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Deep neural decoders for near term fault-tolerant experiments

Christopher Chamberland, Pooya Ronagh

arXiv:1802.06441v2quant-phstat.ML

TL;DR

Efficient decoders suited to realistic experimental noise remain a challenge. The paper evaluates neural-network decoders across fault-tolerant schemes and reports advantages over simpler methods, alongside limits in low-noise and large-code settings.

  • Problem

    Finding efficient decoders for realistic experimental noise is a challenge, motivating alternatives to standard decoding methods.

    Source: §I · p. 1, §V · p. 19

  • Method

    The paper applies multiple neural-network methods to fault-tolerant schemes under a full circuit-level noise model, without requiring knowledge of the underlying noise model.

    Source: §I · p. 2

  • Results

    With enough training data, neural decoders achieve lower logical failure rates than the lookup-table schemes, and the conclusion reports advantages over single-hidden-layer networks and regression methods.

    Source: D. Limitations of deep neural decoders · p. 22, §VI · p. 22

  • Takeaways & Limitations

    For large-scale quantum computation, scalable decoders may be preferable even if they perform less well than neural decoders trained on small-distance codes.

    Source: §VI · p. 22

    Neural-decoder scaling is challenging because better codes and lower physical error rates make training data sparser, while deep-learning classifiers are not viable for sparse classification problems.

    Source: D. Limitations of deep neural decoders · p. 22

Abstract

from arXiv · show

Finding efficient decoders for quantum error correcting codes adapted to realistic experimental noise in fault-tolerant devices represents a significant challenge. In this paper we introduce several decoding algorithms complemented by deep neural decoders and apply them to analyze several fault-tolerant error correction protocols such as the surface code as well as Steane and Knill error correction. Our methods require no knowledge of the underlying noise model afflicting the quantum device making them appealing for real-world experiments. Our analysis is based on a full circuit-level noise model. It considers both distance-three and five codes, and is performed near the codes pseudo-threshold regime. Training deep neural decoders in low noise rate regimes appears to be a challenging machine learning endeavour. We provide a detailed description of our neural network architectures and training methodology. We then discuss both the advantages and limitations of deep neural decoders. Lastly, we provide a rigorous analysis of the decoding runtime of trained deep neural decoders and compare our methods with anticipated gate times in future quantum devices. Given the broad applications of our decoding schemes, we believe that the methods presented in this paper could have practical applications for near term fault-tolerant experiments.

I. Introduction

The paper addresses the need for efficient fault-tolerant decoders suited to realistic noise, including settings where the device noise model is unknown. It evaluates neural decoders across protocols and code distances, and examines training challenges and inference runtime.

I. Introduction

  • Deep neural networks are proposed as efficient decoders that may tolerate large noise rates and learn error probabilities from experimental data, even when the noise model is unknown.Fault-tolerant gate and measurement errors complicate adaptation of some existing decoders to experimental protocols.Source: §I · pp. 1–2
  • The paper investigates whether neural decoders perform well beyond distance three and near or below typical fault-tolerant thresholds, and how their runtime compares with realistic gate times.The introduction also asks what decoding speeds neural networks can achieve.Source: §I · p. 2
  • Neural decoders are evaluated on surface-code and CNOT-exRec protocols using Steane and Knill error correction, for distance-three and distance-five codes under full circuit-level noise.The CNOT-exRec is examined because it limits the underlying code threshold in most cases with Steane and Knill error correction.Source: §I · p. 2
  • The methods are designed for experimental data and require no knowledge of the device’s underlying noise model.Existing approaches discussed in the introduction require some noise knowledge, which can be difficult to obtain for realistic devices.Source: §I · pp. 1–2
  • A proper performance analysis trains the neural network at every considered physical error rate; training at low error rates is difficult because useful training samples become sparse.Training on hardware tuned to a noisier regime may fit a different noise model than the original device’s.Source: §I · p. 2
  • The paper details decoder architectures and training, simulates circuits under full circuit-level depolarizing noise, and analyzes inference hardware and runtime against anticipated gate delays.It also recommends network quantization as a suitable technology for decoding quantum error-correcting codes.Source: §I · pp. 2–3

II. Fault-tolerant protocols

This section sets out the fault-tolerant protocols, code concepts, and noise setting used to study near-term decoding. It contrasts fast lookup-table decoding for small codes with a scalable naive baseline that can be paired with a deep neural decoder.

II. Fault-tolerant protocols

  • The study focuses on near-term experiments and uses lookup-table decoders for small-distance codes alongside a naive decoder that can pair with a deep neural decoder.The proposed schemes are stated to be usable through distance d ≤ 7 with existing resources and the code families considered; a distance-nine color code would require 8.8 exabytes for its lookup table.Source: §II · p. 3
  • A distance-d stabilizer code corrects t = floor((d−1)/2) errors, and its syndrome is a bit string recording commutation with the stabilizer generators.The code’s distance is the minimum weight of a nontrivial logical operator.Source: §II · p. 3
  • Fault-tolerant error correction requires that up to t combined input errors and protocol faults preserve the ideally decoded codeword, and that up to t protocol faults leave at most that many errors.Both criteria are required to prevent errors accumulating across rounds and to ensure error correction extends qubit lifetime below threshold.Source: §II · p. 3
  • Protocol comparisons use a full extended rectangle around a logical gate, with leading error-correction rounds before the gate and trailing rounds after it.The figure illustrates an exRec for a logical CNOT.Source: §II · p. 4, Figure 1 · p. 4
  • Although the decoders are designed for unknown noise, benchmarking uses a full circuit-level depolarizing noise model.The specified model includes Pauli faults after two-qubit gates and resting-qubit locations, as well as errors in state preparation and measurement.Source: §II · p. 4

A. Rotated surface code

The rotated surface-code section develops a lookup-table decoder and a syndrome-repetition procedure for fault tolerance. The lookup table decodes quickly but does not scale with code distance.

A. Rotated surface code

  • The rotated surface code is a [[d^2, 1, d]] stabilizer code on a two-dimensional lattice, with data and stabilizer-measurement ancilla qubits arranged on the lattice.The figure identifies data qubits as white circles and ancillas as black circles; green squares measure Z stabilizers and red squares measure X stabilizers.Source: A. Rotated surface code · p. 4, Figure 2 · p. 4
  • A surface-code measurement cycle uses local data–ancilla interactions, and all stabilizer generators have weight two or four regardless of lattice size.The described full measurement cycle takes six time steps.Source: A. Rotated surface code · p. 4
  • The lookup table assigns each syndrome a correction, accounting for errors from up to t = 2d + 1 faults whose weight exceeds t by choosing a logically equivalent correction.For the stated construction, the required syndrome distinctions hold with the CNOT ordering in Fig. 3.Source: A. Rotated surface code · p. 5
  • Lookup-table decoding converts the measured syndrome to a row index and retrieves its correction, but the number of syndromes grows exponentially with code distance.The construction is therefore not scalable as code distance increases.Source: A. Rotated surface code · p. 5
  • A single syndrome measurement is not fault-tolerant because measurement errors can be indistinguishable from data errors; distance-three decoding instead uses three rounds and a majority syndrome.If all three syndromes differ, the final-round syndrome is used; the described decoder was shown to be fault-tolerant.Source: A. Rotated surface code · p. 5
  • For higher-distance codes, decoding initializes ndiff to 0, repeats syndrome measurements while updating it, and, when ndiff reaches t, repeats measurement to obtain syndrome s and applies correction E(s).Source: A. Rotated surface code · p. 5

B. Steane error correction

Steane error correction uses transversal CNOTs to extract X- and Z-error information from encoded ancillas. Because ancilla preparation can spread errors, verification rejects ancillas with detected syndromes or incorrect logical eigenvalues.

B. Steane error correction

  • Transversal CNOTs couple encoded data to |+⟩ and |0⟩ ancillas, allowing measurements to reveal X- and Z-error information for correction.The data-to-ancilla CNOT copies data X errors to the |+⟩ ancilla; the corresponding phase-error procedure uses measurement in the X basis.Source: B. Steane error correction · p. 6
  • Because encoded |+⟩ and |0⟩ preparation is not generally fault-tolerant, verifier ancillas detect high-weight errors and failed ancillas are discarded.Rejection follows a non-trivial verifier syndrome or a measured −1 eigenvalue of a logical operator.Source: B. Steane error correction · p. 6, Figure 5 · p. 6
  • Ancilla verification can be replaced by a decoding circuit using measurements and Pauli frames, but this analysis focuses on verification.Source: B. Steane error correction · p. 6
  • The order of bit- and phase-flip correction preserves fault-tolerant properties but creates asymmetry in X- and Z-logical failure rates.Source: B. Steane error correction · pp. 6–7
  • The analyzed codes are the [[7, 1, 3]] Steane and [[19, 1, 5]] self-dual CSS color codes, with three weight-six stabilizers of each type in the latter.Source: B. Steane error correction · p. 7, Figure 6 · p. 7
  • To obtain pseudo-thresholds for both codes, the study considers the CNOT-exRec because it has the largest number of locations and thus limits performance; its many CNOT gates result in correlated errors.Source: B. Steane error correction · p. 7

C. Knill error correction

Knill error correction applies to any stabilizer code and uses one transversal CNOT to couple data with an encoded logical Bell pair. Measurements recover error syndromes and logical Bell outcomes while teleporting the encoded data to the second ancilla block.

C. Knill error correction

  • Knill-EC applies to any stabilizer code and requires one transversal CNOT between data and ancilla, unlike Steane-EC’s two.Source: C. Knill error correction · p. 7
  • Measurements of X on data and Z on the first ancilla reveal Pauli eigenvalues, including stabilizer syndromes and logical Bell-measurement outcomes.The Bell measurement teleports encoded data to the second ancilla block.Source: C. Knill error correction · p. 8
  • For pre-existing errors of combined weight at most t, syndrome information identifies an equivalent correctable error and recovers the proper logical Bell-measurement outcome.Source: C. Knill error correction · p. 8
  • Under circuit-level noise, CNOT and measurement faults contribute data-block Z errors and ancilla-block X errors to the final syndrome.Source: C. Knill error correction · p. 8
  • Encoded |0⟩ and |+⟩ preparation is not fault-tolerant, so CSS codes can use the Steane ancilla-verification method in Knill-EC.Each Knill-EC unit has one extra idle-qubit location compared with Steane-EC.Source: C. Knill error correction · p. 8
  • Knill-EC can accommodate leakage-reduction units through teleportation, though LRUs remain necessary after preparing each |0⟩ and |+⟩ state.Source: C. Knill error correction · p. 8

D. Naive decoder

The section defines a simple recovery from a syndrome using pure errors, then motivates scalable decoding and describes how neural decoders can predict residual logical errors. It also analyzes the access and computation costs of the lookup-table and naive approaches.

D. Naive decoder

  • The naive decoder chooses the pure-error operator T(s) as recovery, while decoding more generally seeks the most likely logical operator L(s).The recovery decomposes into logical, pure-error, and stabilizer operators; the stabilizer choice does not affect the recovered state.Source: D. Naive decoder · p. 9

E. Lookup table and naive decoder complexity

  • Array-based lookup-table and inference-mapping decoders take O(1) time under the stated random-access-memory assumption.The complexity analysis treats memory access as constant time.Source: E. Lookup table and naive decoder… · p. 9
  • The naive decoder takes O((n − k) log(n − k)) time with parallel resources to combine the selected pure-error strings.The procedure accesses a pure error for each 1 in the syndrome and combines the strings with a parallel tree adder.Source: E. Lookup table and naive decoder… · p. 9

III. Deep neural decoders

  • Fast decoders are motivated by difficult noise characterization and the need to adapt to unknown experimental noise models.The paper notes that efficient circuit-level simulations may require approximations for noise models not described by Pauli channels.Source: §III · p. 9
  • The neural network is trained to predict residual logical errors from syndromes, and after training inference uses measured syndromes to return the correction vector.For the surface code, inputs include ancilla measurements from each error-correction round; for Steane and Knill EC, they are transversal measurement outcomes from leading and trailing blocks.Source: §III · pp. 9–10

A. Deep learning

This section presents the learning objective, label representation, neural-network layouts, and optimization choices used for deep neural decoding. It also describes hyperparameter-search challenges and Bayesian optimization as a candidate approach.

A. Deep learning

  • One-hot labels for all physical-qubit errors require |E| bits and grow exponentially with the number of physical qubits, motivating prediction of only the residual logical-error vector.The compact target records logical errors remaining after an auxiliary decoding scheme.Source: A. Deep learning · p. 10
  • Training approximates the conditional distribution of error labels given syndromes with a parameterized model, using softmax cross-entropy as the loss.The section describes the dataset as syndrome-label pairs and the objective as minimizing a distance between probability distributions.Source: The objective function · p. 10
  • The paper considers feedforward, recurrent, and convolutional network layouts, with each suited to different structures in the syndrome data.Feedforward networks connect consecutive layers; RNNs process temporal inputs, while CNNs exploit local properties of image-like inputs.Source: Layouts · p. 11
  • Minibatch gradient descent approximates the full-data gradient, while momentum, AdaGrad, and RMSProp provide update variants for optimization.AdaGrad adapts updates using accumulated squared gradients; RMSProp modifies its root-mean-square scaling to slow AdaGrad's aggressive gradient decay.Source: Stochastic gradient descent · pp. 11–12
  • Hyperparameter selection is costly and lacks a known gradient-based solution, and Bayesian optimization is presented as a sample-efficient, gradient-free candidate with its own tuning choices.The paper also identifies parameter sensitivity, interactions, and consistent comparisons across network types as reasons for automated tuning.Source: Hyperparameter tuning · p. 12

B. Steane and Knill EC deep neural decoder for the CNOT-exRec

For Steane and Knill CNOT-exRec decoding, the section describes disjoint X and Z networks that predict logical recovery choices. Its recurrent design processes leading and trailing error-correction rounds sequentially.

B. Steane and Knill EC deep neural decoder for the CNOT-exRec

  • The feedforward design separates X and Z networks, whose output classes encode identity or the corresponding logical error and whose losses are summed.The final layer before each error output uses the identity activation because softmax is included in the softmax cross-entropy loss.Source: B. Steane and Knill EC deep neural… · p. 12, Figure 10 · p. 13
  • The CNOT-exRec RNN processes leading-round syndromes first and trailing-round syndromes second, using two iterations of its hidden cell.The two networks predict whether X- or Z-recovery is needed on the output qubits.Source: The CNOT-exRec RNN · p. 12, Figure 11 · p. 13
  • An LSTM carries temporal information in its internal state, with input, forget, hidden, and output layers controlling information flow across epochs.Peepholes allow the internal state to contribute to the input, forget, and output layers.Source: The CNOT-exRec RNN · p. 13, Figure 12 · p. 13

C. Surface code deep neural decoder

The section proposes recurrent and convolutional architectures for surface-code syndromes, reflecting their temporal sequence and local structure. The CNN treats successive error-correction rounds as a three-dimensional input.

C. Surface code deep neural decoder

  • The surface-code RNN processes consecutive error-correction syndromes for as many epochs as the maximum number of rounds, repeating the last syndrome to pad sequences that terminate early.The stated maxima are three rounds for the distance-three rotated code and six for the distance-five code.Source: The surface code RNN · pp. 13–14
  • The surface-code CNN treats each round as a two-dimensional syndrome image and the sequence of rounds as a three-dimensional animation, with convolution kernels spanning space and time.The design is motivated by local relationships among surface-code errors, syndromes, and recovery operators.Source: The surface code CNN · p. 14
  • The distance-five CNN uses separate X and Z inputs, symmetric outward padding, and two convolutional layers with 3 × 3 × 3 and 4 × 4 × 4 kernels.The X and Z inputs are padded to 4 × 5 × 6 and 5 × 4 × 6, respectively, before the convolutional layers and a fully connected layer.Source: The surface code CNN · p. 14

IV. Numerical experiments

The experiments evaluate neural decoders on distance-three and distance-five fault-tolerant protocols under circuit-level depolarizing noise. Feedforward networks generally perform strongly, with the best architecture varying by protocol and decoder baseline.

Experimental setup

  • The study simulates circuit-level depolarizing noise across physical error rates from p = 1.0 × 10−4 to p = 2.0 × 10−3, with datasets of 2×106 samples for distance three and 2 × 107 for distance five.Each deep-learning experiment was run ten times; the reported results give averages and standard deviations.Source: §IV · p. 14

Distance-three Steane CNOT-exRec

  • For distance-three Steane-EC, two-hidden-layer feedforward networks outperform RNNs, while softmax regression matches lookup-table decoding for LU-DND but not PE-DND.Adding hidden layers helps, but three or more layers perform worse as deeper-network training becomes more challenging.Source: §IV · pp. 15–16
  • A single-hidden-layer PE-DND feedforward network is almost on par with two-hidden-layer LU-DND, despite the pure-error table growing linearly with syndromes while the lookup table grows exponentially.The authors suggest that more frequent logical faults with pure-error recovery make training data less sparse and help the network capture patterns.Source: §IV · p. 16

Distance-three Knill CNOT-exRec

Distance-three rotated surface code

  • For the distance-three rotated surface code, single-hidden-layer feedforward networks perform best, and PE-DND matches or slightly improves on LU-DND.RNNs do not improve the results.Source: Distance-three rotated surface code · p. 16

Distance-five Steane CNOT-exRec

  • For distance-five Steane-EC, LU-DND and PE-DND both significantly improve pseudo-thresholds, while RNNs and three-hidden-layer feedforward networks add no improvement.PE-DND has slightly more difficulty with pattern recognition than LU-DND.Source: Steane-EC CNOT-exRec for the [[19, 1,… · p. 16

Distance-five Knill CNOT-exRec

  • For distance-five Knill-EC, both LU-DND and PE-DND similarly improve the scheme’s pseudo-threshold.The experiment used a two-hidden-layer feedforward network, selected based on its effectiveness in the Steane experiment.Source: Knill-EC CNOT-exRec for the [[19, 1,… · p. 16

Distance-five rotated surface code

  • For distance-five rotated surface-code decoding, two-hidden-layer LU-DND feedforward networks perform best overall; PE-DND feedforward and 3D-CNN options are slightly less performant.The 3D-CNN slightly improves on the PE-DND feedforward network but is only slightly better than lookup-table decoding in the LU-DND case.Source: Distance-five rotated surface code · p. 16

V. Performance analysis

The performance analysis compares trained neural decoders with lookup-table decoding and considers whether decoding can meet the timing needs of fault-tolerant operations. It distinguishes fast lookup from memory-limited scaling and emphasizes the runtime demands of near-term devices.

Decoder efficiency

  • Lookup-table decoding takes effectively O(1) time when its full table fits in memory, but table size grows exponentially with the number of syndromes.For larger codes, neither lookup tables nor inference maps are feasible because of exponentially growing memory use.Source: §V · p. 19

Runtime requirements

  • With anticipated 10 ns gates, CNOT-exRec operations and multiple surface-code error-correction rounds can take as little as 100 ns, making comparable classical decoding times necessary for active correction.The authors caution that polynomial asymptotic complexity alone does not establish a decoder’s viability.Source: §V · p. 19

Neural-decoder inference

  • A trained DND can infer recovery operations through a sequence of highly parallelizable matrix multiplications.The paper discusses this inference approach as an alternative to lookup tables for larger codes.Source: §V · p. 19

A. Inference mapping from a neural decoder

Inference mapping stores the trained neural decoder’s outputs for every possible syndrome, assigning each syndrome to a recovery operator. The method fits distance-three schemes but becomes infeasible for the reported distance-five examples.

Inference-map memory

  • Inference maps require 2.10 megabytes for the considered distance-three schemes, but 590 exabytes for distance-five Steane or Knill color codes and 2.79 × 1024 exabytes for the distance-five rotated surface code.The paper identifies the distance-five memory requirements as making inference mapping infeasible for larger codes.Source: A. Inference mapping from a neural… · p. 19

B. Fast inference from a trained neural network

Quantized inference preserves the two-hidden-layer decoder's improvement over lookup-table methods at 8-bit and 7-bit precision. Its arithmetic can be organized with logarithmic-depth adder trees, but active correction with 10ns gate delays requires very fast classical arithmetic.

B. Fast inference from a trained neural network

  • The LU-DND improves logical fault rate from 1.95 × 10−4 for lookup-table methods to 9.45 × 10−5, with the improvement preserved at 8-bit and 7-bit fixed-point precision.The quantization experiment used the Steane EC dataset at p = 2 × 10−4 and averaged logical error rates over 10 training and cross-validation rounds.Source: Network quantization · p. 20, Figure 26 · p. 20

C. Classical arithmetic performance

  • For networks with at most 1000 neurons per hidden layer, inference cost is logarithmic in syndrome count and hidden-layer size, and linear in the number of hidden layers.The critical path uses full-adder trees of depth log(S + 1) for S input syndromes and log(L1 + 1) for a first hidden layer of L1 neurons.Source: C. Classical arithmetic performance · p. 21, Figure 27 · p. 21
  • For active correction with 10ns gate delays, classical arithmetic must run well below Table VII's adder-time limits, which assume independent arithmetic operations occur in parallel.The table's limits are optimistic; actual adder delays should be well below those values.Source: C. Classical arithmetic performance · p. 21, Table VII · p. 21

D. Limitations of deep neural decoders

Deep neural decoders can outperform single-hidden-layer networks and regression methods, but sparse training data at low error rates makes scaling to larger-distance codes challenging. The paper also reports that its decoders may be less suitable than scalable alternatives for large-scale quantum computation.

D. Limitations of deep neural decoders

  • As codes improve or physical error rates fall, training samples become sparser, and deep networks may learn zeros instead of meaningful dataset patterns.The authors say scaling neural decoders to large-distance codes appears significantly challenging.Source: D. Limitations of deep neural decoders · p. 22
  • Training on the highest physical fault rate, p = 2 × 10−3, worked better for smaller error-rate test sets, but this is not a practical experimental training method.The authors note that device noise is not entirely known and cannot be intentionally intensified for decoder training.Source: D. Limitations of deep neural decoders · p. 22, Figure 28 · p. 22

VI. Conclusion

  • The authors consider multiple fault-tolerant schemes and neural-network architectures under a full circuit-level noise framework.They identify machine-learning algorithms suited to sparse data and application to actual quantum devices as future directions.Source: §VI · pp. 22–23
  • The paper reports a clear advantage for deep networks over single-hidden-layer networks and regression methods, alongside evidence of low-noise training limitations.For large-scale quantum computation, the authors say scalable decoders may be preferable even if they perform less well than neural decoders trained on small-distance codes.Source: §VI · p. 22, D. Limitations of deep neural decoders · p. 22

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