Forbidden subspaces for level-1 QAOA and IQP circuits

TL;DR

This paper characterizes the problem instances solvable by level-1 QAOA and IQP circuits, revealing fundamental differences from quantum and classical annealing methods and establishing limits based on subspace dimensions.

Contribution

It introduces a class of problem Hamiltonians for level-1 QAOA, identifies limitations for certain subspace dimensions, and compares QAOA's capabilities with QA and SA.

Findings

QAOA can exactly solve problems with specific subspace structures.

Certain problem instances are exponentially hard for QA and SA but solvable by QAOA.

No genuine solutions exist for subspaces larger than 2 but smaller than 2^n.

Abstract

We present a thorough investigation of problems that can be solved exactly with the level-1 Quantum Approximate Optimization Algorithm (QAOA). To this end we implicitly define a class of problem Hamiltonians that employed as phase separator in a level-1 QAOA circuit provide unit overlap with a target subspace spanned by a set of computational basis states. For one-dimensional target subspaces we identify instances within the implicitly defined class of Hamiltonians for which Quantum Annealing (QA) and Simulated Annealing (SA) have an exponentially small probability to find the solution. Consequently, our results define a first demarcation line between QAOA, QA and SA, and highlight the fundamental differences between an interference-based search heuristic such as QAOA and heuristics that are based on thermal and quantum fluctuations like SA and QA respectively. Moreover, for two-dimensional solution subspaces we are able to show that the depth of the QAOA circuit grows linearly with the Hamming distance between the two target states. We further show that there are no genuine solutions for target subspaces of dimension higher than $2$ and smaller than $2^n$. We also transfer these results to Instantaneous Quantum Polynomial (IQP) circuits.