A quantum algorithm for Khovanov homology

TL;DR

This paper explores quantum algorithms for computing Khovanov homology, a complex topological invariant, providing complexity results and proposing an efficient quantum approach under certain spectral conditions.

Contribution

It introduces the first quantum algorithms for approximating Khovanov homology and establishes their complexity, addressing computational challenges in topological invariants.

Findings

Additive approximations are DQC1-hard, BQP-hard, and #P-hard.

Proposed quantum algorithm is efficient under spectral gap conditions.

Introduces a pre-thermalization procedure to improve quantum homology computations.

Abstract

Khovanov homology is a topological knot invariant that categorifies the Jones polynomial, recognizes the unknot, and is conjectured to appear as an observable in $4D$ supersymmetric Yang--Mills theory. Despite its rich mathematical and physical significance, the computational complexity of Khovanov homology remains largely unknown. To address this challenge, this work initiates the study of efficient quantum algorithms for Khovanov homology. We provide simple proofs that increasingly accurate additive approximations to the ranks of Khovanov homology are DQC1-hard, BQP-hard, and #P-hard, respectively. For the first two approximation regimes, we propose a novel quantum algorithm. Our algorithm is efficient provided the corresponding Hodge Laplacian thermalizes in polynomial time and has a sufficiently large spectral gap, for which we give numerical and analytical evidence. Our approach introduces a pre-thermalization procedure that allows our quantum algorithm to succeed even if the Betti numbers of Khovanov homology are much smaller than the dimensions of the corresponding chain spaces, overcoming a limitation of prior quantum homology algorithms. We introduce novel connections between Khovanov homology and graph theory to derive analytic lower bounds on the spectral gap.