Quantum error correction will likely be essential for building a large-scale quantum computer, but it comes with significant requirements at the level of classical control software. In particular, a quantum error-correcting code must be supplemented with a fast and accurate classical decoding algorithm. Standard techniques for measuring the parity-check operators of a quantum error-correcting code involve repeated measurements, which both increases the amount of data that needs to be processed by the decoder, and changes the nature of the decoding problem. Knill error correction is a technique that replaces repeated syndrome measurements with a single round of measurements, but requires an auxiliary logical Bell state. Here, we provide a theoretical and numerical investigation into Knill error correction from the perspective of decoding. We give a self-contained description of the protocol, prove its fault tolerance under locally decaying (circuit-level) noise, and numerically benchmark its performance for quantum low-density parity-check codes. We show analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.
Comments
9 + 7 pages, 7 figures. This version: added a link to the GitHub of the package we built for this project, extended the acknowledgements and fixed a few typos
Quantum error correction will likely be essential for building a large-scale quantum computer, but it comes with significant requirements at the level of classical control software. In particular, a quantum error-correcting code must be supplemented with a fast and accurate classical decoding algorithm. Standard techniques for measuring the parity-check operators of a quantum error-correcting code involve repeated measurements, which both increases the amount of data that needs to be processed by the decoder, and changes the nature of the decoding problem. Knill error correction is a technique that replaces repeated syndrome measurements with a single round of measurements, but requires an auxiliary logical Bell state. Here, we provide a theoretical and numerical investigation into Knill error correction from the perspective of decoding. We give a self-contained description of the protocol, prove its fault tolerance under locally decaying (circuit-level) noise, and numerically benchmark its performance for quantum low-density parity-check codes. We show analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.
Quantum computation promises to revolutionize information processing by exploiting superposition and entanglement, enabling applications such as quantum simulation, combinatorial optimization, and secure quantum communication. Beyond individual processors, the quantum internet aims to connect quantum nodes through entanglement distribution, enabling distributed computation. However, quantum states are fragile: environmental interactions, imperfect gate operations, and measurement errors can quickly degrade encoded information. Reliable quantum computation therefore requires carefully designed quantum error-correcting codes and fault-tolerant protocols. Unlike classical redundancy schemes, quantum error correction must preserve delicate phase relationships and relies on indirect error detection through syndrome extraction with ancillary qubits, enabling error identification without collapsing encoded states. Topological codes, in particular, emerged early as the preferred choice for many architectures due to their planar structure, local interactions, and natural support for fault-tolerant logical operations. Color codes, for instance, allow transversal implementation of all Clifford gates, whereas non-Clifford operations require magic-state injection. More recent constructions, including quantum low-density parity-check codes, offer higher encoding rates and reduced resource overhead, but require long-range qubit connectivity and more complex protocols. Despite experimental progress, several challenges remain. Many platforms exhibit biased noise, motivating codes that exploit channel asymmetry. Real-time decoding remains a critical bottleneck, as fault-tolerant computation requires low-latency processing of rapidly generated syndrome data. Fault-tolerant logical state initialization also remains resource-intensive and scales poorly with code distance. Addressing these challenges, this thesis develops a unified framework spanning theory and experiment. It provides precise performance analysis of quantum error-correcting codes in the low physical error-rate regime, introduces codes tailored to biased noise, develops fast low-complexity decoding algorithms for real-time operation, and presents modular and efficient protocols for fault-tolerant state preparation of CSS codes. The latter is demonstrated experimentally through the preparation of a logical zero state of the Golay code on the Quantinuum H2 trapped-ion quantum computer.
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 (QEC) represents the cornerstone technology for realizing fault-tolerant quantum computing, addressing the fundamental challenge of quantum state decoherence in noisy intermediate-scale quantum (NISQ) devices. This comprehensive review examines recent advances in QEC implementations from 2020-2024, spanning theoretical foundations to practical hardware demonstrations. We analyze the evolution from pioneering stabilizer codes to modern topological approaches, with particular emphasis on surface codes and low-density parity-check (LDPC) codes that have demonstrated below-threshold error correction in superconducting and trapped-ion systems. The paper provides a comparative analysis of classical and machine learning-based decoder algorithms, evaluating their performance through complexity analysis and threshold comparisons in real-time error correction scenarios. Through examination of breakthrough experiments by Google, IBM, Quantinuum, and emerging platforms, we assess the current state of fault-tolerant quantum computing and identify critical engineering challenges including correlated noise models, cryogenic control systems, and scalable logical gate implementations. Our quantitative analysis reveals that while significant progress has been made toward practical fault-tolerance thresholds, achieving the resource efficiency and reliability required for large-scale quantum computation remains challenging. We provide detailed resource estimates for key applications including Shor's algorithm and optimization problems, along with analysis of industry roadmaps toward thousand-logical-qubit systems. This review serves as a comprehensive guide for researchers and engineers working toward the next generation of fault-tolerant quantum computers, with particular focus on information-theoretic perspectives relevant to quantum information entropy and syndrome processing.
Quantum error correction (QEC) represents the cornerstone technology for realizing fault-tolerant quantum computing, addressing the fundamental challenge of quantum state decoherence in noisy intermediate-scale quantum (NISQ) devices. This comprehensive review examines recent advances in QEC implementations from 2020-2024, spanning theoretical foundations to practical hardware demonstrations. We analyze the evolution from pioneering stabilizer codes to modern topological approaches, with particular emphasis on surface codes and low-density parity-check (LDPC) codes that have demonstrated below-threshold error correction in superconducting and trapped-ion systems. The paper provides a comparative analysis of classical and machine learning-based decoder algorithms, evaluating their performance through complexity analysis and threshold comparisons in real-time error correction scenarios. Through examination of breakthrough experiments by Google, IBM, Quantinuum, and emerging platforms, we assess the current state of fault-tolerant quantum computing and identify critical engineering challenges including correlated noise models, cryogenic control systems, and scalable logical gate implementations. Our quantitative analysis reveals that while significant progress has been made toward practical fault-tolerance thresholds, achieving the resource efficiency and reliability required for large-scale quantum computation remains challenging. We provide detailed resource estimates for key applications including Shor's algorithm and optimization problems, along with analysis of industry roadmaps toward thousand-logical-qubit systems. This review serves as a comprehensive guide for researchers and engineers working toward the next generation of fault-tolerant quantum computers, with particular focus on information-theoretic perspectives relevant to quantum information entropy and syndrome processing.
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 (QEC) is considered essential for the development of scalable, fault-tolerant quantum computers in the medium- to long-term. By encoding quantum information across many physical qubits, carefully designed measurements can be taken to gain limited knowledge as to errors which may have occurred in the system, known as the error syndrome, with out causing quantum decoherence. In surface codes, qubits are arranged in a two-dimensional topological space, such as a torus or plane. In quantum low-density parity-check (qLDPC) codes, this is generalised to effectively arbitrary arrangements of qubits and syndrome checks. This work studies decoders: the algorithms responsible for inferring error correcting decisions from syndromes. In particular, the union–find decoder is studied in terms of its performance at scale. A simulation of a union–find decoding architecture is used to identify computational bottlenecks and pro pose improvements. In literature, it is assumed that the decoder runs in time scaling quadratically in the number of qubits under a naive implementation, but near-linearly when including two well-known optimisations. A key result is that, under independent and phenomenological noise models on surface codes, this complexity is strictly linear regardless of these optimisations. A supporting analytical argument is presented using percolation theory. Generalisation of the decoding problem to qLDPC codes is also studied, with approaches broadly relying on Gaussian elimination to find appropriate solutions. A strategy is proposed which uses a novel online variant of the Gaussian elimination algorithm to solve this linear system incrementally on growing local clusters, with accompanying complexity analysis and empir ical data demonstrating a reduction in runtime. An investigation into the use of metachecks is also presented, inspired by single-shot decoding, with implications for how such qLDPC decoders could be further improved.