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Towards Minimal Fault-tolerant Error-Correction Sequence with Quantum Hamming Codes

Sha Shi, Xiao-Yang Xu, Min-Quan Cheng, Dong-Sheng Wang, Yun-Jiang Wang

TL;DR

The paper tackles the overhead of fault-tolerant measurement sequences in quantum error correction, focusing on distance-3 FTQEC with quantum Hamming codes. It introduces a minimal FTMS framework achieving length $2r+1$ for codes $[\![2^r-1, 2^r-1-2r, 3]\!]$ with $r=3k+1$, via a constructive cyclic transformation of stabilizer rows that preserves a self-dual CSS-like symmetry. The authors provide a necessary-and-sufficient condition for the existence of such sequences and an explicit method using a cyclic polynomial $g(x)$ to generate transformation matrices $C$, yielding a highly hardware-efficient FTQEC sequence. The findings offer a pathway to substantial reductions in both time and hardware overhead for FTQEC and may generalize to other stabilizer codes.

Abstract

The high overhead of fault-tolerant measurement sequences (FTMSs) poses a major challenge for implementing quantum stabilizer codes. Here, we address this problem by constructing efficient FTMSs for the class of quantum Hamming codes $[\![2^r-1, 2^r-1-2r, 3]\!]$ with $r=3k+1$ ($k \in \mathbb{Z}^+$). Our key result demonstrates that the sequence length can be reduced to exactly $2r+1$-only one additional measurement beyond the original non-fault-tolerant sequence, establishing a tight lower bound. The proposed method leverages cyclic matrix transformations to systematically combine rows of the initial stabilizer matrix and preserving a self-dual CSS-like symmetry analogous to that of the original quantum Hamming codes. This induced symmetry enables hardware-efficient circuit reuse: the measurement circuits for the first $r$ stabilizers are transformed into circuits for the remaining $r$ stabilizers simply by toggling boundary Hadamard gates, eliminating redundant hardware. For distance-3 fault-tolerant error correction, our approach simultaneously reduces the time overhead via shorting the FTMS length and the hardware overhead through symmetry-enabled circuit multiplexing. These results provide an important advance towards the important open problem regarding the design of minimal FTMSs for quantum Hamming codes and may shed light on similar challenges in other quantum stabilizer codes.

Towards Minimal Fault-tolerant Error-Correction Sequence with Quantum Hamming Codes

TL;DR

The paper tackles the overhead of fault-tolerant measurement sequences in quantum error correction, focusing on distance-3 FTQEC with quantum Hamming codes. It introduces a minimal FTMS framework achieving length for codes with , via a constructive cyclic transformation of stabilizer rows that preserves a self-dual CSS-like symmetry. The authors provide a necessary-and-sufficient condition for the existence of such sequences and an explicit method using a cyclic polynomial to generate transformation matrices , yielding a highly hardware-efficient FTQEC sequence. The findings offer a pathway to substantial reductions in both time and hardware overhead for FTQEC and may generalize to other stabilizer codes.

Abstract

The high overhead of fault-tolerant measurement sequences (FTMSs) poses a major challenge for implementing quantum stabilizer codes. Here, we address this problem by constructing efficient FTMSs for the class of quantum Hamming codes with (). Our key result demonstrates that the sequence length can be reduced to exactly -only one additional measurement beyond the original non-fault-tolerant sequence, establishing a tight lower bound. The proposed method leverages cyclic matrix transformations to systematically combine rows of the initial stabilizer matrix and preserving a self-dual CSS-like symmetry analogous to that of the original quantum Hamming codes. This induced symmetry enables hardware-efficient circuit reuse: the measurement circuits for the first stabilizers are transformed into circuits for the remaining stabilizers simply by toggling boundary Hadamard gates, eliminating redundant hardware. For distance-3 fault-tolerant error correction, our approach simultaneously reduces the time overhead via shorting the FTMS length and the hardware overhead through symmetry-enabled circuit multiplexing. These results provide an important advance towards the important open problem regarding the design of minimal FTMSs for quantum Hamming codes and may shed light on similar challenges in other quantum stabilizer codes.
Paper Structure (15 sections, 3 theorems, 29 equations, 1 figure)

This paper contains 15 sections, 3 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Internal faults potentially confused with input errors
  4. Identifying internal faults with quantum Hamming codes using redundant stabilizers
  5. Distance-3 FTQEC with quantum Hamming codes via one additional measurement beyond the original non-FT sequence
  6. A necessary and sufficient condition
  7. Construction of the Cyclic Transformation Matrices
  8. Discussion
  9. Conclusion
  10. Appendix: Proof of Theorem 3
  11. The proof of all columns of $B$ are distinct
  12. The proof of forward consistency
  13. The proof of backward consistency
  14. Proof of no 1-initiated subsequence starting from the $l$-th entry ($l\ge 3$) of $C_{12}v_j$ matches the corresponding subsequence in $C_{11}v_i$
  15. Proof of no 1-initiated subsequence starting from the $l$-th entry ($l\ge 3$) of $C_{11}v_i$ matches the corresponding subsequence in $C_{12}v_j$

Key Result

Theorem 1

(FT Measurement Criteria) The distinguishing capability between an input error and an internal fault in a distance-3 FT stabilizer code is mathematically equivalent to the syndrome matrix satisfying the following two conditions:

Figures (1)

  • Figure 1: Two representative error mechanisms arising from internal faults during the measurement of $ZZXIX$ using flag-based method for 5-qubit code. (Red): The errors caused by $X_1$ occurring on the control qubit during the operation of the first CNOT gate; (Blue): The errors caused by $Z$ occurring on the target (ancilla) qubit during the first CNOT gate operation. Here, black $X$ and $Z$ symbols denote measurement bases, while colored symbols indicate errors.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Example 4
  • proof
  • proof