Fermi Fields, Clifford Alegebras and Path Integrals

Cheolhoi HWANG, Haewon LEE

William Munizzi, Howard J. Schnitzer

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.

M. de Montigny, F.C. Khanna, F.M. Saradzhev

H. Nakajima

Quantized unconstrained fermi systems, Z2 graded Clifford algebras, of finite arbitrary dimension n are analyzed in terms of hermitian fermi operators, where 2[(n+1)/2] dimensional irreducible representations are discussed and odd dimensions are focussed upon. Path integral formula for the system is rigorously presented in terms of real Grassmann numbers via the reducible representation of the system.

Mark S. Swanson