Engineering CSS surgery: compiling any CNOT in any code

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.