We consider a class of decoding algorithms that are applicable to error correction for both Abelian and non-Abelian anyons. This class includes multiple algorithms that have recently attracted attention, including the Bravyi-Haah RG decoder and variants thereof. They are applied to both the problem of correcting a single burst of errors (with perfect syndrome measurements) and active correction of continuously occurring errors (with noisy syndrome measurements). For Abelian models we provide a threshold proof in both cases for all decoders in this class, showing that they can arbitrarily suppress errors when the noise rate is under a finite threshold. For non-Abelian models such a proof is found for a single burst of errors. The reasons why the proof cannot be applied to the case of continuously occurring errors are discussed.
We consider a class of decoding algorithms that are applicable to error correction for both Abelian and non-Abelian anyons. This class includes multiple algorithms that have recently attracted attention, including the Bravyi-Haah RG decoder and variants thereof. They are applied to both the problem of correcting a single burst of errors (with perfect syndrome measurements) and active correction of continuously occurring errors (with noisy syndrome measurements). For Abelian models we provide a threshold proof in both cases for all decoders in this class, showing that they can arbitrarily suppress errors when the noise rate is under a finite threshold. For non-Abelian models such a proof is found for a single burst of errors. The reasons why the proof cannot be applied to the case of continuously occurring errors are discussed.
Hard-decision renormalization group (HDRG) decoders are an important class of decoding algorithms for topological quantum error correction. Due to their versatility, they have been used to decode systems with fractal logical operators, color codes, qudit topological codes, and non-Abelian systems. In this work, we develop a method of performing HDRG decoding which combines strengths of existing decoders and further improves upon them. In particular, we increase the minimal number of errors necessary for a logical error in a system of linear size L from to for any We apply our algorithm to decoding quantum double models and a non-Abelian anyon model with Fibonacci-like fusion rules, and show that it indeed significantly outperforms previous HDRG decoders. Furthermore, we provide the first study of continuous error correction with imperfect syndrome measurements for the quantum double models. The parallelized runtime of our algorithm is for the perfect measurement case. In the continuous case with imperfect syndrome measurements, the averaged runtime is for Abelian systems, while continuous error correction for non-Abelian anyons stays an open problem.
Maximum likelihood (ML) decoders perform quantum error correction by measuring a set of syndromes, classically computing an optimal recovery based on these syndromes and the noise model, and then performing that recovery. ML decoders are well understood for the toric code and generalize straightforwardly to other surface codes with Abelian anyons, where they can saturate information-theoretic decoding thresholds. On the other hand, codes with non-Abelian anyons have become increasingly relevant because they can support transversal or constant-depth non-Clifford gates and admit efficient preparation protocols in modern quantum devices. However, they present new challenges for decoding: non-Abelian anyons cannot in general be deterministically annihilated, and the logical operators cannot be classified by Abelian homology. In this work, we show that refined syndrome sets, including measurements that resolve non-Abelian anyon fusion outcomes, systematically improve the performance of ML decoders. We demonstrate this concretely by constructing explicit decoders for the D4 quantum double model under charge noise, and obtaining numerical results for thresholds via mappings to classical statistical-mechanics models. We show that there is a nonzero noise threshold below which the entire logical qudit can be recovered with a single round of measurements.
Quantum gates in topological quantum computation are performed by braiding non-Abelian anyons. These braiding processes can presumably be performed with very low error rates. However, to make a topological quantum computation architecture truly scalable, even rare errors need to be corrected. Error correction for non-Abelian anyons is complicated by the fact that it needs to be performed on a continuous basis, and further errors may occur while we are correcting existing ones. Here, we prove the feasibility of this task, establishing non-Abelian anyons as a viable platform for scalable quantum computation. We thereby focus on Ising anyons as the most prominent example of non-Abelian anyons and show that for these a finite error rate can indeed be corrected continuously. There is a threshold error rate pc>0 such that for all error rates p<pc the probability of a logical error per time step can be made exponentially small in the distance of a logical qubit.
Topological orders (TOs) are widely used as quantum error-correcting codes, with anyon excitations serving as error syndromes. For certain Abelian TOs, decoding can be performed by independently matching particle-antiparticle pairs of each species. However, matching-based decoders cannot handle more general fusion rules in either Abelian or non-Abelian TOs, nor account for noise that correlates different anyon species. While clustering decoders are more broadly applicable, they typically neglect anyon data and fusion properties, leading to poor performance in practice. In this work, we introduce a fundamentally different decoder for arbitrary TOs based on integer linear programming (ILP). The ILP formulation linearizes the error-correction problem through the introduction of auxiliary variables and encodes fusion rules as linear constraints. Classical optimization then identifies the minimum-weight error configuration. As concrete examples, we determine error-correction thresholds for three TOs: the Abelian Z2 TO under depolarizing noise, where charge and flux errors are correlated; the Abelian Z3 TO, which does not admit a pairwise matching decoder; and the non-Abelian D4 TO under noise channels that generate all anyon species. We demonstrate the versatility of the ILP decoder by showing a clear performance advantage over most existing decoders in all three cases. We further extend the method to incorporate noisy syndrome measurements and propose a just-in-time variant for continuous error correction. Our results establish ILP as a natural framework for handling correlated errors and general anyon fusion rules, and as a powerful and flexible general-purpose decoder for incoherent anyon noise in arbitrary TOs, with applications to fault-tolerant quantum computation.
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.
Topological order (TO) provides a natural platform for storing and manipulating quantum information. However, its stability to noise has only been systematically understood for Abelian TOs. In this Letter, we exploit the nondeterministic fusion of non-Abelian anyons to inform active error-correction and design decoders where the fusion products, instead of flag qubits, herald the noise. This intrinsic heralding offers improved thresholds over those of Abelian counterparts. Furthermore, we use Bayesian inference to obtain a statistical mechanics model for fixed-point non-Abelian TOs with perfect measurements under any noise model, which yields the optimal threshold conditioned on measuring anyon syndromes. We numerically illustrate these results for D_{4}≅Z_{4}⋊Z_{2} TO. In particular, for non-Abelian charge noise and perfect syndrome measurement, we find a conditioned optimal threshold p_{c}=0.218(1), whereas an intrinsically heralded minimum-weight perfect matching (MWPM) decoder already gives p_{c}=0.208 42(2), outperforming standard honeycomb-lattice MWPM with p_{c}=0.158 60(1). Our Letter highlights how non-Abelian properties can enhance stability, rather than reduce it, and discusses potential generalizations for achieving fault tolerance.