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On Error Thresholds for Pauli Channels: Some answers with many more questions

Avantika Agarwal, Alan Bu, Amolak Ratan Kalra, Debbie Leung, Luke Schaeffer, Graeme Smith

Abstract

This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.

On Error Thresholds for Pauli Channels: Some answers with many more questions

Abstract

This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.
Paper Structure (21 sections, 2 theorems, 24 equations, 11 figures, 11 tables)

This paper contains 21 sections, 2 theorems, 24 equations, 11 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Our Results
  4. Some Lessons
  5. Preliminaries
  6. Setup and Construction
  7. Notation.
  8. Review of Weight Enumerators
  9. Weight Enumerators for Repetition Code
  10. Weight Enumerators for Concatenated Repetition Code
  11. Description of threshold estimation algorithm
  12. Overview:
  13. Stabilizer coset entropy:
  14. Normalizer coset entropy:
  15. Search over Stabilizer Codes
  16. ...and 6 more sections

Key Result

theorem 3.1

Consider an $[[n,1]]$ repetition code with $Z$-type stabilizers. Let $M = X^{\otimes k} \otimes I^{\otimes (n-k)}$ be a Pauli error on $n$ qubits. The stabilizer and normalizer coset enumerators for $M$ are: where $[M]_S$ denotes the stabilizer coset of $M$ and $[M]_{N(S)}$ denotes the normalizer coset of $M$.

Figures (11)

  • Figure 1: The depiction of the channel capacity setup divincenzo1998quantum. The input is first encoded using a random stabilizer code, and each output qubit is further encoded by a code C. The final encoded qubits pass through the channel $\mathcal{N}_l = \mathcal{N}^{\otimes l}$. The decoder then performs syndrome measurements of C followed by those of the random stabilizer code.
  • Figure 2: Order of code concatenation
  • Figure 3: Normalizer coset partitioned by stabilizer cosets
  • Figure 4: Pictorial representation of Cases 3 and 4
  • Figure 5: Partitioning of encoded qubits for concatenated repetition codes.
  • ...and 6 more figures

Theorems & Definitions (4)

  • definition 2.1: Weight Enumerator gottesman2024surviving
  • definition 2.2: Complete Weight Enumerator
  • theorem 3.1
  • theorem 3.2