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Explicit Instances of Quantum Tanner Codes

Rebecca Katharina Radebold, Stephen D. Bartlett, Andrew C. Doherty

TL;DR

This work constructs explicit instances of quantum Tanner codes using dihedral groups and random classical codes, achieving high encoding rates and distances that scale linearly with the number of physical qubits. The codes maintain low-weight stabilizers (LDPC) and are decoded with a BP+OSD pipeline, yielding competitive pseudo-thresholds under both phenomenological and circuit-level noise, and showing favorable space-time overheads for small instances. The study provides a detailed numerical assessment, including comparisons to surface codes and a thorough analysis of overhead metrics, highlighting potential near-term applicability around $p \sim 10^{-3}$ with a few hundred qubits. The findings support quantum Tanner codes as a practical qLDPC option for early fault-tolerant quantum computing and point to avenues for further reductions in overhead and improvements via alternative group choices and optimized decoding strategies.

Abstract

We construct several explicit instances of quantum Tanner codes, a class of asymptotically good quantum low-density parity check (qLDPC) codes. The codes are constructed using dihedral groups and random pairs of classical codes and exhibit high encoding rates, relative distances, and pseudo-thresholds. Using the BP+OSD decoder, we demonstrate good performance in the phenomenological and circuit-level noise settings, comparable to the surface code with similar distances. Finally, we conduct an analysis of the space-time overhead incurred by these codes.

Explicit Instances of Quantum Tanner Codes

TL;DR

This work constructs explicit instances of quantum Tanner codes using dihedral groups and random classical codes, achieving high encoding rates and distances that scale linearly with the number of physical qubits. The codes maintain low-weight stabilizers (LDPC) and are decoded with a BP+OSD pipeline, yielding competitive pseudo-thresholds under both phenomenological and circuit-level noise, and showing favorable space-time overheads for small instances. The study provides a detailed numerical assessment, including comparisons to surface codes and a thorough analysis of overhead metrics, highlighting potential near-term applicability around with a few hundred qubits. The findings support quantum Tanner codes as a practical qLDPC option for early fault-tolerant quantum computing and point to avenues for further reductions in overhead and improvements via alternative group choices and optimized decoding strategies.

Abstract

We construct several explicit instances of quantum Tanner codes, a class of asymptotically good quantum low-density parity check (qLDPC) codes. The codes are constructed using dihedral groups and random pairs of classical codes and exhibit high encoding rates, relative distances, and pseudo-thresholds. Using the BP+OSD decoder, we demonstrate good performance in the phenomenological and circuit-level noise settings, comparable to the surface code with similar distances. Finally, we conduct an analysis of the space-time overhead incurred by these codes.

Paper Structure

This paper contains 18 sections, 4 theorems, 16 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Stabilizer and CSS codes
  4. Quantum Tanner Code Construction
  5. Left-Right Cayley Complexes
  6. Classical Codes
  7. Quantum Tanner Codes
  8. Belief Propagation and Ordered-Statistics Decoding
  9. Explicit Instances of Quantum Tanner Codes
  10. Numerical Results
  11. Error Models
  12. Simulation Results and Pseudo-Thresholds
  13. Overhead Comparison
  14. Conclusion and Future Work
  15. Acknowledgments
  16. ...and 3 more sections

Key Result

Lemma 1

Let $G$ be a finite group and let $A, B \subset G$ such that $\langle A, B \rangle = G$ and $|A| = |B| = \Delta$. Let $Cay(G, A)$ and $Cay(G, B)$ denote the Cayley graphs based on left and right actions on $G$, respectively. Let $\lambda_2^A$ and $\lambda_2^B$ denote the second largest eigenvalues o where $\lambda_2^{LRCC}$ denotes the second largest eigenvalue of $\Gamma(G, A, B)$.

Figures (2)

  • Figure 1: Performance of our quantum Tanner codes from Table \ref{['params']} in simulations with (a) phenomenological noise, and (b) circuit-level noise. Logical error rate is plotted as a function of physical error rate for $N=d$ rounds of syndrome extraction, for code distance $d$. Error bars were computed for 95% binomial proportion confidence intervals via $p_L = \hat{p_L} \pm \frac{1.645}{\sqrt{\eta}} \sqrt{\frac{\eta_s\eta_f}{\eta^2}}$, where $\eta$ represents the total number of shots and $\eta_s$ and $\eta_f$ the number of successes and failures, respectively. For the combined $p_L$ as computed in Eq. \ref{['logical error rate def']}, $\eta = \eta_x+\eta_z$. Here, $\eta_x, \eta_z \in [10^5, 10^9]$, depending on the number of errors encountered. Note the distinct ranges of physical error rates in plots (a) and (b).
  • Figure 2: A comparison of the performances of quantum Tanner codes with 10 logical qubits and $k = 10$ copies of various surface codes of comparable distance. The logical error rates of the surface codes were computed via $p'_L = 1- (1-p_L)^k$ for $k =10$, where $p_L$ is the logical error rate of a single surface code encoding one logical qubit.

Theorems & Definitions (10)

  • Definition 1: Left-Right Cayley Complex
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 2.1
  • proof
  • Lemma 3
  • proof