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Efficient fault-tolerant decoding of topological color codes

Ashley M. Stephens

arXiv:1402.3037v1quant-ph

TL;DR

Efficient color-code decoding algorithms had been proposed, but their performance with errors in syndrome measurements was unknown. This paper adapts graph matching to account for those errors in triangular 4.8.8 codes and reports thresholds of 0.0208(1) for a three-dimensional bit-flip channel and 0.00143(1) per gate when circuit details are included.

  • Problem

    The performance of efficient color-code decoding algorithms remained unknown when syndrome-measurement qubits were themselves affected by errors.

    Source: INTRODUCTION · p. 1

  • Method

    The paper modifies a graph-matching algorithm for triangular 4.8.8 color codes to account for syndrome errors using repeated measurements and time-extended subgraphs.

    Source: INTRODUCTION · p. 1, FAULT-TOLERANT DECODING VIA GRAPH… · p. 3

  • Results

    0.0208(1) is the threshold for the three-dimensional bit-flip channel, versus 0.0305(4) previously reported for integer-program-based decoding; including circuit details gives 0.00143(1) per gate.

    Source: INTRODUCTION · p. 1, NUMERICAL RESULTS · p. 4

  • Takeaways & Limitations

    The reported results indicate graph-matching decoding is feasible for fault-tolerant color-code error correction, although color-code thresholds remain significantly lower than surface-code thresholds.

    Source: INTRODUCTION · p. 1, SUMMARY AND FURTHER WORK · p. 5

    The decoder does not account for correlated errors introduced by syndrome-measurement circuits, and some configurations can fail even with fewer than (d+1)/2 errors.

    Source: SUMMARY AND FURTHER WORK · p. 5

Abstract

from arXiv · show

Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.

INTRODUCTION

Fault-tolerant decoding must be efficient and tolerate syndrome uncertainty, but color-code decoding poses a hypergraph-matching challenge. This paper modifies an efficient graph-matching approach for syndrome errors and reports thresholds for triangular 4.8.8 codes.

INTRODUCTION

  • Topological color codes admit transversal implementation of the entire Clifford group, a property that may reduce the overhead of achieving universality via state distillation.The motivation is to seek alternatives amid daunting surface-code overhead.Source: INTRODUCTION · p. 1
  • Color-code decoding is challenging because its equivalent matching problem is a hypergraph-matching problem with no known efficient algorithm.Efficient approximations have been proposed, but their performance when syndrome-measurement qubits are error-prone remained unknown.Source: INTRODUCTION · p. 1
  • 0.0208(1) is the reported threshold for a three-dimensional bit-flip channel, versus 0.0305(4) previously reported for integer-program-based decoding.Accounting for circuit details reduces the threshold to 0.00143(1) per gate, significantly below the surface-code threshold under an identical noise model.Source: INTRODUCTION · p. 1
  • The paper modifies Delfosse’s algorithm, which reduces color-code hypergraph matching to three efficiently solvable graph-matching problems, to account for syndrome errors.It applies the method to triangular 4.8.8 codes encoding one logical qubit.Source: INTRODUCTION · p. 1

TRIANGULAR 4.8.8 TOPOLOGICAL COLOR CODES

The paper studies planar 4.8.8 topological color codes, whose local stabilizers and triangular boundaries define a one-logical-qubit family. The dual graph supports a three-subgraph representation used in decoding.

TRIANGULAR 4.8.8 TOPOLOGICAL COLOR CODES

  • Data qubits occupy graph nodes, and each face carries two stabilizer generators whose simultaneous +1 eigenspace defines the code space.The generators are products of single-qubit Pauli operators on qubits incident on each face.Source: TRIANGULAR 4.8.8 TOPOLOGICAL COLOR CODES · pp. 1–2
  • In the triangular code diagrams, boundaries host logical Pauli operators, while hatched faces mark error terminations without implied qubits or stabilizers.The dual-graph view instead places data qubits on faces and stabilizer generators on nodes, with three colored subgraphs.Source: Figure 1 · p. 2, Figure 2 · p. 2
  • The 4.8.8 family on a planar disc admits transversal implementation of the entire Clifford group; triangular boundaries give one-logical-qubit codes with parameters [[(d^2+2d−1)/2, 1, d]] for odd d.Additional logical qubits can be introduced by puncturing holes while stabilizer generators remain local in two spatial dimensions.Source: TRIANGULAR 4.8.8 TOPOLOGICAL COLOR CODES · p. 2

FAULT-TOLERANT DECODING VIA GRAPH MATCHING

The decoding method treats three colored subgraphs as independent surface codes, then combines their inferred errors to choose a correction. Repeated syndrome measurements extend each subgraph through time so the method can account for syndrome errors.

FAULT-TOLERANT DECODING VIA GRAPH MATCHING

  • The method can succeed when corrections return the system to the code space, but some configurations instead yield a logical operator.The figures illustrate both outcomes: corrections on a three-qubit set can combine with errors to give identity, while an eight-qubit correction set can yield a logical operator.Source: Figure 3 · p. 3, Figure 4 · p. 3
  • For each subgraph, the decoder finds a minimum-weight error set, unions the identified edges, partitions data qubits into two sets, and corrects the lower-weight set.The independent subgraphs are surface-code-equivalent because a single-qubit error anticommutes with only two adjacent stabilizers in each one.Source: FAULT-TOLERANT DECODING VIA GRAPH… · pp. 2–3
  • Correlations between the three subgraphs might improve error identification, but the paper does not investigate that possibility in detail.The method’s examples include both successful decoding and failure.Source: FAULT-TOLERANT DECODING VIA GRAPH… · p. 3
  • Repeated measurements identify data-qubit errors through adjacent stabilizers changing state and syndrome-qubit errors through one stabilizer changing and then changing back.The measurement circuits are repeated d times to resolve syndrome ambiguity.Source: FAULT-TOLERANT DECODING VIA GRAPH… · p. 3
  • With unreliable syndrome measurements, each subgraph extends along a time axis, where data-qubit and syndrome-qubit errors appear as spacelike and timelike edges, respectively.The decoder projects identified spacelike edges into two dimensions; edges appearing an even number of times cancel.Source: FAULT-TOLERANT DECODING VIA GRAPH… · p. 3

SYNDROME MEASUREMENT CIRCUITS

The stabilizer-measurement circuits use cat states for weight-four and weight-eight checks. Their design detects preparation-induced correlated errors, with connectivity requirements for a two-dimensional qubit array.

SYNDROME MEASUREMENT CIRCUITS

  • The circuit choice aims to make high-weight correlated errors relatively unlikely without stochastic verification, while trading off threshold, depth, and connectivity.Standard single-qubit operator measurements are also sufficient given the code structure’s limits on error propagation.Source: SYNDROME MEASUREMENT CIRCUITS · p. 3
  • Weight-four checks use a two-qubit cat state, while weight-eight checks use a four-qubit cat state with post-processing to identify propagated errors.For weight-eight checks, Z measurements detect errors and an X measurement reveals the stabilizer eigenvalue; detected correlated errors receive corrections.Source: SYNDROME MEASUREMENT CIRCUITS · p. 3
  • The weight-eight circuit’s Z measurements reveal weight-two cat-state errors caused by a single preparation error.The circuit diagrams show interactions between ancillary and data qubits.Source: Figure 5 · p. 4
  • On a two-dimensional array, each qubit must interact with either three or five neighbors to execute the specified circuits.Source: Figure 5 · p. 4

NUMERICAL RESULTS

The simulations report thresholds across three noise models, with the circuit-noise threshold compared against the surface code under the same model. They also test a modified identity-gate failure rate and use finite-size scaling to extract thresholds.

NUMERICAL RESULTS

  • Odd code distances 9–21 were simulated, and finite-size scaling fits extracted thresholds from logical-error-rate data.The reported fits had R2 > 0.999; where applicable, the lower of the separate X and Z thresholds sets the overall threshold.Source: NUMERICAL RESULTS · p. 4
  • 0.0760(2) code-capacity and 0.0208(1) phenomenological thresholds were observed for the color code.Code-capacity noise affects data qubits; phenomenological noise also affects syndrome-bit values.Source: NUMERICAL RESULTS · p. 4
  • 0.00143(1) per-gate circuit-noise threshold, versus 0.00672(1) per gate for the surface code under the same noise model.The numerical data for the noise models are shown in Fig. 6.Source: NUMERICAL RESULTS · p. 4, Figure 6 · p. 4
  • 0.00213(1) per-gate threshold was observed when identity-gate failure probability was reduced to p/10.This change reflects the possibility that quantum memories may be more reliable than other gates.Source: NUMERICAL RESULTS · p. 4

SUMMARY AND FURTHER WORK

The reported triangular 4.8.8 color-code thresholds accord with prior results but are significantly lower than surface-code thresholds. The decoding algorithm does not achieve the codes’ full algebraic distance, leaving scope for improvement.

SUMMARY AND FURTHER WORK

  • With syndrome errors, thresholds accord with [20], remain significantly below surface-code thresholds, and exceed the highest reported concatenated-code thresholds for local one- or two-dimensional arrays [44, 45].The authors state that the lower thresholds would largely undermine color codes’ high error-correcting rate and transversal implementation of the entire Clifford group.Source: SUMMARY AND FURTHER WORK · p. 5
  • The decoding algorithm does not achieve the codes’ full algebraic distance because it omits circuit-correlated errors and fails on some configurations, including cases with fewer than (d+1)/2 errors.Addressing these issues is necessary to determine optimal performance; the authors nevertheless describe graph-matching decoding as feasible in this context.Source: SUMMARY AND FURTHER WORK · p. 5
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