Fundamental thresholds for computational and erasure errors via the coherent information

TL;DR

This work develops a unified framework based on the coherent information (CI) of the noisy QEC code state to study both erasure and computational errors in quantum error correction. By deriving exact CI expressions, the authors map optimal decoding problems to classical statistical-mechanics models (e.g., diluted RBIM, eight-vertex, and three-body Ising-type models) and show how erasures modify these mappings via missing bonds or decoupled stabilizers. Numerical CI calculations for 2D toric and color codes reveal 50% erasure thresholds, with finite-size CI crossings accurately matching known thresholds, and extend the approach to a low-density parity-check code—the lift-connected surface code—where new 3D-local spin models are obtained and analyzed. Overall, the CI framework provides a rigorous, scalable tool to estimate optimal thresholds for broad classes of QEC codes under realistic noise and to uncover deep links between quantum error correction and statistical physics.

Abstract

Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the 2D toric and color codes and compute optimal thresholds for erasure errors only, finding a 50 percent threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code, the lift-connected surface code. We find a 50 percent threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.

Figures (18)

• Figure 1: a) Toric code with $Z$ stabilizers defined on plaquettes and $X$ stabilizers centered on nodes of a square lattice. An error configuration is displayed: bit flip errors (stars next to qubits) and erasure errors as open circles replacing filled circles. b) Spin model corresponding to the error configuration shown in a). Spins are located at the center of the $X$ stabilizers, qubits without errors are denoted by a negative coupling. Bit flip and erasure errors are identified as positive and zero couplings, respectively.
• Figure 2: a) Coherent information setup. We start from a generalized Bell state $\sum_{i=1}^{2^k} | \boldsymbol{R}_i\rangle |\boldsymbol{R}_i \rangle /2^{k/2}$ between the $k$ reference qubits and identical $k$ seed qubits. Then we prepare the state $\sum_{i=1}^{2^k} | \boldsymbol{R}_i\rangle |\boldsymbol{L}_i \rangle /2^{k/2}$ where $|\boldsymbol{L}_i\rangle$ are the code words of a QEC code and send the state on $Q$ through an error channel. The CI is computed as $I=S(\rho_{Q})-S(\rho_{RQ})$. b) Correctability phase diagram for the 2D color code under bit/phase flip and erasure errors. The black line is the same shown in Fig. \ref{['fig:phase_diagram']} in Sec. \ref{['sec:results_topo_codes']} for the color code. Below threshold (green region) the CI asymptotically approaches its maximum value of $k\log 2$. Above threshold (red region) the CI approaches its minimum $-k\log 2$.
• Figure 3: Formation of super-plaquettes and super-star operators after losing one qubit. Each qubit is shared between four stabilizers, two $X$ (red regions) and two $Z$ (blue regions). After erasure the product of stabilizers of the same species remains well-defined and forms what has been called super-plaquettes (enclosed by dashed blue line) and super-stars (enclosed by red dashed line) stace_thresholds_2009. Let us note thateach of the former weight-3 stabilizers commutes with the super-plaquette and super-star operators, however its expectation value has been randomized by the qubit erasure.
• Figure 4: Error chains for the depolarizing noise model. Given a fixed error chain in the first replica $C^{(1)}$, the probability of each $C^{(s)}$ is parameterized according to the errors that appear in the first replica. The probability of having the $C^{(s)}$ depicted in the picture is parameterized through the weighted probabilities $Q_x^{(s)}(l)$,$Q_y^{(s)}(l)$, shown in Eqs. \ref{['eq:q_prob']} and \ref{['eq:q_xyz']}.
• Figure 5: Toric code square lattice with qubits on the edges. Error chain $C^{(1)}$ in red and commuting error configuration $v^{(s)}$ in violet. The blue plaquettes are the super-plaquettes formed after erasures (denoted by open circles and missing edges of the square lattice). The syndrome generated by $C^{(1)}$ is shown as four-point stars in the center of the respective stabilizer plaquettes. The error chain $v^{(s)}$ does not create nor remove syndromes.