Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits

TL;DR

This paper shows that universal logical gates on topologically encoded qubits can be implemented with constant-depth local unitary circuits, enabling efficient braiding of non-Abelian anyons without increasing error growth or space overhead.

Contribution

It introduces a method to perform universal logical gates via constant-depth circuits on topologically encoded qubits, reformulating braiding as an effectively instantaneous process.

Findings

Non-Abelian anyons can be braided with constant-depth circuits.

Universal gates on Fibonacci code are achievable without extra space overhead.

Braiding can be viewed as an instantaneous process rather than slow adiabatic evolution.

Abstract

A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the length of error strings do not grow by more than a constant factor. When applied to the Fibonacci code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. These results also apply directly to braiding of topological defects in surface codes. Our results reformulate the notion of braiding in general as an effectively instantaneous process, rather than as an adiabatic, slow process.