Efficient quantum processing of 3-manifold topological invariants

TL;DR

This paper presents a quantum algorithm that efficiently approximates 3-manifold topological invariants within SU(2) Chern-Simons-Witten theory, extending previous link invariant algorithms to broader topological quantum field theory applications.

Contribution

It introduces a quantum algorithm for 3-manifold invariants in CSW theory using a q-deformed spin network model, expanding the scope of quantum topological computations.

Findings

Efficient quantum approximation of 3-manifold invariants.

Extension of link invariant algorithms to 3-manifold invariants.

Connection between quantum invariants and field-theoretic solvability.

Abstract

A quantum algorithm for approximating efficiently 3--manifold topological invariants in the framework of SU(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer in the sense of Wiesner and Crutchfield, where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in Refs 2,3. Thus all the significant quantities - partition functions and observables - of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite k. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and 3-manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.