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Pauli Stabilizer Models for Gapped Boundaries of Twisted Quantum Doubles and Applications to Composite Dimensional Codes

Mohamad Mousa, Amit Jamadagni, Eugene Dumitrescu

TL;DR

This work develops a constructive, Pauli-stabilizer framework to realize gapped boundaries, domain walls, and 0D defects for Abelian twisted quantum doubles by local anyon condensation starting from the bulk $D(\mathbb{Z}_4)$. It presents a concrete six-step algorithm to generate boundary stabilizers, applies it to $\mathbb{Z}_4$ and DS phases, and analyzes resulting codes via ground-state degeneracy and pants-decomposition arguments. The authors introduce DS-$\mathbb{Z}_4$ patch codes that combine $4$- and $2$-dimensional logical spaces, reveal biased error properties, and discuss decoding implications, enabling automation and scalable code design. The approach connects lattice stabilizers with TQFT data, enabling systematic exploration of composite-dimensional codes and providing a path toward near-term architectures that exploit hardware-specific strengths while expanding topological code capabilities.

Abstract

We provide new algorithms and provide example constructions of stabilizer models for the gapped boundaries, domain walls, and $0D$ defects of Abelian composite dimensional twisted quantum doubles. Using the physically intuitive concept of condensation, our algorithm explicitly describes how to construct the boundary and domain-wall stabilizers starting from the bulk model. This extends the utility of Pauli stabilizer models in describing non-translationally invariant topological orders with gapped boundaries. To highlight this utility, we provide a series of examples including a new family of quantum error-correcting codes where the double of $\mathbb{Z}_4$ is coupled to instances of the double semion (DS) phase. We discuss the codes' utility in the burgeoning area of quantum error correction with an emphasis on the interplay between deconfined anyons, logical operators, error rates and decoding. We also augment our construction, built using algorithmic tools to describe the properties of explicit stabilizer layouts at the microscopic lattice-level, with dimensional counting arguments and macroscopic-level constructions building on pants decompositions. The latter outlines how such codes' representation and design can be automated. Going beyond our worked out examples, we expect our explicit step-by-step algorithms to pave the path for new higher-algebraic-dimensional codes to be discovered and implemented in near-term architectures that take advantage of various hardware's distinct strengths.

Pauli Stabilizer Models for Gapped Boundaries of Twisted Quantum Doubles and Applications to Composite Dimensional Codes

TL;DR

This work develops a constructive, Pauli-stabilizer framework to realize gapped boundaries, domain walls, and 0D defects for Abelian twisted quantum doubles by local anyon condensation starting from the bulk . It presents a concrete six-step algorithm to generate boundary stabilizers, applies it to and DS phases, and analyzes resulting codes via ground-state degeneracy and pants-decomposition arguments. The authors introduce DS- patch codes that combine - and -dimensional logical spaces, reveal biased error properties, and discuss decoding implications, enabling automation and scalable code design. The approach connects lattice stabilizers with TQFT data, enabling systematic exploration of composite-dimensional codes and providing a path toward near-term architectures that exploit hardware-specific strengths while expanding topological code capabilities.

Abstract

We provide new algorithms and provide example constructions of stabilizer models for the gapped boundaries, domain walls, and defects of Abelian composite dimensional twisted quantum doubles. Using the physically intuitive concept of condensation, our algorithm explicitly describes how to construct the boundary and domain-wall stabilizers starting from the bulk model. This extends the utility of Pauli stabilizer models in describing non-translationally invariant topological orders with gapped boundaries. To highlight this utility, we provide a series of examples including a new family of quantum error-correcting codes where the double of is coupled to instances of the double semion (DS) phase. We discuss the codes' utility in the burgeoning area of quantum error correction with an emphasis on the interplay between deconfined anyons, logical operators, error rates and decoding. We also augment our construction, built using algorithmic tools to describe the properties of explicit stabilizer layouts at the microscopic lattice-level, with dimensional counting arguments and macroscopic-level constructions building on pants decompositions. The latter outlines how such codes' representation and design can be automated. Going beyond our worked out examples, we expect our explicit step-by-step algorithms to pave the path for new higher-algebraic-dimensional codes to be discovered and implemented in near-term architectures that take advantage of various hardware's distinct strengths.

Paper Structure

This paper contains 25 sections, 67 equations, 58 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. The $\mathbb{Z}_4$ Family of Codes
  4. Description of $\mathbb{Z}_4$
  5. Bulk
  6. External Boundaries
  7. Domain walls and defects
  8. Topological Codes
  9. Double Semion Model
  10. Bulk
  11. External Boundaries
  12. DS-$\mathbb{Z}_4$ Code
  13. Patch Error Rates
  14. Microscopic Boundary Formulation
  15. Boundaries by Local Condensation
  16. ...and 10 more sections

Figures (58)

  • Figure 1: Strings for creating e and m anyons in $\mathbb{Z}_4$. (b) Braiding and $e$ and an $m$ anyons, the two strings overlap at the purple edge.
  • Figure 2: A logical-qubit $\mathbb{Z}_4$ surface code composed of three different boundaries. The logical $\widetilde{Z}$ operator is a vertical string operator, transporting an $e^2$ particle between the rough and even boundaries. The logical $\widetilde{X}$ operator is a horizontal string operator transporting an $m$ particle between the two smooth boundaries.
  • Figure 3: Stabilizers layout for the $3$ non-trivial invertible $\mathbb{Z}_4$ domain walls ending with $0$D defects. Definitions of symbols are given in Eqs. \ref{['eq:defectlegos1']} and \ref{['eq:defectlegos2']}.
  • Figure 4: Left: a two-logical-qudit $\mathbb{Z}_4$ surface code with $N=3$ smooth punctures. The logical $\widetilde{Z}$ operators can be taken to be $e$ loops around $N-1$ holes. The logical $\widetilde{X}$ operators are $m$ strings from the $N-1$ holes to the first one. Right: another two-logical-qudit $\mathbb{Z}_4$ surface code with $N=3$ ($e\leftrightarrow m$) twists. The logical $\widetilde{Z}$ operators can be taken to be $e$ loops around $N-1$ twists. The logical $\widetilde{X}$ operators are hybrid $m$-$e$ loops between the $N-1$ twists and the first one.
  • Figure 5: A four-logical-qubit $\mathbb{Z}_4$ surface code with $N=3$ even punctures. The logical $\widetilde{Z}$ operators can be taken to be $e$ and $m$ loops around $N-1$ holes. The logical $\widetilde{X}$ operators are $m^2$ or $e^2$ strings from the $N-1$ holes to the first one. In general, we have $2(N-1)$ qubits.
  • ...and 53 more figures

Theorems & Definitions (25)

  • Remark 3.1
  • Example 3.1: $\mathbb{Z}_4$ on a Surface
  • Example 4.1: Smooth boundary for the $\mathbb{Z}_4$ surface code
  • Example 4.2: Even boundary for the $\mathbb{Z}_4$ surface code
  • Remark 4.1
  • Example 4.3: Boundary for the Doubled Semion (DS) phase
  • Example 4.4: Domain Wall $(e\leftrightarrow m)$ for $\mathbb{Z}_4$ (or $\mathbb{Z}_n$)
  • Example 4.5: Domain wall between $D(\mathbb{Z}_4)$ and DS
  • Example 5.1: $D(\mathbb{Z}_2)$ on a Disk I
  • Example 5.2: $\mathbb{Z}_2$ Surface Code
  • ...and 15 more