Classically Optimal Variational Quantum Algorithms

Jonathan Wurtz,Peter Love

doi:10.1109/tqe.2021.3122568

Abstract

Hybrid quantum-classical algorithms, such as variational quantum algorithms (VQA), are suitable for implementation on NISQ computers. In this Article we expand an implicit step of VQAs: the classical pre-computation subroutine which can non-trivially use classical algorithms to simplify, transform, or specify problem instance-specific variational quantum circuits. In VQA there is a trade-off 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 QAOA, with tractable circuit construction and optimization but poor approximation ratios. Combining these two we define the Spanning Tree QAOA (ST-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 pre-computation subroutines into VQA to improve heuristic or guaranteed performance.

Highlights

Today’s noisy intermediate scale quantum computers (NISQ) are bounded in power by size, noise and decoherence [Pre18]
By choosing the angles γc = 0, γT = π/4, β = π/4, the unitary is equivalent to Eq (7) for the particular choice of spanning tree generated by P, and so Spanning Tree QAOA (ST-QAOA) can give the same solution as the classical subroutine
Can ST-QAOA exceed the performance of the best classical algorithms for worst case graphs? This would be the case if further variational optimization is possible in ST-QAOA for all graphs when the best classical algorithm is used as a subroutine, and would represent quantum advantage

Summary

Introduction

Today’s noisy intermediate scale quantum computers (NISQ) are bounded in power by size, noise and decoherence [Pre18]. We use the VQA pre-computation step (Fig. 1B) to generate problem instance-specific circuits that use problem structure from the form and solutions of classical algorithms as well as from the objective function. This algorithm uses approximate solutions from a classical MAXCUT solver as a subroutine to construct a problem instance-specific circuit with r rounds of gates.

Results

Conclusion