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Fault-tolerance of the [[8,1,3]] non-CSS code

Pranav Maheshwari, Ankur Raina

TL;DR

The work introduces an eight-qubit non-CSS [[8,1,3]] quantum error-correcting code with a unitary encoder adapted from stabilizer formalism and a modified bare-ancilla syndrome extraction method to achieve fault-tolerant error correction. It analyzes two realistic noise models—standard depolarising and anisotropic—via Qiskit-based simulations to extract pseudo-thresholds and leading-order error terms, comparing a practical method against a modified, projection-free approach. The results show single-qubit errors can be corrected and certain second errors detected, with higher logical pseudo-thresholds under depolarising noise and when using the modified simulation, indicating robust fault tolerance. The study underscores the importance of stabilizer reordering and phase-aware encoding for non-CSS codes, and suggests future work to generalize the method to other non-CSS codes and noise environments, potentially improving benchmarks for fault-tolerant quantum computation.

Abstract

We present a fault-tolerant [[8, 1, 3]] non-CSS quantum error correcting code and study its logical error rates. We choose the unitary encoding procedure for stabilizer codes given by Gottesman and modify it to suit the setting of a class of non- CSS codes. Considering two types of noise models for this study, namely the depolarising noise and anisotropic noise, to depict the logical error rates obtained in decoding, we adopt the procedure of the bare ancilla method presented by Brown et al. to reorder the measurement sequence in the syndrome extraction step and upgrade it to obtain higher pseudo-thresholds and lower leading order terms of logical error rates.

Fault-tolerance of the [[8,1,3]] non-CSS code

TL;DR

The work introduces an eight-qubit non-CSS [[8,1,3]] quantum error-correcting code with a unitary encoder adapted from stabilizer formalism and a modified bare-ancilla syndrome extraction method to achieve fault-tolerant error correction. It analyzes two realistic noise models—standard depolarising and anisotropic—via Qiskit-based simulations to extract pseudo-thresholds and leading-order error terms, comparing a practical method against a modified, projection-free approach. The results show single-qubit errors can be corrected and certain second errors detected, with higher logical pseudo-thresholds under depolarising noise and when using the modified simulation, indicating robust fault tolerance. The study underscores the importance of stabilizer reordering and phase-aware encoding for non-CSS codes, and suggests future work to generalize the method to other non-CSS codes and noise environments, potentially improving benchmarks for fault-tolerant quantum computation.

Abstract

We present a fault-tolerant [[8, 1, 3]] non-CSS quantum error correcting code and study its logical error rates. We choose the unitary encoding procedure for stabilizer codes given by Gottesman and modify it to suit the setting of a class of non- CSS codes. Considering two types of noise models for this study, namely the depolarising noise and anisotropic noise, to depict the logical error rates obtained in decoding, we adopt the procedure of the bare ancilla method presented by Brown et al. to reorder the measurement sequence in the syndrome extraction step and upgrade it to obtain higher pseudo-thresholds and lower leading order terms of logical error rates.
Paper Structure (22 sections, 2 theorems, 10 equations, 9 figures, 6 tables)

This paper contains 22 sections, 2 theorems, 10 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Quantum error correction
  4. Preliminaries
  5. Error Correction Protocol
  6. Encoder Design
  7. Efficient encoding mechanism
  8. Enhanced Encoder for non-CSS Codes
  9. Single Qubit Fault Tolerance
  10. Reordering Stabilizers
  11. Details of [[8,1,3]] Code
  12. Merit of Fault Tolerance
  13. Errors and Syndrome Values
  14. Quantum Circuits
  15. Noise Models
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\mathcal{S} = \langle g_1, \dots, g_{n-k}\rangle$ be generated by $n-k$ independent and commuting elements from $\mathcal{G}_n$, such that $-I \notin \mathcal{S}$, then $\mathcal{V}_{\mathcal{S}}$ is a $2^k$ dimensional vector space.

Figures (9)

  • Figure 1: Visualization of encoded operators in $\mathcal{G}_n$ corresponding to stabilizer formalism.
  • Figure 2: Block diagram description of an $[[n,k]]$ quantum error correcting code.
  • Figure 3: An schematic of the efficient encoder example for 2 stabilizers $(M_1, M_2)$ in standard form.
  • Figure 4: Propagation of error on data qubits when two qubit gates are followed by a $P \otimes X$ error, where $P \in \{X,Y,Z\}$. Errors of the kind $P \otimes Z$ lead only to $P$ error on respective data qubit as $Z$ error on ancilla qubit does not propagate to data qubits. Each color denotes the gate that gets faulty and the corresponding error on data qubits.
  • Figure 5: Encoder Circuit for $[[8,1,3]]$ code. The last $k=1$ qubit is initialized in $\ket{\psi_i}$ state.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Definition 1
  • Theorem 2