The x+y Floquet code: A simple example for topological quantum computation in the path integral approach

TL;DR

This paper introduces a new fault-tolerant quantum circuit based on the path integral approach, demonstrating its application to topological quantum error correction with a focus on the toric code in a cubic lattice.

Contribution

It presents a novel fault-tolerant circuit construction using the path integral method for topological codes, including boundary handling and logical gate implementation.

Findings

Constructed a fault-tolerant circuit for the toric code in a cubic lattice.

Showed how to incorporate boundaries and corners into the circuit.

Demonstrated topologically protected logical operations like lattice surgery.

Abstract

The path-integral approach to topological quantum error correction provides a unified way to construct and analyze fault-tolerant circuits in spacetime. In this work, we demonstrate its utility and versatility at hand of a simple example: We construct a new fault-tolerant circuit for the toric-code phase by traversing its path integral on a $(x,y,z)$ cubic lattice in the $x+y$ direction. The circuit acts on qubits on a square lattice, and alternates between horizontal nearest-neighbor $CX$ gates and vertical nearest-neighbor $ZZ$ and $XX$ measurements. We show how to incorporate boundaries and corners into the fault-tolerant circuit and how to perform topologically protected logic gates. As a specific example, we consider performing a fault-tolerant logical $ZZ$ measurement via lattice surgery of two spatial rectangular blocks of our fault-tolerant circuit.