Dequantizing Short-Path Quantum Algorithms

TL;DR

This paper dequantizes certain short-path quantum algorithms for constraint satisfaction problems, showing they do not outperform classical algorithms super-quadratically, but offering new quantum-inspired classical methods.

Contribution

It identifies a classical mechanism underlying short-path quantum algorithms and provides dequantized algorithms with comparable or better classical complexity.

Findings

Classical algorithms run in time 2^{(1-c')n} for the same problems.

Quantum algorithms do not achieve super-quadratic advantage over classical methods.

Provides a new quantum-inspired approach to classical algorithm design.

Abstract

The short-path quantum algorithm introduced by Hastings (Quantum 2018, 2019) is a variant of adiabatic quantum algorithms that enables an easier worst-case analysis by avoiding the need to control the spectral gap along a long adiabatic path. Dalzell, Pancotti, Campbell, and Brand\~{a}o (STOC 2023) recently revisited this framework and obtained a clear analysis of the complexity of the short-path algorithm for several classes of constraint satisfaction problems (MAX-$k$-CSPs), leading to quantum algorithms with complexity $2^{(1-c)n/2}$ for some constant $c>0$. This suggested a super-quadratic quantum advantage over classical algorithms. In this work, we identify an explicit classical mechanism underlying a substantial part of this line of work, and show that it leads to clean dequantizations. As a consequence, we obtain classical algorithms that run in time $2^{(1-c')n}$, for some constant $c'>c$, for the same classes of constraint satisfaction problems. This shows that current short-path quantum algorithms for these problems do not achieve a super-quadratic advantage. On the positive side, our results provide a new ``quantum-inspired'' approach to designing classical algorithms for important classes of constraint satisfaction problems.