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Quantum Error Detection II: Bounds

A. Ashikhmin, A. Barg, E. Knill, S. Litsyn

TL;DR

This work derives both existence-based and LP-based bounds on the probability of undetected error for quantum codes over depolarizing channels. It shows that stabilizer codes can achieve exponentially small undetected-error probabilities by bounding weight enumerators via self-orthogonal quaternary codes, and provides two asymptotic upper bounds on the error exponent E(R_Q,p) that converge for high code rates. Complementing these, two lower-bound methodologies are developed: an Aaltonen–MRRW-type LP bound and a Hamming-type bound, applicable in different p and rate regimes, grounded in $q$-ary MacWilliams identities and Krawtchouk polynomial analysis. Collectively, the results extend classical coding bounds to quantum error detection, offering guidance for high-rate quantum code design under a completely depolarizing channel.

Abstract

In Part II we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent.The lower (existence) bound for stabilizer codes is proved by a counting argument for classical self-orthogonal quaternary codes. Upper bounds for any quantum codes are proved by linear programming. We present two general solutions of the LP problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1.

Quantum Error Detection II: Bounds

TL;DR

This work derives both existence-based and LP-based bounds on the probability of undetected error for quantum codes over depolarizing channels. It shows that stabilizer codes can achieve exponentially small undetected-error probabilities by bounding weight enumerators via self-orthogonal quaternary codes, and provides two asymptotic upper bounds on the error exponent E(R_Q,p) that converge for high code rates. Complementing these, two lower-bound methodologies are developed: an Aaltonen–MRRW-type LP bound and a Hamming-type bound, applicable in different p and rate regimes, grounded in -ary MacWilliams identities and Krawtchouk polynomial analysis. Collectively, the results extend classical coding bounds to quantum error detection, offering guidance for high-rate quantum code design under a completely depolarizing channel.

Abstract

In Part II we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent.The lower (existence) bound for stabilizer codes is proved by a counting argument for classical self-orthogonal quaternary codes. Upper bounds for any quantum codes are proved by linear programming. We present two general solutions of the LP problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1.

Paper Structure

This paper contains 7 sections, 14 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Upper (existence) bounds on $P_{ue}(n,K,p)$
  3. A linear program for quantum undetected error
  4. Lower Bounds on $P_{ue}(n,K,p)$
  5. An Aaltonen-MRRW-type bound
  6. Hamming-type bound
  7. Appendix

Key Result

Theorem 1

Figures (1)

  • Figure 1: Bounds on $E(R_Q,p)\!:$ Existence bound, Theorem \ref{['GV']}; Aaltonen-MRRW-type bound, Theorem \ref{['thm:qlp1']}; Hamming-type bound, Theorem \ref{['thm:qham']}; $p=0.1.$

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 4 more