Magic and Non-Clifford Gates in Topological Quantum Field Theory

TL;DR

This paper demonstrates how topological quantum field theories naturally produce non-Clifford gates essential for universal quantum computing, linking path integrals with quantum gate construction.

Contribution

It reveals that topological path integrals generate gates across the Clifford hierarchy in various field theories, advancing topological quantum computing.

Findings

Constructed Ising interaction gate via path integration in Chern-Simons theory.

Showed the Toffoli gate is obstructed in SU(2)_1 but supported in SU(2)_3.

Derived the T gate from Dijkgraaf-Witten theory with a boundary Dehn twist.

Abstract

Non-Clifford gates, used to generate quantum magic, are essential for universal quantum computation. We show that non-Clifford gates arise naturally from path integrals in topological quantum field theories, where their magic-generating properties are determined by the algebraic data of the theory. In Chern-Simons theory, we construct the Ising interaction gate, whose generator is prepared by path integration over simple three-boundary manifolds, and show that it produces non-local magic away from discrete Clifford points. We show that the Toffoli gate is obstructed in $SU(2)_1$ by the $\mathbb{Z}_2$ fusion structure, while $SU(2)_3$ is the minimal theory supporting the required conditional logic, given the density of the mapping class group in the projective unitary group on the manifold boundary. Finally, we demonstrate that the T gate arises as a path integral in Dijkgraaf-Witten theory, with gauge group $\mathbb{Z}_4$, where a single Dehn twist on the boundary torus produces the gate without approximation. These results show that topological path integrals construct gates in multiple levels of the Clifford hierarchy, and across distinct classes of field theories, with implications for topological quantum computing.