A quantum algorithm for Khovanov homology

TL;DR

This work initiates a formal study of quantum algorithms for Khovanov homology, the categorification of the Jones polynomial, with the goal of achieving quantum speedups for knot invariants and unknot detection. It introduces a quantum algorithm that encodes the Khovanov complex into the ground state of the Hodge Laplacian and uses a novel pre-thermalization mechanism (Gibbs sampling and SWAP testing) to extract Betti numbers even when the homology has small dimension relative to the chain space. The paper also establishes complexity-theoretic hardness for increasingly precise additive approximations to Kh ranks ($ ext{DQC1}$-hard, $ ext{BQP}$-hard, and $ ext{ extsf{ extbackslash#P}}$-hard in successive regimes) and analyzes spectral gaps via homological perturbation theory, supported by extensive numerical studies and graph-theoretic connections. While efficient thermalization cannot be guaranteed in general, the authors provide analytic and numerical evidence that the spectral gap scales at most polynomially in the crossing number for typical knots, and they outline strategies to enhance the gap. Overall, these results position Khovanov homology as a promising candidate for exponential quantum speedups and lay groundwork for broader quantum-homology algorithms in low-dimensional topology and related areas.

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.

Figures (2)

• Figure 1: Three different planar diagrams for a trefoil knot.
• Figure 2: Left: Implementing the recursive algorithm that produces the Kauffmann bracket polynomial on a Hopf link with $m=2$ crossings. At each step, a crossing (circled red) is replaced by two possible smoothings. The final step produces a collection of unknotted loops, each contributing a factor of $q+q^{-1}$ to the polynomial. Right: The resulting states $\ket{r}$ and number of loops $\ell(r)$ associated with the resolution $r$. Here we enumerate the crossings from the top of the diagram down.

Theorems & Definitions (2)

• Remark 1
• Example 2