Engineering CSS surgery: compiling any CNOT in any code

TL;DR

This paper presents a systematic framework for implementing logical CNOT gates in CSS quantum error correction codes using surgery protocols, with a focus on fault-tolerance and general applicability.

Contribution

It introduces a novel methodology employing subcodes and homological algebra to design and analyze surgery protocols for any CSS code without assumptions on logical operator structure.

Findings

Developed a universal surgery protocol for CNOT in CSS codes

Ensured fault-tolerance in the proposed surgery method

Validated the approach with a proof-of-concept implementation

Abstract

We introduce a framework for implementing logic in CSS quantum error correction codes, building on the surgery methods of Cowtan and Burton [CB24]. Our approach offers a systematic methodology for designing and analysing surgery protocols. At the physical level, we introduce the concept of subcodes, which encapsulate all the necessary data for performing surgery. At the logical level, leveraging homological algebra, subcodes enable us to track the logical operations induced by any surgery protocol, regardless of the choice of logical operator basis. In particular, we make no assumptions on the structure of the logical operators of the code. As a proof of concept, we develop a surgery protocol inspired by lattice surgery that implements a logical CNOT gate between any two logical qubits of any CSS code, with fault-tolerance guarantees.