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Topological defects in general quantum LDPC codes

Pak Kau Lim, Kirill Shtengel, Leonid P. Pryadko

Abstract

We consider the structure of defects carrying quantum information in general quantum low-density parity-check (LDPC) codes. These generalize the corresponding constructions for topological quantum codes, without the need for locality. Relation of such defects to (generalized) topological entanglement entropy is also discussed.

Topological defects in general quantum LDPC codes

Abstract

We consider the structure of defects carrying quantum information in general quantum low-density parity-check (LDPC) codes. These generalize the corresponding constructions for topological quantum codes, without the need for locality. Relation of such defects to (generalized) topological entanglement entropy is also discussed.

Paper Structure

This paper contains 9 sections, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Defect construction
  3. Distance bounds for a defect in a CSS code
  4. Code with locally linearly-independent generators
  5. Stabilizer group with an expansion
  6. Defect codes with arbitrary large distances
  7. Relation with topological entanglement entropy
  8. Conclusions
  9. All the proofs

Figures (1)

  • Figure 1: (a) Homologically trivial hole on a torus. (b) Surface code with a smooth boundary. Removing qubits (edges) inside of the circle we get a non-trivial defect. (c) This circle contains a boundary edge with no neighboring plaquette; removing the corresponding edges we again get a trivial defect.

Theorems & Definitions (3)

  • proof : Proof of Statement \ref{['th:CSS-decomposition']}
  • proof : Proof of Statement \ref{['th:lower-dX-bound']}
  • proof : Proof of Statement \ref{['th:lower-dZ-bound-new']}