Analysing correlated noise on the surface code using adaptive decoding algorithms

Laboratory hardware is rapidly progressing towards a state where quantum error-correcting codes can be realised. As such, we must learn how to deal with the complex nature of the noise that may occur in real physical systems. Single qubit Pauli errors are commonly used to study the behaviour of error-correcting codes, but in general we might expect the environment to introduce correlated errors to a system. Given some knowledge of structures that errors commonly take, it may be possible to adapt the error-correction procedure to compensate for this noise, but performing full state tomography on a physical system to analyse this structure quickly becomes impossible as the size increases beyond a few qubits. Here we develop and test new methods to analyse blue a particular class of spatially correlated errors by making use of parametrised families of decoding algorithms. We demonstrate our method numerically using a diffusive noise model. We show that information can be learnt about the parameters of the noise model, and additionally that the logical error rates can be improved. We conclude by discussing how our method could be utilised in a practical setting blue and propose extensions of our work to study more general error models.Comment: 19 pages, 8 figures, comments welcome; v2 - minor typos corrected some references added; v3 - accepted to Quantu

Multi-path Summation for Decoding 2D Topological Codes

Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of $17.76 \pm 0.02 \%$. This is larger than the threshold achieved by previous matching-based decoders ($14.88 \pm 0.02 \%$), though still below the known upper bound of $\sim 18.9 \%$.Comment: 19 pages, 13 figures, published in Quantum, available at https://quantum-journal.org/papers/q-2018-10-19-102

Principles and Parameters: a coding theory perspective

We propose an approach to Longobardi's parametric comparison method (PCM) via the theory of error-correcting codes. One associates to a collection of languages to be analyzed with the PCM a binary (or ternary) code with one code words for each language in the family and each word consisting of the binary values of the syntactic parameters of the language, with the ternary case allowing for an additional parameter state that takes into account phenomena of entailment of parameters. The code parameters of the resulting code can be compared with some classical bounds in coding theory: the asymptotic bound, the Gilbert-Varshamov bound, etc. The position of the code parameters with respect to some of these bounds provides quantitative information on the variability of syntactic parameters within and across historical-linguistic families. While computations carried out for languages belonging to the same family yield codes below the GV curve, comparisons across different historical families can give examples of isolated codes lying above the asymptotic bound.Comment: 11 pages, LaTe

Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States

A long-standing open question about Gaussian continuous-variable cluster states is whether they enable fault-tolerant measurement-based quantum computation. The answer is yes. Initial squeezing in the cluster above a threshold value of 20.5 dB ensures that errors from finite squeezing acting on encoded qubits are below the fault-tolerance threshold of known qubit-based error-correcting codes. By concatenating with one of these codes and using ancilla-based error correction, fault-tolerant measurement-based quantum computation of theoretically indefinite length is possible with finitely squeezed cluster states.Comment: (v3) consistent with published version, more accessible for general audience; (v2) condensed presentation, added references on GKP state generation and a comparison of currently achievable squeezing to the threshold; (v1) 13 pages, a few figure