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Physical Review A

Learning to learn with an evolutionary strategy applied to variational quantum algorithms

Lucas Friedrich, Jonas Maziero

Published Feb 27, 2025• 4 citations
Publisher

Abstract

Variational quantum algorithms (VQAs) employ parametrized quantum circuits optimized using classical methods to minimize a cost function. While VQAs have found broad applications, certain challenges persist. Notably, a significant computational burden arises during parameter optimization. The prevailing ``parameter shift rule'' mandates a double evaluation of the cost function for each parameter. In this paper, we introduce an optimization approach named ``learning to learn with an evolutionary strategy'' (LLES). LLES unifies ``learning to learn'' and ``evolutionary strategy'' methods. Learning to learn treats optimization as a learning problem, utilizing recurrent neural networks to iteratively propose VQA parameters. Conversely, evolutionary strategy employs gradient searches to estimate function gradients. Our optimization method is applied to two distinct tasks: determining the ground state of an Ising Hamiltonian and training a quantum neural network. The obtained results underscore the efficacy of this approach. Additionally, we identify a key hyperparameter that significantly influences gradient estimation using the evolutionary strategy method.

DOI
10.1103/physreva.111.022630

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Variational quantum algorithms (VQAs) employ parametrized quantum circuits optimized using classical methods to minimize a cost function. While VQAs have found broad applications, certain challenges persist. Notably, a significant computational burden arises during parameter optimization. The prevailing parameter shift rule'' mandates a double evaluation of the cost function for each parameter. In this paper, we introduce an optimization approach named learning to learn with an evolutionary strategy'' (LLES). LLES unifies learning to learn'' and evolutionary strategy'' methods. Learning to learn treats optimization as a learning problem, utilizing recurrent neural networks to iteratively propose VQA parameters. Conversely, evolutionary strategy employs gradient searches to estimate function gradients. Our optimization method is applied to two distinct tasks: determining the ground state of an Ising Hamiltonian and training a quantum neural network. The obtained results underscore the efficacy of this approach. Additionally, we identify a key hyperparameter that significantly influences gradient estimation using the evolutionary strategy method.
Variational Quantum Algorithms (VQAs) employ quantum circuits parameterized by UUU, optimized using classical methods to minimize a cost function. While VQAs have found broad applications, certain challenges persist. Notably, a significant computational burden arises during parameter optimization. The prevailing ``parameter shift rule'' mandates a double evaluation of the cost function for each parameter. In this article, we introduce a novel optimization approach named ``Learning to Learn with an Evolutionary Strategy'' (LLES). LLES unifies ``Learning to Learn'' and ``Evolutionary Strategy'' methods. ``Learning to Learn'' treats optimization as a learning problem, utilizing recurrent neural networks to iteratively propose VQA parameters. Conversely, ``Evolutionary Strategy'' employs gradient searches to estimate function gradients. Our optimization method is applied to two distinct tasks: determining the ground state of an Ising Hamiltonian and training a quantum neural network. Results underscore the efficacy of this novel approach. Additionally, we identify a key hyperparameter that significantly influences gradient estimation using the ``Evolutionary Strategy'' method.
Recent research has demonstrated that parametric quantum circuits (PQCs) are affected by gradients that progressively vanish to zero as a function of the number of qubits. We show that using a combination of gradient-free natural evolutionary strategy and gradient descent can mitigate the possibility of optimizing barren plateaus in the landscape. We implemented 2 specific methods: natural evolutionary strategy stochastic gradient descent (NESSGD) and natural evolutionary strategy adapting the step size according to belief in observed gradients (NESAdaBelief) to optimize PQC parameter values. They were compared with standard stochastic gradient descent, adaptive moment estimation, and a version of adaptive moment estimation adapting the step size according to belief in observed gradients in 5 classification tasks. NESSGD and NESAdaBelief demonstrated some superiority in 4 of the tasks. NESAdaBelief showed higher accuracy than AdaBelief in all 5 tasks. In addition, we investigated the applicability of NESSGD under the parameter shift rule and demonstrated that NESSGD can adapt to this rule, which means that our proposed method could also optimize the parameters of PQCs on quantum computers.
Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.
Variational quantum circuits (VQCs) are flexible models for optimization and learning based on parameterized quantum circuits trained via cost minimization. A key challenge in their optimization is the presence of barren plateaus, where gradients vanish exponentially with the number of qubits, severely limiting trainability. Local cost functions mitigate this issue by restricting evaluation to qubit subsets, improving gradient scaling. In this work, we propose an evolutionary strategy-inspired method to automatically identify informative qubit subsets for constructing local cost functions. Operating at the level of qubit selection rather than parameter optimization, the approach enables adaptive, circuit-aware objective design. Numerical experiments using the PennyLane framework show that the method alleviates barren plateaus and improves optimization performance in the identity-gate learning problem.
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.
Variational quantum algorithms (VQAs) have emerged as the leading strategy for using these noisy devices to achieve an advantage over their classical counterparts.VQAs use a hybrid quantum-classical framework, in which one employs the quantum computer to prepare a parameterized quantum state and outsource the optimization to a classical subroutine. VQAs have been proposed for applications in various fields, including physics, quantum chemistry, machine learning, and optimization. Nevertheless, many challenges must be addressed before we can use them for practical tasks. In this dissertation, we propose strategies to overcome some challenges in implementing VQAs. We first look at the issue of trainability for VQAs and propose three methods for optimizing them.The first method is based on natural evolutionary strategies (NES) and can alleviate the issue of vanishing gradients for randomly initialized parameterized quantum circuits (PQCs). The second method is based on the notion of information flow in PQCs and uses a mutual information-based distance metric to define paths for stochastic optimization of VQAs. Finally, we develop gradient evaluation procedures for optimizing objectives with the unitary coupled-cluster ansatz. Next, we study the issue of expressibility of PQCs used in variational algorithms.First, using the theory of quantum optimal control we define a notion of reachability of general ans\"atze used in VQAs. Next, using a variational autoencoder (VAE) coupled with a regression model, we try to gain a better understanding of the parameterization of PQCs. Finally, we attempt to remedy the inefficient usage of quantum resources for storing quantum data using evolutionary strategies to design quantum circuits for compressing quantum data. We present numerical evidence that the work presented here can increase trainability, efficiency, and understanding of VQAs.We believe our work is of interest to the quantum computing community as we continue to push the limits of NISQ devices for solving practical problems.