Parallel decoding of multiple logical qubits in tensor-network codes

Terry Farrelly, Nicholas Milicevic, Robert J. Harris, Nathan A. McMahon, and Thomas M. Stace

Phys. Rev. A 105, 052446 – Published 31 May 2022

Abstract

We consider tensor-network stabilizer codes and show that their tensor-network decoder has the property that independent logical qubits can be decoded in parallel. As long as the error rate is below threshold, we show that this parallel decoder is essentially optimal. Holographic codes are interesting examples of tensor-network codes, so we first test out this parallel decoding strategy on the max-rate holographic Steane (heptagon) code. For holographic codes with a constant number of logical qubits, the tensor-network decoder was shown to be efficient with complexity polynomial in $n$ , the number of physical qubits. Here we show that, by using the parallel decoding scheme, the complexity is also linear in $k$ , the number of logical qubits. We then calculate the bulk threshold (the threshold for logical qubits a fixed distance from the code center) under depolarizing noise for the max-rate holographic Steane code to be $9.4 %$ . We also introduce some further holographic error-correcting codes and calculate their thresholds under depolarizing noise. One example is based on an 11-qubit code due to Gottesman, which has asymptotic rate 0.114 and threshold 13.8%. Another code we consider is an asymptotically zero-rate holographic code, which performs extremely well under dephasing noise in the $X, Y$ , or $Z$ basis with a threshold of $50 %$ .

Received 2 November 2021
Accepted 5 May 2022

DOI:https://doi.org/10.1103/PhysRevA.105.052446

Physics Subject Headings (PhySH)

Quantum error correction

Quantum Information, Science & Technology

Authors & Affiliations

Terry Farrelly ^1,*, Nicholas Milicevic¹, Robert J. Harris¹, Nathan A. McMahon^1,2, and Thomas M. Stace¹

¹ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia
²Department of Physics, Friedrich-Alexander University Erlangen-Nürnberg, 91058 Erlangen, Germany

^*farreltc@tcd.ie

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Vol. 105, Iss. 5 — May 2022

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Figure 1
(a) Tensor-network code composed of two Steane-code tensors, giving a [12,2,3] code. The contraction of two indices of the Steane code is possible because the Steane code can distinguish single-qubit Pauli errors on any site (because it can correct any single-qubit error). The reason this allows us to contract tensor indices to get a new code is discussed around Eq. (3). (b) Uniform tiling of hyperbolic space with regular heptagons where four heptagons meet at any vertex. (c) Holographic Steane code. This is also called the heptagon code because each tensor is naturally associated with a heptagonal tile (though one could conceive of other codes corresponding to a heptagonal tiling not based on the Steane code). The code can be constructed by starting from the central tile, which has a Steane-code tensor $T^{1} (L)$ associated with it. Then at radius 2 we add seven further tensors (the same tensors, but now represented by green heptagons), each with one leg contracted to one of the outgoing legs of the central tensor. Finally, we add further $T^{1} (L)$ tensors (blue disks) at radius 3, with one or two outgoing legs contracted with outgoing legs of tensors at radius 2. This is a radius-3 holographic code. The uncontracted legs at the boundary correspond to physical qubits. The resulting tensor describes a stabilizer code, which follows using Eq. (3) and noting that each Pauli error localized only on qubits 6 and 7 of the Steane code has a unique syndrome. (d) Tensor-network contraction used for the maximum-likelihood decoder. The central tensor describes the logical cosets of the code, while the boundary (single-leg) tensors describe the (uncorrelated) noise model on the physical qubits.
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Figure 2
All points correspond to 1000 samples and error bars are standard errors. Lines between points are to guide the eye between data for a given code radius. (a) Monte Carlo estimates for the probability of a logical error for the central logical qubit versus the single-qubit error probability for different code radii. The estimates of $p_{failure} = 1 - p_{success}$ used the final line of Eq. (11). The error bars are generally smaller than the error bars when $p_{failure}$ is estimated using Eq. (9), which is shown in Fig. 4 in Appendix pp2. (b) Plot of $p_{failure} (p)$ versus $d_{est} = n^{0.54}$ , where the latter is the estimate from [26] for the code distance. Each line corresponds to a different value of the depolarizing noise strength $p$ . This plot is consistent with exponential decay of the logical error rate with the code distance. (c) Fraction of cases (instances of the randomly chosen error $E$ ) for which all eight logical qubits satisfy the necessary condition ${max}_{L_{i}} {Prob}_{i} (L_{i} | \vec{s}) > K / (K + 1) = 8 / 9$ , which guarantees that the marginal decoder is optimal. (The marginal decoder can still be correct even when this condition is not satisfied.) (d) Monte Carlo estimates for the word failure probability of the central eight logical qubits.
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Figure 3
Monte Carlo estimates for the probability of a logical error for the logical qubit of a zero-rate holographic code (built from the seven-qubit tailored code [35]) versus the single-qubit error probability for different code radii. The error model is dephasing in the $X$ basis, but similar behavior is seen for this code for dephasing in both the $Y$ and $Z$ bases. Each code reaches a logical failure rate of 0.5 at $p = 0.5$ . All points in the above plots correspond to 1000 samples and error bars are standard errors. Lines between points are to guide the eye between data for a given code radius and are not necessarily an accurate interpolation.
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Figure 4
Estimates of the probability of a logical error $p_{failure} = 1 - p_{success}$ when decoding the central logical qubit versus the single-qubit error probability for different code radii. For each point we have taken 1000 samples. This was obtained by using Monte Carlo sampling to estimate $p_{failure}$ via Eq. (9). Error bars correspond to standard errors, but are generally larger than in Fig. 2. Lines between points are to guide the eye between data for a given code radius.
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Figure 5
Monte Carlo estimates of the probability of failing to correct an error for the central logical qubit of the holographic Steane code as a function of the single-qubit depolarizing probability $p$ . Each point corresponds to 1000 samples and error bars are standard errors. Lines between points are to guide the eye between data for a given code radius.
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Figure 6
Failure rate for the Steane holographic code's central logical qubit now plotted against the rescaled error probability $x = (p - p_{th}) n^{1 / ν}$ for code radii from 1 to 8. We fit a quadratic (using least-squares fitting) to the radius-8 data to estimate $f (x)$ because the radius-8 code is closest to the large-system limit. This allowed us to find values of $p_{th}$ and $ν$ such that the data for all radii are as close as possible to the curve. The resulting values are $ν = 2.96$ and $p_{th} = 0.094 47 (5)$ .
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