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arXiv:2204.07331

Accelerating Noisy VQE Optimization with Gaussian Processes

Mueller, Juliane, Wim Lavrijsen, Costin Iancu, Wibe Albert de Jong

Published Apr 15, 2022• 0 citations
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Abstract

Hybrid variational quantum algorithms, which combine a classical optimizer with evaluations on a quantum chip, are the most promising candidates to show quantum advantage on current noisy, intermediate-scale quantum (NISQ) devices. The classical optimizer is required to perform well in the presence of noise in the objective function evaluations, or else it becomes the weakest link in the algorithm. We introduce the use of Gaussian Processes (GP) as surrogate models to reduce the impact of noise and to provide high quality seeds to escape local minima, whether real or noise-induced. We build this as a framework on top of local optimizations, for which we choose Implicit Filtering (ImFil) in this study. ImFil is a state-of-the-art, gradient-free method, which in comparative studies has been shown to outperform on noisy VQE problems. The result is a new method: "GP+ImFil". We show that when noise is present, the GP+ImFil approach finds results closer to the true global minimum in fewer evaluations than standalone ImFil, and that it works particularly well for larger dimensional problems. Using GP to seed local searches in a multi-modal landscape shows mixed results: although it is capable of improving on ImFil standalone, it does not do so consistently and would only be preferred over other, more exhaustive, multistart methods if resources are constrained.

DOI
10.48550/arxiv.2204.07331

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Hybrid variational quantum algorithms, which combine a classical optimizer with evaluations on a quantum chip, are the most promising candidates to show quantum advantage on current noisy, intermediate-scale quantum (NISQ) devices. The classical optimizer is required to perform well in the presence of noise in the objective function evaluations, or else it becomes the weakest link in the algorithm. We introduce the use of Gaussian Processes (GP) as surrogate models to reduce the impact of noise and to provide high quality seeds to escape local minima, whether real or noise-induced. We build this as a framework on top of local optimizations, for which we choose Implicit Filtering (ImFil) in this study. ImFil is a state-of-the-art, gradient-free method, which in comparative studies has been shown to outperform on noisy VQE problems. The result is a new method: "GP+ImFil". We show that when noise is present, the GP+ImFil approach finds results closer to the true global minimum in fewer evaluations than standalone ImFil, and that it works particularly well for larger dimensional problems. Using GP to seed local searches in a multi-modal landscape shows mixed results: although it is capable of improving on ImFil standalone, it does not do so consistently and would only be preferred over other, more exhaustive, multistart methods if resources are constrained.
Hybrid variational quantum algorithms, which combine a classical optimizer with evaluations on a quantum chip, are the most promising candidates to show quantum advantage on current noisy, intermediate-scale quantum (NISQ) devices. The classical optimizer is required to perform well in the presence of noise in the objective function evaluations, or else it becomes the weakest link in the algorithm. We introduce the use of Gaussian Processes (GP) as surrogate models to reduce the impact of noise and to provide high quality seeds to escape local minima, whether real or noise-induced. We build this as a framework on top of local optimizations, for which we choose Implicit Filtering (ImFil) in this study. ImFil is a state-of-the-art, gradient-free method, which in comparative studies has been shown to outperform on noisy VQE problems. The result is a new method: "GP+ImFil". We show that when noise is present, the GP+ImFil approach finds results closer to the true global minimum in fewer evaluations than standalone ImFil, and that it works particularly well for larger dimensional problems. Using GP to seed local searches in a multi-modal landscape shows mixed results: although it is capable of improving on ImFil standalone, it does not do so consistently and would only be preferred over other, more exhaustive, multistart methods if resources are constrained.
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.
Variational Quantum Eigensolvers (VQEs) are a powerful class of hybrid quantum-classical algorithms designed to approximate the ground state of a quantum system described by its Hamiltonian. VQEs hold promise for various applications, including lattice field theory. However, the inherent noise of Noisy Intermediate-Scale Quantum (NISQ) devices poses a significant challenge for running VQEs as these algorithms are particularly susceptible to noise, e.g., measurement shot noise and hardware noise. In a recent work, it was proposed to enhance the classical optimization of VQEs with Gaussian Processes (GPs) and Bayesian Optimization, as these machine-learning techniques are well-suited for handling noisy data. In these proceedings, we provide additional insights into this new algorithm and present further numerical experiments. In particular, we examine the impact of hardware noise and error mitigation on the algorithm's performance. We validate the algorithm using classical simulations of quantum hardware, including hardware noise benchmarks, which have not been considered in previous works. Our numerical experiments demonstrate that GP-enhanced algorithms can outperform state-of-the-art baselines, laying the foundation for future research on deploying these techniques to real quantum hardware and lattice field theory setups.
We present a collection of optimizers tuned for usage on Noisy Intermediate-Scale Quantum (NISQ) devices. Optimizers have a range of applications in quantum computing, including the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization (QAOA) algorithms. They are also used for calibration tasks, hyperparameter tuning, in machine learning, etc. We analyze the efficiency and effectiveness of different optimizers in a VQE case study. VQE is a hybrid algorithm, with a classical minimizer step driving the next evaluation on the quantum processor. While most results to date concentrated on tuning the quantum VQE circuit, we show that, in the presence of quantum noise, the classical minimizer step needs to be carefully chosen to obtain correct results. We explore state-of-the-art gradient-free optimizers capable of handling noisy, black-box, cost functions and stress-test them using a quantum circuit simulation environment with noise injection capabilities on individual gates. Our results indicate that specifically tuned optimizers are crucial to obtaining valid science results on NISQ hardware, and will likely remain necessary even for future fault tolerant circuits.
Currently available quantum processors are noisy intermediate-scale quantum (NISQ) devices. The variational quantum eigensolver (VQE) is an algorithm that is closer to near-term applicability due to lower quantum hardware requirements. In VQE, trial states with variational parameters are prepared by quantum computers, and the optimal parameters are determined by a classical optimizer. This optimization is known to be an NP-hard problem. In this work, we make a comparison between various gradient-free optimizers in terms of approximation ratio and function evaluations. As a result, we find that COBYLA method is the best to find the approximated solution with a lower number of executions of quantum computers.
Variational quantum eigensolvers are touted as a near-term algorithm capable of impacting many applications. However, the potential has not yet been realized, with few claims of quantum advantage and high resource estimates, especially due to the need for optimization in the presence of noise. Finding algorithms and methods to improve convergence is important to accelerate the capabilities of near-term hardware for VQE or more broad applications of hybrid methods in which optimization is required. To this goal, we look to use modern approaches developed in circuit simulations and stochastic classical optimization, which can be combined to form a surrogate optimization approach to quantum circuits. Using an approximate (classical CPU/GPU) state vector simulator as a surrogate model, we efficiently calculate an approximate Hessian, passed as an input for a quantum processing unit or exact circuit simulator. This method will lend itself well to parallelization across quantum processing units. We demonstrate the capabilities of such an approach with and without sampling noise and a proof-of-principle demonstration on a quantum processing unit utilizing 40 qubits.