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arXiv:2005.04447 · Machine Learning Science and Technology

Natural evolution strategies and variational Monte Carlo

Tianchen Zhao, Giuseppe Carleo, James Stokes, Shravan Veerapaneni

Published Nov 18, 2020• 21 citations
arXivHTMLPDF

Abstract

Abstract A notion of quantum natural evolution strategies is introduced, which provides a geometric synthesis of a number of known quantum/classical algorithms for performing classical black-box optimization. The recent work of Gomes et al (2019 arXiv:1910.10675) on heuristic combinatorial optimization using neural quantum states is pedagogically reviewed in this context, emphasizing the connection with natural evolution strategies (NES). The algorithmic framework is illustrated for approximate combinatorial optimization problems, and a systematic strategy is found for improving the approximation ratios. In particular, it is found that NES can achieve approximation ratios competitive with widely used heuristic algorithms for Max-Cut, at the expense of increased computation time.

DOI
10.1088/2632-2153/abcb50

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IEEE International Conference on Electronics and Information Technologies (ELIT)•Sep 26, 2023•0 citations
Abstract A notion of quantum natural evolution strategies is introduced, which provides a geometric synthesis of a number of known quantum/classical algorithms for performing classical black-box optimization. The recent work of Gomes et al (2019 arXiv:1910.10675) on heuristic combinatorial optimization using neural quantum states is pedagogically reviewed in this context, emphasizing the connection with natural evolution strategies (NES). The algorithmic framework is illustrated for approximate combinatorial optimization problems, and a systematic strategy is found for improving the approximation ratios. In particular, it is found that NES can achieve approximation ratios competitive with widely used heuristic algorithms for Max-Cut, at the expense of increased computation time.
A notion of quantum natural evolution strategies is introduced, which provides a geometric synthesis of a number of known quantum/classical algorithms for performing classical black-box optimization. Recent work of Gomes et al. [2019] on combinatorial optimization using neural quantum states is pedagogically reviewed in this context, emphasizing the connection with natural evolution strategies. The algorithmic framework is illustrated for approximate combinatorial optimization problems, and a systematic strategy is found for improving the approximation ratios. In particular it is found that natural evolution strategies can achieve state-of-art approximation ratios for Max-Cut, at the expense of increased computation time.
Abstract Natural evolutionary strategies (NES) are a family of gradient-free black-box optimization algorithms. This study illustrates their use for the optimization of randomly initialized parameterized quantum circuits (PQCs) in the region of vanishing gradients. We show that using the NES gradient estimator the exponential decrease in variance can be alleviated. We implement two specific approaches, the exponential and separable NES, for parameter optimization of PQCs and compare them against standard gradient descent. We apply them to two different problems of ground state energy estimation using variational quantum eigensolver and state preparation with circuits of varying depth and length. We also introduce batch optimization for circuits with larger depth to extend the use of ES to a larger number of parameters. We achieve accuracy comparable to state-of-the-art optimization techniques in all the above cases with a lower number of circuit evaluations. Our empirical results indicate that one can use NES as a hybrid tool in tandem with other gradient-based methods for optimization of deep quantum circuits in regions with vanishing gradients.
One of the leading candidates for near-term quantum advantage is the class of Variational Quantum Algorithms, but these algorithms suffer from classical difficulty in optimizing the variational parameters as the number of parameters increases. Therefore, it is important to understand the expressibility and power of various ansätze to produce target states and distributions. To this end, we apply notions of emulation to Variational Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA) to show that QAOA is outperformed by variational annealing schedules with equivalent numbers of parameters. Our Variational Quantum Annealing schedule is based on a novel polynomial parameterization that can be optimized in a similar gradient-free way as QAOA, using the same physical ingredients. In order to compare the performance of ansätze types, we have developed statistical notions of Monte-Carlo methods. Monte-Carlo methods are computer programs that generate random variables that approximate a target number that is computationally hard to calculate exactly. While the most well-known Monte-Carlo method is Monte-Carlo integration (e.g. Diffusion Monte-Carlo or path-integral quantum Monte-Carlo), QAOA is itself a Monte-Carlo method that finds good solutions to NP-complete problems such as Max-cut. We apply these statistical Monte-Carlo notions to further elucidate the theoretical framework around these quantum algorithms.
In this work, we develop a framework aiming at designing quantum algorithms for combinatorial optimization problems while providing theoretical guarantees on their approximation ratios. The principal innovative aspect of our work is the construction of a time-dependent Lyapunov function that naturally induces a controlled Schrödinger evolution with a time dependent Hamiltonian for maximizing approximation ratios of algorithms. Because the approximation ratio depends on the optimal solution, which is typically elusive and difficult to ascertain a priori, the second novel component is to construct the upper bound of the optimal solution through the current quantum state. By enforcing the non-decreasing property of this Lyapunov function, we not only derive a class of quantum dynamics that can be simulated by quantum devices but also obtain rigorous bounds on the achievable approximation ratio. As a concrete demonstration, we apply our framework to Max-Cut problem, implementing it as an adaptive variational quantum algorithm based on a Hamiltonian ansatz. This algorithm avoids ansatz and graph structural assumptions and bypasses parameter training through a tunable parameter function integrated with measurement feedback.
NP-hard problems are one of the most computationally intensive challenges in computer science, appearing in many domains such as logistics, cybersecurity, drug design, and financial modelling. Classical algorithms become impractical as problem size increases due to combinatorial explosion. This paper presents a structured framework for analysing quantum algorithms targeting NP-hard optimization problems, like the Traveling Salesman Problem (TSP), Max-Cut, and Boolean Satisfiability (SAT). We have developed a taxonomy which categorizes quantum approaches—quantum annealing, variational algorithms (QAOA/VQE), and Grover-based search—based on problem encoding strategies, resource requirements, and theoretical guarantees. From the comparative analysis of existing literature, we: 1) Map algorithm classes to problem domains (e.g., QAOA for Max-Cut, annealing for TSP) 2) Identify performance trade-offs between solution quality, qubit counts, and circuit depth 3) Analyse hardware compatibility constraints across NISQ platforms 4) Reveal fundamental scalability barriers including noise susceptibility and embedding overhead The framework establishes implementation guidelines for near- term quantum hardware and proposes hybrid quantum-classical pathways to bridge theoretical potential and practical deployment. This systematic analysis prioritizes research directions for achieving quantum advantage in combinatorial optimization.
Obtaining accurate solutions to many-electron quantum systems is of critical importance, in principle, such solutions allow computing most of possible chemical and physical properties from first principles. Exact solutions require utilizing exponentially scaling algorithms, which are not practical for large systems. Neural networks provide a new approach to approximating the solution while keeping a high level of accuracy. In this work, we explore a novel method of optimizing neural network wavefunctions by employing Evolution Strategies (ES) within a Variational Monte Carlo (VMC) framework. This allows us to more robustly find solutions while keeping a small number of parameters, making it an efficient alternative to traditional methods.