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Variational Quantum Algorithm Landscape Reconstruction by Low-Rank Tensor Completion (2405.10941v2)

Published 17 May 2024 in quant-ph, cs.AR, and cs.ET

Abstract: Variational quantum algorithms (VQAs) are a broad class of algorithms with many applications in science and industry. Applying a VQA to a problem involves optimizing a parameterized quantum circuit by maximizing or minimizing a cost function. A particular challenge associated with VQAs is understanding the properties of associated cost functions. Having the landscapes of VQA cost functions can greatly assist in developing and testing new variational quantum algorithms, but they are extremely expensive to compute. Reconstructing the landscape of a VQA using existing techniques requires a large number of cost function evaluations, especially when the dimension or the resolution of the landscape is high. To address this challenge, we propose a low-rank tensor-completion-based approach for local landscape reconstruction. By leveraging compact low-rank representations of tensors, our technique can overcome the curse of dimensionality and handle high-resolution landscapes. We demonstrate the power of landscapes in VQA development by showcasing practical applications of analyzing penalty terms for constrained optimization problems and examining the probability landscapes of certain basis states.

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Citations (4)

Summary

  • The paper presents a low-rank tensor completion method that reconstructs local VQA landscapes from limited samples, significantly reducing computational costs.
  • It converts high-dimensional cost landscapes into a compact tensor train format, effectively mitigating the curse of dimensionality inherent in variational quantum algorithms.
  • Numerical evaluations on cases like QAOA and UCCSD demonstrate improved resolution and scalability, offering practical insights for optimizing quantum algorithms.

Analyzing and Enhancing Variational Quantum Algorithms through Low-Rank Tensor Completion

The paper "Variational Quantum Algorithm Landscape Reconstruction by Low-Rank Tensor Completion" addresses challenges and opportunities in optimizing variational quantum algorithms (VQAs), a class of algorithms with potential for solving complex problems on near-term quantum devices. VQAs typically involve parameterized quantum circuits and require careful tuning of various components for efficacy. However, the computational expense associated with VQA cost landscape evaluation poses a significant barrier. This paper presents a novel methodology utilizing low-rank tensor completion techniques to efficiently reconstruct local VQA landscapes, facilitating the understanding and improvement of these algorithms.

Context and Motivation

VQAs, including the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA), show promise for solving problems ranging from quantum chemistry to graph optimization. These algorithms necessitate a parametrization dependent on quantum circuits and classical optimization, aiming to minimize a problem-specific cost function. However, due to the complexity and heuristic nature of VQAs, analyzing and tuning them to achieve optimal performance remains challenging. This complexity is exacerbated by the difficulty in computing the landscapes of VQA cost functions, which could otherwise provide valuable insights into their optimization dynamics.

Approach: Low-Rank Tensor Completion

The authors propose leveraging low-rank tensor completion to reconstruct local VQA landscapes from a limited number of sample evaluations. The method hinges on the observation that local VQA landscapes can be represented as low-rank tensors, allowing for significant reductions in dimensionality and computation. By converting the VQA landscape into a tensor train format, the authors address the curse of dimensionality and reduce the computational and storage cost associated with landscape evaluations.

This methodology is particularly pertinent given the typical exponential growth of landscape evaluations with the number of parameters. By underpinning the landscape in a compact low-rank representation, the authors' approach facilitates detailed landscape reconstruction with minimal sampling—only marginally increasing resource demands with higher resolutions.

Numerical Evaluation

Empirical evaluations demonstrate the validity of the proposed methodology across various VQA instances, such as QAOA and UCCSD in quantum chemistry. The tensor-completion-based approach consistently outperforms traditional techniques by enabling high-resolution landscape reconstructions with comparatively low overhead. The approach's scalability with respect to landscape resolution and dimensionality is another key advantage, contributing to its practical applicability.

Implications and Applications

The paper explores applications of landscape reconstruction in advancing VQA design and analysis. Notable use cases include understanding penalty terms in constrained optimization and examining the probability landscapes of specific basis states within quantum circuits. These applications highlight the approach's potential to streamline VQA tuning processes and enhance optimization strategies. Additionally, by capturing and exploiting the landscape details, researchers can more effectively identify optimization obstacles such as local minima and saddle points.

Future Directions

The approach presents several avenues for future research, both in refining the reconstruction algorithms and in extending their application. For instance, integrating prior knowledge about specific VQAs could further reduce sample complexity. Furthermore, this methodology's utility in addressing quantum noise and error mitigation in VQA execution holds promise.

Overall, the paper offers a substantive methodological advancement for analyzing and optimizing VQAs, effectively bridging a crucial gap between theoretical potential and practical feasibility of quantum algorithms on near-term quantum devices. The integration of low-rank tensor completion with quantum landscaping evidences a significant step forward in harnessing the power of quantum computing for complex problem-solving.

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