Classical and Quantum Algorithms for Topological Invariants of Torus Bundles

TL;DR

This paper develops classical and quantum algorithms for efficiently computing Witten-Reshetikhin-Turaev invariants of torus bundles by leveraging algebraic structures, achieving exponential space advantages with quantum methods.

Contribution

It introduces a novel approach using non-commutative torus structures to enable polynomial-time classical and exponential-space quantum algorithms for topological invariants.

Findings

Classical algorithms run in polynomial time with $O(N^2)$ space.

Quantum algorithms use only $O( ext{log} N)$ qubits, offering exponential space savings.

Extracting individual coefficients is #P-complete, but quantum algorithms can approximate many.

Abstract

Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative torus structure to embed the skein algebra of the closed torus into its symmetric subalgebra at roots of unity. This yields a fixed $N^2$-dimensional representation that supports polynomial-time classical computation with $O(N^2)$ space, and a quantum algorithm using only $O(\log N)$ qubits -- an exponential space advantage. We further prove that extracting individual expansion coefficients is #P-complete, yet there is a quantum algorithm that can efficiently approximate these coefficients for a non-negligible fraction of configurations.