Limitations of Local Quantum Algorithms on Random Max-k-XOR and Beyond

TL;DR

This paper investigates the limitations of local quantum algorithms, including QAOA, on random constraint satisfaction problems like MAX-k-XOR, showing they cannot achieve arbitrary approximation under certain geometric properties, and confirms landscape independence at logarithmic depth.

Contribution

It introduces a generalized notion of local quantum algorithms, proves their limitations on random CSPs with the overlap-gap property, and confirms the landscape independence of QAOA at logarithmic depth.

Findings

QAOA and similar algorithms cannot arbitrarily approximate certain CSPs.

MAX-k-XOR with even k ≥ 4 exhibits the overlap-gap property.

QAOA landscape is independent of specific parameters at logarithmic depth.

Abstract

We introduce a notion of \emph{generic local algorithm} which strictly generalizes existing frameworks of local algorithms such as \emph{factors of i.i.d.} by capturing local \emph{quantum} algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). Motivated by a question of Farhi et al. [arXiv:1910.08187, 2019] we then show limitations of generic local algorithms including QAOA on random instances of constraint satisfaction problems (CSPs). Specifically, we show that any generic local algorithm whose assignment to a vertex depends only on a local neighborhood with $o(n)$ other vertices (such as the QAOA at depth less than $\epsilon\log(n)$) cannot arbitrarily-well approximate boolean CSPs if the problem satisfies a geometric property from statistical physics called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. We show that the random MAX-k-XOR problem has this property when $k\geq4$ is even by extending the corresponding result for diluted $k$-spin glasses. Our concentration lemmas confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth -- in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. One of these concentration lemmas is a strengthening of McDiarmid's inequality, applicable when the random variables have a highly biased distribution, and may be of independent interest.