We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette $\Z2$ gauge models) as well as unexplored earlier generally non-local disordered spin models with non-trivial phase diagrams. The decoding transition corresponds to a transition from the ordered phase by proliferation of ``post-topological'' extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with finite rates the number of distinct classes of such extended defects is exponentially large, corresponding to extensive ground state entropy of these codes. Here, the transition can be driven by the entropy of the extended defects, a mechanism distinct from that in the local spin models where the number of defect types (domain walls) is always finite.
We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette Z2 gauge models) as well as unexplored earlier generally non-local disordered spin models with non-trivial phase diagrams. The decoding transition corresponds to a transition from the ordered phase by proliferation of ``post-topological'' extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with finite rates the number of distinct classes of such extended defects is exponentially large, corresponding to extensive ground state entropy of these codes. Here, the transition can be driven by the entropy of the extended defects, a mechanism distinct from that in the local spin models where the number of defect types (domain walls) is always finite.
In this paper, the LDPC (Low Density Parity Check Codes) decoding algorithm has been investigated using statistical mechanics properties. The main advantage of LDPC codes is their performance that works in near Shannon limit, and the ability to use iterative decoding algorithms. In 1989 N. Sourlas showed that error-correcting codes can be considered as Spin-Glass systems thus making it possible to model LDPC codes as an Ising model, opening the way for information theorists to solve coding problems with the power of statistical mechanics. According to N. Sourlas, the decoding problem can be solved by finding the ground state of the corresponding spin-system Hamiltonian. The main goal of this paper is to review as simple as possible the statistical properties of LDPC codes, and how it is used especially facing the decoding problem.
We explore a class of quantum circuit models with local decoherence (noise) and local projective measurements, each respecting a global Z_2 symmetry. The model supports a spin glass phase where the Z_2 symmetry is spontaneously broken (not possible in an equilibrium 1d system), a paramagnetic phase characterized by a divergent susceptibility, and an intermediate trivial phase. All three phases are also stable to Z_2-symmetric local unitary gates, and the dynamical phase transitions between the phases are in the percolation universality class. The open circuit dynamics can be purified by explicitly introducing a bath with its own scrambling dynamics, as in [Bao, Choi, Altman, arXiv:2102.09164], which does not change any of the universal physics. Within the spin glass phase the circuit dynamics can be interpreted as a quantum repetition code, with each stabilizer of the code measured stochastically at a finite rate, and the decoherences as effective bit-flip errors. Motivated by the geometry of the spin glass phase, we devise a novel decoding algorithm for recovering an arbitrary initial qubit state in the code space, assuming knowledge of the history of the measurement outcomes, and the ability of performing local Pauli measurements and gates on the final state. For a circuit with L^d qubits running for time T, the time needed to execute the decoder scales as O(L^d T) (with dimensionality d). With this decoder in hand, we find that the information of the initial encoded qubit state can be retained (and then recovered) for a time logarithmic in L for a 1d circuit, and for a time at least linear in L in 2d below a finite error threshold. For both the repetition and toric codes, we compare and contrast our decoding algorithm with earlier algorithms that map the error model to the random bond Ising model.
Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analysing QEC performance. However, predicting the behaviour of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly-solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansatze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the domain wall energy cost distributions. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.
We demonstrate the equivalence between thermodynamic observables of Ising spin-glass models on small-world lattices and the decoding properties of error-correcting low-density parity-check codes on channels with memory. In particular, the self-consistent equations for the effective field distributions in the spin-glass model within the replica symmetric ansatz are equivalent to the density evolution equations forr Gilbert–Elliott channels. This relationship allows us to present a belief-propagation decoding algorithm for finite-state Markov channels and to compute its performance at infinite block lengths from the density evolution equations. We show that loss of reliable communication corresponds to a first order phase transition from a ferromagnetic phase to a paramagnetic phase in the spin glass model. The critical noise levels derived for Gilbert–Elliott channels are in very good agreement with existing results in coding theory. Furthermore, we use our analysis to derive critical noise levels for channels with both memory and asymmetry in the noise. The resulting phase diagram shows that the combination of asymmetry and memory in the channel allows for high critical noise levels: in particular, we show that successful decoding is possible at any noise level of the bad channel when the good channel is good enough. Theoretical results at infinite block lengths using density evolution equations aree compared with average error probabilities calculated from a practical implementation of the corresponding decoding algorithms at finite block lengths.
Topologically ordered phases are stable to local perturbations, and topological quantum error-correcting codes enjoy thresholds to local errors. We connect the two notions of stability by constructing classical statistical mechanics models for decoding general Calderbank-Shor-Steane codes and classical linear codes. Our construction encodes correction success probabilities under uncorrelated bit-flip and phase-flip errors, and simultaneously describes a generalized Z 2 lattice-gauge theory with quenched disorder. We observe that the clean limit of the latter is precisely the imaginary-time path integral of the corresponding quantum code Hamiltonian when the errors are turned into a perturbative X or Z magnetic field. Motivated by error-correction considerations, we define general order parameters for all such generalized Z 2 lattice-gauge theories, and show that they are generally lower bounded by success probabilities of error correction. For CSS codes satisfying the low-density parity-check condition and with a sufficiently large code distance, we prove the existence of a low-temperature ordered phase of the corresponding lattice-gauge theories, particularly for those lacking Euclidean spatial locality and/or when there is a nonzero code rate. We further argue that these results provide evidence for stable phases in the corresponding perturbed quantum Hamiltonians, obtained in the limit of continuous imaginary time. To do so, we distinguish space- and timelike defects in the lattice-gauge theory. A high free-energy cost of spacelike defects corresponds to a successful “memory experiment” and suppresses the energy splitting among the ground states, while a high free-energy cost of timelike defects corresponds to a successful “stability experiment” and points to a nonzero gap to local excitations.
We investigate the statistical physics of quantum error correction in U(1) symmetry-enriched topological quantum memories. Starting from a phenomenological error model of charge-conserving noise, we study the optimal decoder assuming the local charges of each anyon can be measured. The error threshold of the optimal decoder corresponds to a continuous phase transition in a disordered two-dimensional integer loop model on the Nishimori line. Using an effective replica field theory analysis and Monte Carlo numerics, we show that the optimal decoding transition exhibits Berezinskii-Kosterlitz-Thouless universality with a modified universal jump in winding number variance. We further generalize the model beyond the Nishimori line, which defines a large class of suboptimal decoders. At low nonzero temperatures and strong disorder, we find numerical evidence of a disorder-dominated loop-glass phase which corresponds to a "confidently incorrect" decoder. The zero-temperature limit defines the minimum-cost flow decoder, which serves as the U(1) analog of minimum-weight perfect matching in Z2 topological codes. Both the optimal and minimum-cost flow decoders are shown to dramatically outperform the charge-agnostic optimal decoder in symmetry-enriched topological codes.