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SDP bounds on quantum codes: rational certificates

Gerard Anglès Munné, Felix Huber

Abstract

A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon $18$ upper bounds on the maximum size of $n$-qubit codes with $6 \leq n \leq 19$. Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.

SDP bounds on quantum codes: rational certificates

Abstract

A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon upper bounds on the maximum size of -qubit codes with . Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.
Paper Structure (15 sections, 3 theorems, 33 equations, 1 table)

This paper contains 15 sections, 3 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. Linear programming bounds on quantum codes
  3. Semidefinite Programming bounds on quantum codes
  4. Main SDP
  5. Lovász SDP for self-dual codes
  6. Symmetry-reduced SDP
  7. Code bounds
  8. Methods
  9. Table
  10. Terwiliger Algebra factors
  11. Dual programs
  12. General codes
  13. Pure codes
  14. Self-dual codes
  15. Additive codes

Key Result

Proposition 7

For pure codes the dual of SDP eq:sdpx_relax with the additional constraint Eq. eq:klred_pure_xijtp consists of SDP eq:dual_solall with the following modifications:

Theorems & Definitions (12)

  • Example 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Example 6
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • ...and 2 more