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List Decodable Quantum LDPC Codes

Thiago Bergamaschi, Fernando Granha Jeronimo, Tushant Mittal, Shashank Srivastava, Madhur Tulsiani

Abstract

We give a construction of Quantum Low-Density Parity Check (QLDPC) codes with near-optimal rate-distance tradeoff and efficient list decoding up to the Johnson bound in polynomial time. Previous constructions of list decodable good distance quantum codes either required access to a classical side channel or were based on algebraic constructions that preclude the LDPC property. Our construction relies on new algorithmic results for codes obtained via the quantum analog of the distance amplification scheme of Alon, Edmonds, and Luby [FOCS 1995]. These results are based on convex relaxations obtained using the Sum-of-Squares hierarchy, which reduce the problem of list decoding the distance amplified codes to unique decoding the starting base codes. Choosing these base codes to be the recent breakthrough constructions of good QLDPC codes with efficient unique decoders, we get efficiently list decodable QLDPC codes.

List Decodable Quantum LDPC Codes

Abstract

We give a construction of Quantum Low-Density Parity Check (QLDPC) codes with near-optimal rate-distance tradeoff and efficient list decoding up to the Johnson bound in polynomial time. Previous constructions of list decodable good distance quantum codes either required access to a classical side channel or were based on algebraic constructions that preclude the LDPC property. Our construction relies on new algorithmic results for codes obtained via the quantum analog of the distance amplification scheme of Alon, Edmonds, and Luby [FOCS 1995]. These results are based on convex relaxations obtained using the Sum-of-Squares hierarchy, which reduce the problem of list decoding the distance amplified codes to unique decoding the starting base codes. Choosing these base codes to be the recent breakthrough constructions of good QLDPC codes with efficient unique decoders, we get efficiently list decodable QLDPC codes.

Paper Structure

This paper contains 45 sections, 26 theorems, 106 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Improved error correction via list decoding.
  3. Our Results
  4. Technical Overview
  5. Quantum Extensions and challenges
  6. Related work
  7. Proofs to Algorithms paradigm
  8. Lower bounds
  9. Discussion and Open Problems.
  10. Outline
  11. Preliminaries
  12. Quantum CSS codes and list decoding
  13. Vector space CSS codes
  14. Folding
  15. List decoding
  16. ...and 30 more sections

Key Result

theorem 1.1

For any constant $0 <\rho < 1$, and small enough $\varepsilon_1,\varepsilon_2 > 0$, there is an infinite family of quantum LDPC codes over a constant-sized alphabet, $q(\varepsilon_1)$, such that:

Theorems & Definitions (72)

  • theorem 1.1: Informal version of \ref{['thm:near_mds_main']}
  • theorem 1.2: Informal version of \ref{['thm:list_decoding_ael']}
  • definition 2.1: CSS Codes
  • definition 2.2: Vector Space CSS code
  • remark 2.3
  • definition 2.4: List of codewords
  • definition 2.5: List Decodable vector space CSS codes, also in BGG22
  • proof
  • definition 2.8: Dual Systems and Basis
  • definition 2.9: Duality preserving map
  • ...and 62 more