The necessity for quantum error correction arises from the need to safeguard quantum information from errors induced by decoherence and other noise forms. However, the optimal decoding of quantum error-correcting codes is a computationally challenging task. In our research, we introduce an ensemble decoder that amalgamates the advantages of various decoders to attain lower logical error rates and the capability to decode multiple errors. We employed statistical methodologies to allocate a specific error syndrome to the decoder with the highest likelihood of accurate decoding. We applied diverse variants of the Noise model to simulate the impact of noise on transmitted quantum states accurately. The efficacy of the ensemble decoder was assessed in both Symmetric and Asymmetric Noise Models. In the Symmetric Noise Model, bit-flip and phase-flip errors are presumed to have equal probabilities, while in the Asymmetric Noise Model, these probabilities can vary. The comparison aimed to evaluate the ensemble decoder's proficiency in rectifying these error types compared to traditional decoding techniques. Our results indicate that the ensemble decoder surpasses conventional decoders in the Symmetric Noise model, particularly for higher code distances. It is also apparent that augmenting the lattice size diminishes the logical error rate and that threshold and pseudo-threshold values escalate with incremental changes in noise asymmetry. The study's findings underscore the substantial potential of the ensemble decoder in quantum error correction and communication realms.
The necessity for quantum error correction arises from the need to safeguard quantum information from errors induced by decoherence and other noise forms. However, the optimal decoding of quantum error-correcting codes is a computationally challenging task. In our research, we introduce an ensemble decoder that amalgamates the advantages of various decoders to attain lower logical error rates and the capability to decode multiple errors. We employed statistical methodologies to allocate a specific error syndrome to the decoder with the highest likelihood of accurate decoding. We applied diverse variants of the Noise model to simulate the impact of noise on transmitted quantum states accurately. The efficacy of the ensemble decoder was assessed in both Symmetric and Asymmetric Noise Models. In the Symmetric Noise Model, bit-flip and phase-flip errors are presumed to have equal probabilities, while in the Asymmetric Noise Model, these probabilities can vary. The comparison aimed to evaluate the ensemble decoder's proficiency in rectifying these error types compared to traditional decoding techniques. Our results indicate that the ensemble decoder surpasses conventional decoders in the Symmetric Noise model, particularly for higher code distances. It is also apparent that augmenting the lattice size diminishes the logical error rate and that threshold and pseudo-threshold values escalate with incremental changes in noise asymmetry. The study's findings underscore the substantial potential of the ensemble decoder in quantum error correction and communication realms.
The issue of error-prone quantum information obstructs the progress of quantum computing, notwithstanding its capacity to transform complex problem-solving. This study examines the impact of depolarizing noise on a 5-qubit, distance-2 surface code, representing a fundamental quantum error correcting (QEC) frame- work. The primary objective is to identify a breakeven error rate beneath which error correction remains effective. We assess logical error rates through simu- lation across different noise levels, identifying a crucial breakeven threshold at around a 10% error rate. Below this level, the surface code exhibits proficient mistake detection; but, beyond this point, the effectiveness of repair diminishes significantly. We examine quantum error mitigation (QEM) techniques such as syndrome amplification and post-selection to evaluate their efficacy in minimiz- ing errors within the threshold range. We also noted spatial connections in error patterns, suggesting possible pathways for focused mistake correction. These find- ings elucidate the functioning of quantum error correction (QEC) in small-scale codes and underscore the necessity for noise-aware QEC techniques in scalable quantum systems.
Quantum computers are highly sensitive to noise due to interactions with the environment, which severely limits the utility of near-term devices. Quantum error correction addresses this challenge by encoding logical information over multiple qubits in order to preserve information in the presence of noise. This facilitates the realization of the theoretical advantages of quantum computing such as exponential speed-up compared to classical computers for certain algorithms. Physical realizations of qubits may have different noise profiles, including biased noise, where phase-flip errors may be more likely than bit-flip errors. We propose a quantum error-correcting code, the XYZ² code, which is tailored to biased noise. The code has high error thresholds against biased noise, below which logical errors are exponentially suppressed with the linear dimension of the code. The important classical task of deriving corrections given partial information about errors is called decoding. We investigate several decoding strategies tailored to specific challenges, ranging from a novel maximum-likelihood sampling decoder, to a model-free, data-driven, neural-network-based decoder, to an efficient multi-step decoder for concatenated codes. We also study the link between quantum error-correcting codes and their statistical-mechanical counterparts by mapping to generalized random-bond Ising models and subsequently deriving the exact results for finite-size corrections under biased noise for moderate code sizes. Quantum error correction is still in its developing stages, and this work provides a useful contribution to the field by bridging a gap between theory and practice. By adapting well-studied topological codes to error models that are closer to reality, this work paves the way for the realization of a fault-tolerant quantum architecture.
Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We classify each error syndrome to the decoder which is more likely to decode it correctly using machine learning techniques. We apply our framework to an ensemble of minimum-weight perfect matching (MWPM) and hard-decision re-normalization-group decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudothreshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.
Quantum error correcting (QEC) codes protect quantum information from decoherence as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, suboptimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered, we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close agreement with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models. Published by the American Physical Society 2024
As quantum computers remain susceptible to noise, QEC is essential for preserving logical information during computations. However, the performance of QEC codes breaks down beyond certain noise thresholds, revealing fundamental limits on their ability to protect quantum information. These limits can be characterized using information-theoretic measures such as the coherent information, which quantifies the maximum rate at which logical information can be reliably transmitted through a noisy quantum channel. In this work, we establish a direct connection between the CI and the binary cross-entropy loss used when training neural network decoders. Specifically, we show that the CI constitutes a sharp lower bound on the achievable loss for decoders that track logical operators across noisy channels. To this end, we develop a transformer-based neural network model based on maximum likelihood decoding. We train this network to estimate the CI and evaluate its performance on the surface code under three noise models: code capacity, phenomenological, and circuit-level noise. Our results demonstrate that the network accurately predicts CI and yields threshold estimates that closely match known theoretical limits. When used as a decoder, the network significantly outperforms the minimum weight perfect matching decoder in terms of logical error rate. We also introduce a novel soft post-selection scheme that independently treats uncertainty in both logical operators and relies on confidence-based filtering of the network's output. We prove that such post-selection strategies, based on the MLD cosets, are optimal, and demonstrate their scalability in terms of both logical error rate and abort probability. These findings establish transformer-based architectures as powerful tools for QEC and provide the first numerical evidence supporting the optimality and scalability of MLD-based post-selection.
Quantum error correcting (QEC) codes protect quantum information from decoherence, as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, sub-optimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close accordance with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive practical tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models.