[Paper Review] Mathematics of Topological Quantum Computing

Eric C. Rowell, Zhenghan Wang|arXiv (Cornell University)|May 17, 2017

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper surveys the mathematical foundations of topological quantum computing (TQC), establishing connections between unitary modular categories, (2+1)-TQFTs, and topological phases of matter. It demonstrates how non-Abelian anyons in topological quantum field theories enable fault-tolerant quantum computation via topological invariants, with key results including the link between Jones polynomial evaluations and quantum complexity, and conjectures on (3+1)-TQFTs and loop braid groups.

ABSTRACT

In topological quantum computing, information is encoded in "knotted" quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of $10^{-10}$, and harnessed to stabilize quantum memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.

Motivation & Objective

• To unify and clarify the mathematical structures underpinning topological quantum computing, particularly unitary modular categories and (2+1)-TQFTs.
• To explain how topological invariants in topological phases of matter—such as ground state degeneracy in non-Abelian anyon systems—enable fault-tolerant quantum computation.
• To explore the computational power of topological quantum field theories, especially the connection between Jones polynomial evaluations and #P-hardness.
• To propose open problems and future directions, including the role of (3+1)-TQFTs, loop braid groups, and the realization of conformal field theories from unitary modular categories.
• To investigate the implications of TQC for foundational mathematics, including complexity classes as axioms and quantum logic.

Proposed method

• Modeling anyon systems via unitary modular categories (UMCs), which encode fusion and braiding rules of anyons.
• Establishing equivalence between unitary (2+1)-TQFTs and UMCs, enabling topological invariants to be computed via category-theoretic structures.
• Analyzing the computational complexity of evaluating the Jones polynomial at roots of unity, showing it is #P-hard for r ≠ 1,2,3,4,6.
• Using state-sum constructions from G-crossed braided fusion categories to build (3+1)-TQFTs, with lattice realizations in topological phases of matter.
• Studying representations of motion groups in 3-manifolds, particularly the loop braid group LBₙ, as generalizations of braid groups in 3D.
• Conjecturing that unitary (3+1)-TQFTs yield homotopy invariants and may generalize Dijkgraaf-Witten TQFTs via higher homotopy types.

Experimental results

Research questions

• RQ1 How can unitary modular categories and (2+1)-TQFTs be used to model topological quantum computation and ensure fault tolerance?
• RQ2 What is the computational complexity of evaluating the Jones polynomial at roots of unity, and how does this relate to quantum supremacy?
• RQ3 Can (3+1)-TQFTs be constructed from spherical 2-fusion categories or G-crossed braided fusion categories, and what are their physical and topological implications?
• RQ4 How do representations of loop braid groups in 3D topological phases generalize braid group representations in 2D TQC?
• RQ5 Can unitary (3+1)-TQFT partition functions be homotopy invariants, and do they generalize Dijkgraaf-Witten TQFTs?

Key findings

• The Jones polynomial evaluation at q = e²πi/r for r ≠ 1,2,3,4,6 is #P-hard, providing a foundational link between topological invariants and quantum computational complexity.
• Unitary (2+1)-TQFTs and unitary modular categories are categorically equivalent, providing a dual mathematical framework for modeling topological phases of matter.
• Non-Abelian anyons in topological phases—such as in fractional quantum Hall liquids—support topological degeneracy that enables topological quantum memory with experimental verification at 10⁻¹⁰ precision.
• The loop braid group LBₙ, generated by leapfrog and interchange moves of loops in 3D, generalizes the braid group and provides a new class of representations in 3D TQC.
• Conjecture: a unitary (3+1)-TQFT's partition function is a homotopy invariant, suggesting deeper connections to smooth 4-manifold classification.
• A consistent lifting of topological twist exponents (mod 1) to rational conformal weights {hᵢ} may determine a corresponding unitary chiral CFT within a given genus, linking UMCs to VOAs.