String Diagrams for Defect-Based Surface Code Computing

TL;DR

This paper formalizes a graphical calculus called KNOT for defect braiding in surface codes, using ZX-calculus, enabling analysis of fault-tolerant quantum computing operations.

Contribution

It introduces the KNOT calculus and its subtheories, linking defect braiding effects to ZX-calculus, and provides a formal framework for analyzing surface code operations.

Findings

Defined the KNOT graphical calculus for defect braiding effects.

Established the soundness and completeness of subtheories for the (0, pi)-fragment.

Connected defect braiding operations to ZX-calculus for quantum computing analysis.

Abstract

Surface codes are a popular choice for implementing fault-tolerant quantum computing. Two-qubit gates may be realised in these codes using only nearest-neighbour interactions, either by lattice surgery or by braiding defects around each other. The effect of lattice surgery operations may be simply described using the ZX-calculus: a graphical language that has proven effective for program design and optimisation. In this work, we formalise a similar description via the ZX-calculus of defect braiding, as it is conventionally described. We define a graphical calculus KNOT, denoting the logical effects (in the absence of byproduct operations) of defect braiding in surface codes: we show how these effects may be described via a fragment of ZX-calculus which we call the (0, pi)-fragment. We then use a doubling construction to define a subtheory of KNOT, more specialised to standard encoding techniques in the defect braiding literature. Within this subtheory, we encompass standard braiding techniques by families of ribbon-like and tangle-like diagrams, each with semantics distinct from KNOT, in terms of the (0, pi)-fragment of ZX diagrams (again in the absence of byproducts). These subtheories may be used interoperably, and are each sound and complete for the (0, pi)-fragment of ZX diagrams. This provides a starting point to use the formal diagrammatics to analyse the operational effects of defect braiding procedures.