Discretized Quantum Exhaustive Search for Variational Quantum Algorithms

TL;DR

This paper introduces a discretized quantum exhaustive search method to enhance variational quantum algorithms, aiming to address optimization challenges in noisy, limited-qubit quantum devices by leveraging classical exhaustive search principles.

Contribution

It adapts classical exhaustive search techniques to quantum algorithms, providing a new approach to improve optimization in variational quantum algorithms for small and larger problems.

Findings

Demonstrated energy landscapes for various problems.

Provided examples of partial exhaustive search for larger problems.

Showed potential for understanding complex problem hardness.

Abstract

Quantum computers promise a great computational advantage over classical computers, yet currently available quantum devices have only a limited amount of qubits and a high level of noise, limiting the size of problems that can be solved accurately with those devices. Variational Quantum Algorithms (VQAs) have emerged as a leading strategy to address these limitations by optimizing cost functions based on measurement results of shallow-depth circuits. However, the optimization process usually suffers from severe trainability issues as a result of the exponentially large search space, mainly local minima and barren plateaus. Here we propose a novel method that can improve variational quantum algorithms -- ``discretized quantum exhaustive search''. On classical computers, exhaustive search, also named brute force, solves small-size NP complete and NP hard problems. Exhaustive search and efficient partial exhaustive search help designing heuristics and exact algorithms for solving larger-size problems by finding easy subcases or good approximations. We adopt this method to the quantum domain, by relying on mutually unbiased bases for the $2^n$-dimensional Hilbert space. We define a discretized quantum exhaustive search that works well for small size problems. We provide an example of an efficient partial discretized quantum exhaustive search for larger-size problems, in order to extend classical tools to the quantum computing domain, for near future and far future goals. Our method enables obtaining intuition on NP-complete and NP-hard problems as well as on Quantum Merlin Arthur (QMA)-complete and QMA-hard problems. We demonstrate our ideas in many simple cases, providing the energy landscape for various problems and presenting two types of energy curves via VQAs.