Graphical CSS Code Transformation Using ZX Calculus

TL;DR

This paper introduces a ZX calculus-based method for transforming CSS quantum error-correcting codes diagrammatically, enabling code morphing and gauge fixing with explicit graphical derivations, exemplified by the Steane and Reed-Muller codes.

Contribution

It presents a novel diagrammatic approach using ZX calculus for transforming CSS codes, including explicit rules and techniques like code morphing and gauge fixing.

Findings

Explicit graphical transformations between Steane and Reed-Muller codes.

A bidirectional rewrite rule for physical implementation of logical ZX diagrams.

Demonstration of code morphing and gauge fixing techniques with graphical derivations.

Abstract

In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code. Then we focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]] code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.