Untangling Surface Codes: Bridging Braids and Lattice Surgery

TL;DR

This work introduces a ZX-calculus–driven framework to translate fault-tolerant surface-code circuits between braiding and lattice surgery representations, enabling semantics-preserving, bidirectional conversions for arbitrary circuits. By defining two reduced instruction sets (LSIS for lattice surgery and RBIS for braids) and casting both formalisms as ICM-like circuits, the authors show how LS and braided operations reduce to multibody measurements expressed via $XX$- and $ZZ$-type measurements, and how the Raussendorf compression rule subsumes existing braid/bridge optimizations. The methodology leverages translations through loops, spiders, and bridges, and uses the bialgebra rule to realize multibody measurements that implement CNOTs in either paradigm. A new LS circuit for CNOT is demonstrated and proven equivalent to its braided counterpart, establishing a foundation for automated verification, compilation, and benchmarking toward a unified language for topological quantum computation. The work also outlines future directions for a formal categorical treatment to solidify the theoretical underpinnings and broad applicability of these translations.

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.

Figures (22)

• Figure 1: The two-body measurement circuits used throughout this manuscript. The XX-measurement (top) uses an ancilla qubit initialized in $\ket{+}$ and measured in the X-basis and has an equivalent ZX-diagram where two red spiders are touching the measured wires. The XX-measurement can be abstracted in the language of lattice surgery (LS) as two grey patch interacting along their red boundaries. The ZZ-measurement (bottom) uses an ancilla initialized in $\ket{0}$ and measured in the Z-basis. The LS diagram uses two grey patches interacted along their green boundary.
• Figure 2: There are different ways of implementing CNOTs with surface code lattice surgery. In the following the color of the square patches is indicative of the patch boundary that is about to be used. The $+/Z$ notation means that the patch will be initialized in $\ket{+}$ and measured in the Z-basis. The indices on the two-body measurements (cf. Fig. \ref{['fig:zzxx']}) indicates their order. For example, in (a) the green ZZ-measurements is performed first, and the red XX-measurement is executed afterwards. (a) The canonical patch arrangement for LS CNOT horsman2012surface; (b) extending one of the patches (e.g. the control $C$) does not change the computation; (c) after extending and rotating the ancilla patch such that it has two operator boundaries on the same side litinski2019game; (d) the ancilla is now initialised in $\ket{0}$ and not measured, the control patch is measured in the Z basis after all three patches performed a three-body red X-measurement. The correctness of (d) is shown using ZX calculus in Fig. \ref{['fig:correctlspatchcnots']}.
• Figure 3: Elements of braided circuits paler2017fault: pairs of defects (here primal, red) form logical qubits and are braided with loops of the opposite type (here dual, blue) for performing CNOTs. The circuit herein does not include bridges.
• Figure 4: CNOT implementations: (a) using braided surface codes fowler2012bridge between two primary logical qubits (here white formed by pairs of defects) mediated by a dual loop (here dark grey); (b) lattice surgery. Time flows in both diagrams from the bottom to the top. The diagrams are visually extremely similar. However, this is just the result of the white bridge that is connecting the pair of vertically parallel defects in (a). In (b), we used green and red and to highlight the different correlation surfaces spanned along the patches.
• Figure 5: Connecting correlation surfaces. Here we show how primal-red can be connected to dual-green. The reverse are also valid. a) two tubes of the same type; b) a tube and a sheet of opposite type; c) two sheets of the same type. Herein, the red strings correspond to the white defects from and the green loops to the grey defect loop from Fig. \ref{['fig:cnots']}.

Theorems & Definitions (10)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6