Instantaneous braids and Dehn twists in topologically ordered states

TL;DR

This paper demonstrates that braids and Dehn twists in topologically ordered states can be implemented by constant depth quantum circuits, enabling fault-tolerant logical gates with minimal resource scaling.

Contribution

It introduces a method to realize topological operations via constant depth circuits, independent of code distance or system size, advancing topological quantum computation.

Findings

Braid and Dehn twist operations can be implemented with constant depth circuits.

The method involves local geometry deformation followed by qubit permutation.

Universal logical gates can be achieved with only constant depth unitary circuits.

Abstract

A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with non-trivial topology. These degenerate states form a representation of the mapping class group (MCG) of the space, which is generated by braids of defects or anyons, and by Dehn twists along non-contractible cycles. These operations can be viewed as fault-tolerant logical gates in the context of topological quantum error correcting codes and topological quantum computation. Here we show that braids and Dehn twists can in general be implemented by a constant depth quantum circuit, with a depth that is independent of code distance $d$ and system size. The circuit consists of a constant depth local quantum circuit (LQC) implementing a local geometry deformation of the quantum state, followed by a permutation on (relabelling of) the qubits. The permutation requires permuting qubits that are separated by a distance of order $d$; it can be implemented by collective classical motion of mobile qubits or as a constant depth circuit using long-range SWAP operations (with a range set by $d$) on immobile qubits. Applying these results to certain non-Abelian quantum error correcting codes demonstrates how universal logical gate sets can be implemented on encoded qubits using only constant depth unitary circuits.