Fault tolerant quantum computers will require efficient co-processors for real-time decoding of their adopted quantum error correction protocols. In this work we examine the possibility of using specialised Ising model hardware to perform this decoding task. Examples of Ising model hardware include quantum annealers such as those produced by D-Wave Systems Inc., as well as classical devices such as those produced by Hitatchi and Fujitsu and optical devices known as coherent Ising machines. We use the coherent parity check (CPC) framework to derive an Ising model mapping of the quantum error correction decoding problem for an uncorrelated quantum error model. A specific advantage of our Ising model mapping is that it is compatible with maximum entropy inference techniques which can outperform maximum likelihood decoding in some circumstances. We use numerical calculations within our framework to demonstrate that maximum entropy decoding can not only lead to improved error suppression, but can also shift threshold values for simple codes. As a high value problem for which a small advantage can lead to major gains, we argue that decoding quantum codes is an ideal use case for quantum annealers. In addition, the structure of quantum error correction codes allows application specific integrated circuit (ASIC) annealing hardware to reduce embedding costs, a major bottleneck in quantum annealing. Finally, we also propose a way in which a quantum annealer could be optimally used as part of a hybrid quantum-classical decoding scheme.
Fault tolerant quantum computers will require efficient co-processors for real-time decoding of their adopted quantum error correction protocols. In this work we examine the possibility of using specialised Ising model hardware to perform this decoding task. Examples of Ising model hardware include quantum annealers such as those produced by D-Wave Systems Inc., as well as classical devices such as those produced by Hitatchi and Fujitsu and optical devices known as coherent Ising machines. We use the coherent parity check (CPC) framework to derive an Ising model mapping of the quantum error correction decoding problem for an uncorrelated quantum error model. A specific advantage of our Ising model mapping is that it is compatible with maximum entropy inference techniques which can outperform maximum likelihood decoding in some circumstances. We use numerical calculations within our framework to demonstrate that maximum entropy decoding can not only lead to improved error suppression, but can also shift threshold values for simple codes. As a high value problem for which a small advantage can lead to major gains, we argue that decoding quantum codes is an ideal use case for quantum annealers. In addition, the structure of quantum error correction codes allows application specific integrated circuit (ASIC) annealing hardware to reduce embedding costs, a major bottleneck in quantum annealing. Finally, we also propose a way in which a quantum annealer could be optimally used as part of a hybrid quantum-classical decoding scheme.
Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.
Quantum computation has the promise to solve problems that are not efficiently solvable by classical means. Yet, due to the fragile nature of quantum states, it is unavoidable that quantum systems are subject to decoherence, gate imperfections, and measurement errors. These problems stand as a major roadblock to the construction of large-scale fault-tolerant quantum computers. Conventional quantum error correction (QEC) codes, such as the surface code or Shor code, need to use an extensive amount of redundancy and classical control, which can be both computationally expensive and challenging to scale. In this context, we propose to investigate the use of machine learning (ML) techniques by realizing on-line identification, localization and correction of these quantum errors for fault-tolerant quantum computation by means of fault-tolerant quantum computing processes. Our model introduces a hybrids ML-QEC architecture that combines classical neural networks and decision-tree classifiers with quantum syndrome decoding. These models are trained on error syndromes produced using the approximation of common noise models (for example, depolarizing noise, dephasing and amplitude damping), simulated under different QEC schemes. The machine learning methods are trained with quantum error patterns and have a capability to rapidly infer where errors are likely to occur, as well as recommend correction operations with high confidence. As a result, the system reduces dramatically the delay and computational costs when compared to conventional decoding methods. There is also a self-adaptive learning module in the work, which recommences training on fresh quantum error distributions, meaning the system can adapt to changes in the environment (and also the decay of the hardware) as time progresses. The study further carries out ML predictions inside a real-time loop and cross-checks them with quantum state tomography and entanglement fidelity such as indicators. Experimental characterization on dedicated devices as well as noise models show that our ML-enhanced decoders reach higher fidelity and lower logical error rates compared to baseline decoders, especially in non-uniform or time-dependent noise. This work is a significant stride towards realizing practical fault-tolerant quantum computation by combining quantum error correction with the versatility and scalability of machine learning. With intelligent models to dictate learning, our proposed framework is able to eliminate a lot of trial-and-error manual processes in the conventional heuristic-based approach, and points to a potential path to achieve scalable quantum information processing platforms.
Quantum error correction (QEC) is indispensable for realizing fault-tolerant quantum computation, yet its effectiveness hinges critically on the classical decoding algorithm that interprets noisy syndrome measurements. Among all possible decoding strategies, maximum likelihood decoding (MLD) is provably optimal, since it identifies the logical group with largest likelihood by summing over all possible errors within logical class consistent with the observed syndrome. Despite its optimality, MLD is computationally intractable in general (#P-hard), motivating a rich landscape of exact and approximate algorithms. In this topical review, we provide a unified perspective on MLD by surveying recent advances through three complementary lenses: statistical mechanics, tensor networks, and artificial intelligence. From the statistical mechanics viewpoint, the MLD problem maps onto evaluating partition functions of disordered spin models, enabling exact solutions for certain codes and noise models as well as threshold estimation via phase-transition analysis. From the tensor network perspective, approximate contraction of tensor networks on the code's factor graph yields decoders that closely approach MLD accuracy with polynomial computational cost. From the artificial intelligence perspective, neural-network-based decoders, including autoregressive generative models and recurrent transformers, learn to approximate the MLD distribution from data, achieving high accuracy with the parallelism afforded by modern hardware accelerators. We discuss the connections among these three approaches, review their application to both simulated and experimental quantum hardware, and outline open challenges including real-time decoding, scalability to large code distances, and generalization to high-rate quantum low-density parity-check codes.
Quantum annealing provides a promising route for the development of quantum optimization devices, but the usefulness of such devices will be limited in part by the range of implementable problems as dictated by hardware constraints. To overcome constraints imposed by restricted connectivity between qubits, a larger set of interactions can be approximated using minor embedding techniques whereby several physical qubits are used to represent a single logical qubit. However, minor embedding introduces new types of errors due to its approximate nature. We introduce and study quantum annealing correction schemes designed to improve the performance of quantum annealers in conjunction with minor embedding, thus leading to a hybrid scheme defined over an encoded graph. We argue that this scheme can be efficiently decoded using an energy minimization technique provided the density of errors does not exceed the per-site percolation threshold of the encoded graph. We test the hybrid scheme using a D-Wave Two processor on problems for which the encoded graph is a two-level grid and the Ising model is known to be NP-hard. The problems we consider are frustrated Ising model problem instances with ``planted'' (a priori known) solutions. Applied in conjunction with optimized energy penalties and decoding techniques, we find that this approach enables the quantum annealer to solve minor embedded instances with significantly higher success probability than it would without error correction. Our work demonstrates that quantum annealing correction can and should be used to improve the robustness of quantum annealing not only for natively embeddable problems but also when minor embedding is used to extend the connectivity of physical devices.
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 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.