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arXiv:1412.6172 · Physical Review Letters

Thresholds for Correcting Errors, Erasures, and Faulty Syndrome Measurements in Degenerate Quantum Codes

Ilya I. Dumer, Alexey A. Kovalev, Leonid P. Pryadko

Published Jul 31, 2015• 33 citations
arXivHTMLPDF

Abstract

We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors for quantum low-density parity-check codes with logarithmic or larger distances. These threshold estimates are parametrically better than the existing analytical bound based on percolation.

DOI
10.1103/physrevlett.115.050502

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We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors for quantum low-density parity-check codes with logarithmic or larger distances. These threshold estimates are parametrically better than the existing analytical bound based on percolation.
We study the fault tolerance of quantum low-density parity check (LDPC) codes, such as generalized toric codes with a finite rate suggested by Tillich and Z\'emor [in ISIT 2009: IEEE International Symposium on Information Theory (IEEE, New York, 2009)]. We show that any family of quantum LDPC codes where each syndrome measurement involves a limited number of qubits and each qubit is involved in a limited number of measurements (as well as any similarly limited family of classical LDPC codes), in which distance scales as a positive power $\ensuremath{\alpha}$ of the number of physical qubits ($\ensuremath{\alpha}<1$ for ``bad'' codes), has a finite error probability threshold. We conclude that for sufficiently large quantum computers, quantum LDPC codes can offer an advantage over the toric codes.
Abstract We develop a general framework for bounding logical error probabilities at finite code distance in stabilizer quantum error-correcting codes. Rather than relying on asymptotic threshold argu ments, the analysis isolates decoder locality and control over syndrome connectivity as sufficient conditions for exponential suppression of logical errors at finite size. Under a stochastic Pauli noise model, we show that component-local decoders admit an explicit finite-size bound determined by the probability of large connected syndrome structures. The framework is formulated for stabilizer codes defined on bounded-degree syndrome graphs and is independent of any specific decoding algorithm. We prove a general finite-size decoding theorem and illustrate its application to the two-dimensional surface code, where the bound scales exponentially with the code distance. We further discuss the extension of the framework to quantum low-density parity-check and hypergraph-product codes. A minimal numerical illustration confirms the qualitative finite-size scaling predicted by the theory. These results provide a complementary perspective to fault-tolerance threshold theorems by mak ing explicit the mechanisms governing logical error suppression at finite code distance, with direct relevance to near- and intermediate-scale quantum devices.
Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D -dimensional hyperbolic space.
A key appeal of quantum low-density parity check (qLDPC) codes is their ability to suppress stochastic Pauli noise below nonzero thresholds. Coherent errors are fundamentally different: they produce superpositions of error patterns whose amplitudes can interfere even after syndrome measurement. Rigorous understanding of coherent errors remains limited. Here we show that general qLDPC codes admit a nonzero code capacity threshold against local coherent noise and more generally local channel noise. For any family of qLDPC codes with distance d=Ω(log⁡n)d=Ω(\log n)d=Ω(logn), we show that there is a constant noise strength below which the logical recovery error in diamond distance decays exponentially with the code distance. The result is established for optimal recovery as well as the minimum-weight decoder. The key technical ingredient is what we call a \emph{cluster resummation}: rather than bounding superposed error configurations one by one, we isolate a large connected error cluster in the channel expansion and exactly resum all errors disconnected from it before taking norms. Standard cluster counting then yields exponential suppression. This work resolves a longstanding challenge in fault tolerance theory, providing general robustness guarantees for qLDPC codes against coherent noise and laying a rigorous foundation for future studies of fault-tolerant quantum technologies.
Abstract The accumulation of physical errors 1–3 prevents the execution of large-scale algorithms in current quantum computers. Quantum error correction 4 promises a solution by encoding k logical qubits onto a larger number n of physical qubits, such that the physical errors are suppressed enough to allow running a desired computation with tolerable fidelity. Quantum error correction becomes practically realizable once the physical error rate is below a threshold value that depends on the choice of quantum code, syndrome measurement circuit and decoding algorithm 5 . We present an end-to-end quantum error correction protocol that implements fault-tolerant memory on the basis of a family of low-density parity-check codes 6 . Our approach achieves an error threshold of 0.7% for the standard circuit-based noise model, on par with the surface code 7–10 that for 20 years was the leading code in terms of error threshold. The syndrome measurement cycle for a length- n code in our family requires n ancillary qubits and a depth-8 circuit with CNOT gates, qubit initializations and measurements. The required qubit connectivity is a degree-6 graph composed of two edge-disjoint planar subgraphs. In particular, we show that 12 logical qubits can be preserved for nearly 1 million syndrome cycles using 288 physical qubits in total, assuming the physical error rate of 0.1%, whereas the surface code would require nearly 3,000 physical qubits to achieve said performance. Our findings bring demonstrations of a low-overhead fault-tolerant quantum memory within the reach of near-term quantum processors.
Quantum error correction protects logical information only when every physical operation remains below the fault-tolerance threshold, a condition that must be maintained continuously rather than only at the initial calibration. In practice, however, analog control parameters inevitably drift because of environmental fluctuations. As future fault-tolerant quantum computations are expected to run for days or even months, interrupting computation for repeated recalibration becomes fundamentally impractical. A promising alternative is to integrate calibration directly into computation by repurposing syndrome measurements as a calibration signal (Sivak et al, Nature 2026), but whether such self-calibration can be achieved with provable efficiency remains an open question. Here we establish a theoretical framework for such self-calibrating quantum fault tolerance. We prove that, for a broad class of control-induced errors, the detection rate defines a locally strongly convex surrogate objective for analog calibration with high probability. This geometric property enables a simple and efficient online optimization algorithm using only syndrome measurements collected during normal error correction. We prove convergence to an ε\varepsilonε detection rate within O(1/ε2)O(1/\varepsilon^2)O(1/ε2) epochs for time-independent drifts and also establish guarantees for time-dependent drifts. We further show that the convergence rate is independent of the code distance for quantum low-density parity-check (LDPC) codes. Pulse-level simulations of neutral-atom arrays and large-scale circuit-level Clifford simulations confirm these theoretical predictions. Our results establish self-calibrating fault tolerance as a provably efficient paradigm in which the same syndrome measurements simultaneously protect logical information and stabilize the underlying hardware.