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Quantum dual extended Hamming code immune to collective coherent errors

En-Jui Chang

TL;DR

The paper addresses collective coherent (CC) errors arising from intrinsic qubit Hamiltonians, which can dominate stochastic noise in quantum memories and communication systems. It proposes a family of constant-excitation (CE) stabilizer codes with parameters $[[2^{r+1}, 2^{r}-(r+1), 3]]$, formed by encoding a classical extended Hamming code into CE subspaces via an inner CDR code. The smallest instance, $[[8,1,3]]$, achieves a higher code rate with asymptotic rate approaching $\tfrac{1}{2}$ and a code-capacity threshold scaling $p_{th} \le \tfrac{1}{n(n-2)}$, outperforming prior CE constructions. This CC-immune, high-rate error-correcting scheme is particularly suited for quantum memories and short-range communications where qubit resources are limited.

Abstract

Collective coherent (CC) errors are inevitable, as every physical qubit undergoes free evolution under its kinetic Hamiltonian. These errors can be more damaging than stochastic Pauli errors because they affect all qubits coherently, resulting in high-weight errors that standard quantum error-correcting (QEC) codes struggle to correct. In quantum memories and communication systems, especially when storage durations are long, CC errors often dominate over stochastic noise. Trapped-ion platforms, for example, exhibit strong CC errors with minimal stochastic Pauli components. In this work, we address the regime where immunity to CC errors, high code rate (due to limited qubit availability), and moderate distance (sufficient for correcting low-weight errors) are all essential. We construct a family of constant-excitation (CE) stabilizer codes with parameters $[[2^{r+1}, 2^r - (r+1), 3]]$. The smallest instance, the $[[8,1,3]]$ code, improves the code rate and error threshold of the best previously known CE code by factors of approximately two and four, respectively.

Quantum dual extended Hamming code immune to collective coherent errors

TL;DR

The paper addresses collective coherent (CC) errors arising from intrinsic qubit Hamiltonians, which can dominate stochastic noise in quantum memories and communication systems. It proposes a family of constant-excitation (CE) stabilizer codes with parameters , formed by encoding a classical extended Hamming code into CE subspaces via an inner CDR code. The smallest instance, , achieves a higher code rate with asymptotic rate approaching and a code-capacity threshold scaling , outperforming prior CE constructions. This CC-immune, high-rate error-correcting scheme is particularly suited for quantum memories and short-range communications where qubit resources are limited.

Abstract

Collective coherent (CC) errors are inevitable, as every physical qubit undergoes free evolution under its kinetic Hamiltonian. These errors can be more damaging than stochastic Pauli errors because they affect all qubits coherently, resulting in high-weight errors that standard quantum error-correcting (QEC) codes struggle to correct. In quantum memories and communication systems, especially when storage durations are long, CC errors often dominate over stochastic noise. Trapped-ion platforms, for example, exhibit strong CC errors with minimal stochastic Pauli components. In this work, we address the regime where immunity to CC errors, high code rate (due to limited qubit availability), and moderate distance (sufficient for correcting low-weight errors) are all essential. We construct a family of constant-excitation (CE) stabilizer codes with parameters . The smallest instance, the code, improves the code rate and error threshold of the best previously known CE code by factors of approximately two and four, respectively.

Paper Structure

This paper contains 6 sections, 18 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Composite error model
  4. Classical extended Hamming code
  5. Quantum dual extended Hamming code
  6. Conclusion

Figures (1)

  • Figure 1: Encoding circuit for the $[[8,1,3]]$ CE QEC code. This stabilizer code encodes one logical qubit into eight physical qubits, offering protection against all weight-1 Pauli errors. It achieves a high encoding rate and is designed to be immune to CC errors.