Classically Optimal Variational Quantum Algorithms

Abstract

Hybrid quantum-classical algorithms, such as variational quantum algorithms (VQAs), are suitable for implementation on noisy intermediate-scale quantum computers. In this article, we expand an implicit step of VQAs: the classical precomputation subroutine, which can nontrivially use classical algorithms to simplify, transform, or specify problem instance-specific variational quantum circuits. In VQA, there is a tradeoff between quality of solution and difficulty of circuit construction and optimization. In one extreme, we find VQA for MAXCUT, which are exact, but circuit design or variational optimization is NP-hard. At the other extreme are low-depth VQA, such as the Quantum Approximate Optimization Algorithm (QAOA), with tractable circuit construction and optimization but poor approximation ratios. Combining these two, we define the Spanning Tree QAOA to solve MAXCUT, which uses an ansatz whose structure is derived from an approximate classical solution and achieves the same performance guarantee as the classical algorithm and, hence, can outperform QAOA at low depth. In general, we propose integrating these classical precomputation subroutines into VQA to improve heuristic or guaranteed performance.