures in the optimization literature, the particular nature of noise in near-term devices invalidates the claim of applicability of off-the-shelf analyses of these algorithms. Specifically, noise makes the evaluations of the objective function via quantum circuits biased . Commonly used optimization procedures, such as SPSA and the parameter shift rule, can thus be seen as derivative-free optimization algorithms with biased function evaluations, for which there are currently no iteration complexity guarantees in the literature. We derive the missing guarantees and find that the rate of convergence is unaffected. However, the level of bias contributes unfavorably to both the constant therein, and the asymptotic distance to stationarity, i.e., the more bias, the farther one is guaranteed, at best, to reach a stationary point of the VQA objective.
Variational quantum algorithms (VQAs) offer a promising path toward using near-term quantum hardware for applications in academic and industrial research. These algorithms aim to find approximate solutions to quantum problems by optimizing a parametrized quantum circuit using a classical optimization algorithm. A successful VQA requires fast and reliable classical optimization algorithms. Understanding and optimizing how off-the-shelf optimization methods perform in this context is important for the future of the field. In this work, we study the performance of four commonly used gradient-free optimization methods [sequential least-squares quadratic programming, constrained optimization by linear approximations, the covariance matrix adaptation evolutionary strategy (CMA-ES), and the simultaneous perturbation stochastic approximation (SPSA)] to find ground-state energies of a range of small chemistry and material science problems. We test a telescoping sampling scheme (where the accuracy of the cost-function estimate provided to the optimizer is increased as the optimization converges) for all methods, demonstrating mixed results across our range of optimizers and problems chosen. We further hyperparameter tune two of the four optimizers (CMA-ES and SPSA) across a large range of models and demonstrate that with appropriate hyperparameter tuning, CMA-ES is competitive with and sometimes outperforms SPSA (which is not observed in the absence of hyperparameter tuning). Finally, we investigate the ability of an optimizer to beat the ``sampling-noise floor'' given by the sampling noise of each cost-function estimate provided to the optimizer. Our results demonstrate the necessity for tailoring and hyperparameter tuning known optimization techniques for inherently noisy variational quantum algorithms and that the variational landscape that one finds in a VQA is highly problem and system dependent. This provides guidance for future implementations of these algorithms in experiments.
Abstract We present a new optimization strategy for small-to-intermediate scale variational quantum algorithms (VQAs) on noisy near-term quantum processors which uses a Gaussian process surrogate model equipped with a classically-evaluated quantum kernel. VQAs are typically optimized using gradient-based approaches however these are difficult to implement on current noisy devices, requiring large numbers of objective function evaluations. Our approach shifts this computational burden onto the classical optimizer component of these hybrid algorithms, greatly reducing the number of quantum circuit evaluations required from the quantum processor. We focus on the variational quantum eigensolver (VQE) algorithm and demonstrate numerically that these surrogate models are particularly well suited to the algorithm’s objective function. Next, we apply these models to both noiseless and noisy VQE simulations and show that they exhibit better performance than widely-used classical kernels in terms of final accuracy and convergence speed. Compared to the typically-used stochastic gradient-descent approach to VQAs, our quantum kernel-based approach is found to consistently achieve significantly higher accuracy while requiring less than an order of magnitude fewer quantum circuit executions. We analyze the performance of the quantum kernel-based models in terms of the kernels’ induced feature spaces and explicitly construct their feature maps. Finally, we describe a scheme for approximating the best-performing quantum kernel using a classically-efficient tensor network representation of its input state and so provide a pathway for scaling this strategy to larger systems.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on variational quantum algorithms.
Fault-tolerant quantum computers, when realized, guarantee significant speedups for particular computational tasks. Quantum hardware in the near term, however, is limited to low hundreds of noisy qubits that are not error-corrected. To leverage such devices as well as accommodate rapid developments in both hardware and algorithms, variational quantum algorithms (VQAs) were introduced. These algorithms have the potential to be useful in areas such as chemistry, optimization, machine learning, among others. Despite this potential, the largest experiment to date has employed just twelve qubits to implement the Variational Quantum Eigensolver (VQE), a VQA for simulating ground state energies of fermionic systems. In this dissertation, we present several methods for extending and further developing VQAs to step closer towards realizing applications in which quantum computers can provide meaningful advantages over classical computers. In the first part, we present methods for analyzing and optimizing parameterized quantum circuits (PQCs), a core component in VQAs for approximating target quantum states, e.g. ground state wavefunctions in VQE. To compare among PQCs and guide their design and selection, we introduce a set of algorithm-independent descriptors that can be estimated from simulations. Using these descriptors, we rule out circuits with limited capabilities in expressive and entangling powers. Next, we present a strategy for optimizing PQCs with large parameter counts and high redundancy by removing circuit operations that are less important for optimizing the algorithm objective, effectively reducing the search space. We demonstrate that this method enables larger VQE simulations by converging previously difficult optimizations while additionally reducing circuit resources. In the second part, we present methods for developing VQAs that are related to the quantum hardware. We introduce a new algorithm for simulating components of a quantum processor using an existing quantum computer. The goal of this application is to leverage near-term devices and algorithms to aid in the design of more powerful next-generation quantum hardware. Lastly, we present a guide for algorithm prototyping by deploying instances of VQAs on quantum computers via the cloud. We demonstrate instances of two algorithms, the quantum autoencoder and classifier, on an eight-qubit device. While further advancements in hardware and algorithms lie ahead, our work, along with developments from previous studies, represents significant progress towards realizing practical quantum advantage.
In recent years, variational quantum algorithms (VQA) have been receiving a great deal of attention, demonstrating their applicability in a wide range of tasks and implementable even in the noisy quantum processors currently available [1]. A VQA is a hybrid quantum-classical algorithm where a problem is encoded in a cost function that can be efficiently evaluated on a parametric quantum device and the minimum represents the desired solution. A classical optimizer can then be used to tune the parameters of the quantum device in order to minimize the loss function and find the optimal solution (Fig. 1). In the field of quantum computing, one of the main promising platforms is based on the use of single photons and integrated photonic circuits since photons are a natural choice for transferring information and have different degree of freedom that can be to encode the information. Moreover the use of integrated photonic allow the realization of compact and table complex interferometers with a high number of control parameters. Even if there are few applications [2], [3], VQA with photonic platforms remains a relatively unexplored subject. One of the main problems is the application of the classical optimizer to the photonic platform since most of the algorithms are based on the estimation of the gradient of the cost function with the finite difference method but that can not be applied to a noisy apparatus.
Variational quantum algorithms (VQAs) have attracted a lot of attention from the quantum computing community for the last few years. Their hybrid quantum-classical nature with relatively shallow quantum circuits makes them a promising platform for demonstrating the capabilities of noisy intermediate scale quantum (NISQ) devices. Although the classical machine learning community focuses on gradient-based parameter optimization, finding near-exact gradients for variational quantum circuits (VQCs) with the parameter-shift rule introduces a large sampling overhead. Therefore, gradient-free optimizers have gained popularity in quantum machine learning circles. Among the most promising candidates is the simultaneous perturbation stochastic approximation (SPSA) algorithm, due to its low computational cost and inherent noise resilience. We introduce a novel approach that uses the approximated gradient from SPSA in combination with state-of-the-art gradient-based classical optimizers. We demonstrate numerically that this out-performs both standard SPSA and the parameter-shift rule in terms of convergence rate and absolute error in simple regression tasks. The improvement of our novel approach over SPSA with stochastic gradient descent (SGD) is even amplified when shot- and hardware-noise are taken into account. We also demonstrate that error mitigation does not significantly affect our results.