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Preserving MWPM-Decodability in Fault-Equivalent Rewrites

Maximilian Schweikart, Linnea Grans-Samuelsson, Aleks Kissinger, Benjamin Rodatz

Abstract

Decoding a quantum error correction code is generally NP-hard, but corrections must be applied at a high frequency to suppress noise successfully. Matchable codes, like the surface code, exhibit a special structure that makes it possible to efficiently, approximately solve the decoding problem through minimum-weight perfect matching (MWPM). However, this efficiency-enabling property can be lost when constructing implementations for fault-tolerant gadgets such as syndrome-extraction circuits or logical operations. In this work, we take a circuit-centric perspective to formalise how the decoding problem changes when applying ZX rewrites to a ZX diagram with a given detector basis. We demonstrate a set of rewrites that preserve MWPM-decodability of circuits and show that these matchability-preserving rewrites can be used to fault-tolerantly extract quantum circuits from phase-free ZX diagrams. In particular, this allows us to build efficiently decodable, fault-tolerant syndrome-extraction circuits for matchable codes.

Preserving MWPM-Decodability in Fault-Equivalent Rewrites

Abstract

Decoding a quantum error correction code is generally NP-hard, but corrections must be applied at a high frequency to suppress noise successfully. Matchable codes, like the surface code, exhibit a special structure that makes it possible to efficiently, approximately solve the decoding problem through minimum-weight perfect matching (MWPM). However, this efficiency-enabling property can be lost when constructing implementations for fault-tolerant gadgets such as syndrome-extraction circuits or logical operations. In this work, we take a circuit-centric perspective to formalise how the decoding problem changes when applying ZX rewrites to a ZX diagram with a given detector basis. We demonstrate a set of rewrites that preserve MWPM-decodability of circuits and show that these matchability-preserving rewrites can be used to fault-tolerantly extract quantum circuits from phase-free ZX diagrams. In particular, this allows us to build efficiently decodable, fault-tolerant syndrome-extraction circuits for matchable codes.
Paper Structure (11 sections, 15 theorems, 26 equations, 1 figure)

This paper contains 11 sections, 15 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matchability preservation
  4. Detector-aware rewrites
  5. Matchability-preserving rewrites
  6. Matchability-preserving circuit extraction
  7. Worked example
  8. Conclusion
  9. Properties of Detector-Aware Rewrites
  10. Other matchability-preserving rewrites in the surface code
  11. Rewrite variants

Key Result

Proposition 3.10

Let $C(D)$ be a ZX diagram with a subdiagram $D$ and let $\mathcal{S}$ be a matchable Pauli web multiset of $C(D)$. Then applying a matchability-preserving rewrite $(D, \mathcal{S}[D]) \underset{R}{\circeq} (E, \mathcal{T})$ to $C(D)$ gives $C(E)$ along with a new Pauli web multiset $\mathcal{S}'$ t

Figures (1)

  • Figure 1: This figure demonstrates two core components in the ZX calculus. (a) shows how a few common building blocks from the quantum circuit model can be represented with few spiders in the ZX calculus. (b) summarises some important ZX rewriting rules.

Theorems & Definitions (43)

  • Definition 2.1: Spider
  • Definition 2.2: ZX Rewrite
  • Definition 2.3: Pauli web
  • Definition 2.4: Matchability
  • Definition 3.1: Stabilising and detecting webs, local detector basis
  • Definition 3.2: Local restriction
  • Example 3.3
  • Definition 3.4: Boundary-respecting coupling
  • Example 3.5
  • Definition 3.6: Detector-aware rewrite
  • ...and 33 more