Untangling Surface Codes: Bridging Braids and Lattice Surgery

TL;DR

This paper introduces a systematic method using ZX calculus to convert and verify fault-tolerant quantum circuits between braiding and lattice surgery representations in surface codes, facilitating automation and optimization.

Contribution

It provides a verified, bidirectional translation framework between braiding and lattice surgery paradigms, unifying topological quantum computation representations.

Findings

Established equivalence between braiding and lattice surgery using ZX calculus

Demonstrated that both operations can be expressed as multibody measurements

Introduced a new CNOT circuit with lattice surgery

Abstract

We present a systematic method for translating fault-tolerant quantum circuits between their braiding and lattice surgery (LS) representations within the surface code. Our approach employs the ZX calculus to establish an equivalence between these two paradigms, enabling verified, bidirectional conversion of arbitrary surface-code-level circuits. We show that both braiding and LS operations can be uniformly expressed as compositions of multibody measurements and demonstrate that the Raussendorf compression rule encompasses all known braid and bridge optimizations. We also introduce a novel CNOT circuit with LS. Our framework provides a foundation for the automated verification, compilation, and benchmarking of large-scale surface code computations, advancing toward a unified formal language for topological quantum computation.