Quantum error correction codes (QECCs) play a central role in both quantum communications and quantum computation. Practical quantum error correction codes, such as stabilizer codes, are generally structured to suit a specific use, and present rigid code lengths and code rates. This paper shows that it is possible to both construct and decode QECCs that can attain the maximum performance of the finite blocklength regime, for any chosen code length when the code rate is sufficiently high. A recently proposed strategy for decoding classical codes called GRAND (guessing random additive noise decoding) opened doors to efficiently decode classical random linear codes (RLCs) performing near the maximum rate of the finite blocklength regime. By using noise statistics, GRAND is a noise-centric efficient universal decoder for classical codes, provided that a simple code membership test exists. These conditions are particularly suitable for quantum systems, and therefore the paper extends these concepts to quantum random linear codes (QRLCs), which were known to be possible to construct but whose decoding was not yet feasible. By combining QRLCs and a newly proposed quantum-GRAND, this work shows it is possible to decode QECCs that are easy to adapt to changing conditions. The paper starts by assessing the minimum number of gates in the coding circuit needed to reach the QRLCs’ asymptotic performance, and subsequently proposes a quantum-GRAND algorithm that makes use of quantum noise statistics, not only to build an adaptive code membership test, but also to efficiently implement syndrome decoding.
Quantum error correction codes (QECCs) play a central role in both quantum communications and quantum computation. Practical quantum error correction codes, such as stabilizer codes, are generally structured to suit a specific use, and present rigid code lengths and code rates. This paper shows that it is possible to both construct and decode QECCs that can attain the maximum performance of the finite blocklength regime, for any chosen code length when the code rate is sufficiently high. A recently proposed strategy for decoding classical codes called GRAND (guessing random additive noise decoding) opened doors to efficiently decode classical random linear codes (RLCs) performing near the maximum rate of the finite blocklength regime. By using noise statistics, GRAND is a noise-centric efficient universal decoder for classical codes, provided that a simple code membership test exists. These conditions are particularly suitable for quantum systems, and therefore the paper extends these concepts to quantum random linear codes (QRLCs), which were known to be possible to construct but whose decoding was not yet feasible. By combining QRLCs and a newly proposed quantum-GRAND, this work shows it is possible to decode QECCs that are easy to adapt to changing conditions. The paper starts by assessing the minimum number of gates in the coding circuit needed to reach the QRLCs’ asymptotic performance, and subsequently proposes a quantum-GRAND algorithm that makes use of quantum noise statistics, not only to build an adaptive code membership test, but also to efficiently implement syndrome decoding.
This work addresses the open question of implementing fault-tolerant QRLCs with feasible computational overhead. We present a new decoder for quantum random linear codes (QRLCs) capable of dealing with imperfect decoding operations. A first approach, introduced by Cruz et al., only considered channel errors, and perfect gates at the decoder. Here, we analyze the fault-tolerant characteristics of QRLCs with a new noise guessing decoding technique, when considering preparation, measurement, and gate errors in the syndrome extraction procedure, while also accounting for error degeneracy. Our findings indicate a threshold error rate (pthreshold) of approximately 2×10-5in the asymptotic limit, while considering realistic noise levels in the mentioned physical procedures.
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.
Quantum computers require error correction to achieve universal quantum computing. However, current decoding of quantum error-correcting codes relies on classical computation, which is slower than quantum operations in superconducting qubits. This discrepancy makes the practical implementation of real-time quantum error correction challenging. In this work, we propose a decoding scheme that leverages the operations of the quantum circuit itself. Given a noisy quantum circuit A, we train a decoding quantum circuit B using syndrome measurements to identify the logical operators needed to correct errors in circuit A. The trained quantum circuit B can be deployed on quantum devices, such as superconducting qubits, to perform real-time decoding and error correction. Our approach is applicable to general quantum codes with multiple logical qubits and operates efficiently under various noise conditions, and the decoding speed matches the speed of the quantum circuits being corrected. We have conducted numerical experiments using surface codes up to distance 7 under circuit-level noise, demonstrating performance on par with the classical minimum-weight perfect matching algorithm. Interestingly, our method reveals that the traditionally classical task of decoding error-correcting codes can be accomplished without classical devices or measurements. This insight paves the way for the development of self-correcting quantum computers.
Quantum error correction codes (QECC) are a key component for realizing the potential of quantum computing. QECC, as its classical counterpart (ECC), enables the reduction of error rates, by distributing quantum logical information across redundant physical qubits, such that errors can be detected and corrected. In this work, we efficiently train novel end-to-end deep quantum error decoders. We resolve the quantum measurement collapse by augmenting syndrome decoding to predict an initial estimate of the system noise, which is then refined iteratively through a deep neural network. The logical error rates calculated over finite fields are directly optimized via a differentiable objective, enabling efficient decoding under the constraints imposed by the code. Finally, our architecture is extended to support faulty syndrome measurement, by efficient decoding of repeated syndrome sampling. The proposed method demonstrates the power of neural decoders for QECC by achieving state-of-the-art accuracy, outperforming for small distance topological codes, the existing end-to-end neural and classical decoders, which are often computationally prohibitive.
Noise is one of the central obstacles to building useful quantum computers, and quantum error correction (QEC) provides the framework for protecting quantum information against it. Unlike classical error correction, QEC must preserve fragile quantum states without copying them, measuring them directly, or destroying the information they encode. Driven by rapid progress in both theory and experiment, this challenge has grown into one of the most active areas of quantum information science. This tutorial gives a guided introduction to modern QEC, developing the core concepts of codes, syndromes, stabilizers, decoding, and fault tolerance before connecting them to major code families and current research directions. We cover both established constructions and newer developments, including topological and subsystem codes, bosonic and qudit codes, dynamical codes, and quantum low-density parity-check (qLDPC) codes. The emphasis is on building operational understanding: explaining not only what the main objects are, but how they are used in code design, error diagnosis, decoding, and fault-tolerant computation. The tutorial is intended for newcomers seeking a first path through QEC, as well as researchers looking for a coherent reference for the concepts, code families, and tools that arise in current work.
Quantum error correction allows to actively correct errors occurring in a quantum computation when the noise is weak enough.To make this error correction competitive information about the specific noise is required.Traditionally, this information is obtained by benchmarking the device before operation.We address the question of what can be learned from only the measurements done during decoding.Such estimation of noise models was proposed for surface codes, exploiting their special structure, and in the limit of low error rates, also for other codes.However, so far it has been unclear under what general conditions noise models can be estimated from the syndrome measurements.In this work, we derive a general condition for identifiability of the error rates.For general stabilizer codes, we prove identifiability under the assumption that the rates are small enough.Without this assumption, we prove a result for perfect codes.Finally, we propose a practical estimation method with linear runtime for concatenated codes.We demonstrate that it outperforms other recently proposed methods and that the estimation is optimal in the sense that it reaches the Cramér-Rao bound.Our method paves the way for practical calibration of error corrected quantum devices during operation.