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Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold

Jong Yeon Lee

Abstract

The toric code is a canonical example of a topological error-correcting code. Two logical qubits stored within the toric code are robust against local decoherence, ensuring that these qubits can be faithfully retrieved as long as the error rate remains below a certain threshold. Recent studies have explored such a threshold behavior as an intrinsic information-theoretic transition, independent of the decoding protocol. These studies have shown that information-theoretic metrics, calculated using the Renyi (replica) approximation, demonstrate sharp transitions at a specific error rate. However, an exact analytic expression that avoids using the replica trick has not been shown, and the connection between the transition in information-theoretic capacity and the random bond Ising model (RBIM) has only been indirectly established. In this work, we present the first analytic expression for the coherent information of a decohered toric code, thereby establishing a rigorous connection between the fundamental error threshold and the criticality of the RBIM.

Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold

Abstract

The toric code is a canonical example of a topological error-correcting code. Two logical qubits stored within the toric code are robust against local decoherence, ensuring that these qubits can be faithfully retrieved as long as the error rate remains below a certain threshold. Recent studies have explored such a threshold behavior as an intrinsic information-theoretic transition, independent of the decoding protocol. These studies have shown that information-theoretic metrics, calculated using the Renyi (replica) approximation, demonstrate sharp transitions at a specific error rate. However, an exact analytic expression that avoids using the replica trick has not been shown, and the connection between the transition in information-theoretic capacity and the random bond Ising model (RBIM) has only been indirectly established. In this work, we present the first analytic expression for the coherent information of a decohered toric code, thereby establishing a rigorous connection between the fundamental error threshold and the criticality of the RBIM.
Paper Structure (5 sections, 55 equations, 3 figures)

This paper contains 5 sections, 55 equations, 3 figures.

Table of Contents

  1. Review on Random Bond Isind Model
  2. Nishimori Line, Free Energy, and Variance across Disorder Ensemble
  3. Coherent information
  4. Free energy as lower bound
  5. Correlated models

Figures (3)

  • Figure 1: Coherent Information under Pauli-$Z$ and $X$ errors for (a) Two raw physical qubits $I_c^\textrm{raw}$ and (b) Two logical qubits of the toric code state $I_c^{\textrm{tc}}$ in the thermodynamic limit. The dashed line in (b) is the contour of $I_c^\textrm{raw} = 0$, which passes the point $(0.1100, 0.1100)$. This point is very close to the critical point of the Nishimori line $(0.1094,0.1094)$NishimoriPoint.
  • Figure 2: Conventions.(a) Cycles along the edges of the original (blue) and dual (red) lattices $C_i$ and $C_i^\perp$ respectively. (b) A set of red plaquettes ${\bm{m}}$ in the original lattice to denote $m$-anyons and corresponding string ${\bm{l}}^{\bm{m}}$ of Pauli-$Z$s. (c) A set of blue plaquettes $\tilde{{\bm{m}}}$ in the dual lattice to denote $e$-anyons and corresponding string (thick black lines) ${\bm{l}}^{\tilde{\bm{m}}}$ of Pauli-$X$s.
  • Figure S1: Random bond Ising model at $T=0$. The plot shows (a) order parameter and (b) its variance across bond configurations as the function of disorder probability $p$ at $T=0$ in the RBIM, adapted from LeeNishimori2022. The order parameter $O$ is squared magnetization normalized to be within $[0,1]$. The black dashed line is the critical point at $p_c\,{=}\,0.103$WangPreskill2003.