Random quantum dynamics generates quantum error correcting codes that are robust against observers and errors. Here, the authors describe a simple physical picture of such codes, mapping characteristic code properties to the statistical mechanics of fluctuating ``entanglement domain walls''. The error correcting criterion is understood as a decoupling condition of domain walls, which leads to a universal scaling law of the ``code distance''.
Random quantum dynamics generates quantum error correcting codes that are robust against observers and errors. Here, the authors describe a simple physical picture of such codes, mapping characteristic code properties to the statistical mechanics of fluctuating entanglement domain walls''. The error correcting criterion is understood as a decoupling condition of domain walls, which leads to a universal scaling law of the code distance''.
We introduce the domain wall color code, a new variant of the quantum error-correcting color code that exhibits exceptionally high code-capacity error thresholds for qubits subject to biased noise. In the infinite bias regime, a two-dimensional color code decouples into a series of repetition codes, resulting in an error-correcting threshold of 50%. Interestingly, at finite bias, our color code demonstrates thresholds identical to those of the noise-tailored XZZX surface code for all single-qubit Pauli noise channels. The design principle of the code is that it introduces domain walls which permute the code's excitations upon domain crossing. For practical implementation, we supplement the domain wall code with a scalable restriction decoder based on a matching algorithm. The proposed code is identified as a comparably resource-efficient quantum error-correcting code highly suitable for realistic noise.
A powerful method for analyzing quantum error-correcting codes is to map them onto classical statistical mechanics models. Such mappings have thus far mostly focused on static codes, possibly subject to repeated syndrome measurements. Recent progress in quantum error correction, however, has prompted new paradigms where codes emerge from stabilizer circuits in spacetime -- a unifying perspective encompassing syndrome extraction circuits of static codes, dynamically generated codes, and logical operations. We show how to construct statistical mechanical models for stabilizer circuits subject to independent Pauli errors, by mapping logical equivalence class probabilities of errors to partition functions using the spacetime subsystem code formalism. We also introduce a modular language of spin diagrams for constructing the spin Hamiltonians, which we describe in detail focusing on independent circuit-level X-Z error channels. With the repetition and toric codes as examples, we use our approach to analytically rank logical error rates and thresholds between code implementations with standard and dynamic syndrome extraction circuits, describe the effect of transversal logical Clifford gates on logical error rates, and perform Monte Carlo simulations to estimate maximum likelihood thresholds. Our framework offers a universal prescription to analyze, simulate, and compare the decoding properties of any stabilizer circuit, while revealing the innate connections between dynamical quantum systems and noise-resilient phases of matter.
Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted Z rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent Z-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.
We introduce the domain wall color code, a new variant of the quantum error-correcting color code that exhibits exceptionally high code-capacity error thresholds for qubits subject to biased noise. In the infinite bias regime, a two-dimensional color code decouples into a series of repetition codes, resulting in an error-correcting threshold of 50%. Interestingly, at finite bias, our color code demonstrates thresholds identical to those of the noise-tailored XZZX surface code for all single-qubit Pauli noise channels. The design principle of the code is that it introduces domain walls which permute the code's excitations upon domain crossing. For practical implementation, we supplement the domain wall code with a scalable restriction decoder based on a matching algorithm. The proposed code is identified as a comparably resource-efficient quantum error-correcting code highly suitable for realistic noise.
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.
Quantum systems evolving under unitary dynamics with a low rate of projective measurements can realize a highly entangled steady state. However, the stability of this phase against the disentangling action of measurements has not been fully understood. Here, the authors quantify the ability of this highly entangled phase to hide local quantum information, thereby relating its stability to its behavior as an emergent quantum error-correcting code. The authors estimate the tolerable measurement rate, above which this quantum code becomes unstable.