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Triangular color codes on trivalent graphs with flag qubits

Christopher Chamberland, Aleksander Kubica, Theodore J. Yoder, Guanyu Zhu

arXiv:1911.00355v3quant-ph

TL;DR

The paper addresses how to extend the triangular color code's competitiveness to circuit-level noise, where stabilizer measurements are part of the challenge. It adapts the Restriction Decoder and uses flag information in a fault-tolerant measurement design. The estimated circuit-level threshold is 0.2%, and the authors prove that 1-flag circuits preserve the full effective distance.

  • Problem

    Extending the color code's error-correction competitiveness to circuit-level noise is a challenge because its stabilizer generators are higher weight and generally require more ancillas.

    Source: §I · p. 1

  • Method

    The paper adapts the Restriction Decoder to the triangular color code and incorporates flag-qubit information into fault-tolerant stabilizer measurement and decoding.

    Source: §I · p. 2

  • Results

    0.2% is the estimated threshold for the triangular color code under the full circuit-level depolarizing noise model.

    Source: §I · p. 2, §IV · p. 13

  • Takeaways & Limitations

    The authors prove that 1-flag stabilizer measurement circuits suffice to preserve the code's full effective distance.

    Source: §I · p. 2

Abstract

from arXiv · show

The color code is a topological quantum error-correcting code supporting a variety of valuable fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the stabilizer measurement circuits, which incorporates flag qubits. We show how information from flag qubits can be used with the Restriction Decoder to maintain the effective distance of the code. We numerically estimate the threshold of the triangular color code to be 0.2%, which is competitive with the thresholds of other topological quantum codes. We also prove that 1-flag stabilizer measurement circuits are sufficient to preserve the full code distance, which may be used to find simpler syndrome extraction circuits of the color code.

I. INTRODUCTION

The paper addresses the challenge of making the color code competitive under circuit-level noise, building on a decoder with good performance but previously limited scope. It adapts decoding for the triangular color code and combines a low-connectivity measurement design with flag information; the estimated circuit-level threshold is 0.2%.

I. INTRODUCTION

  • The color code supports transversal logical Clifford gates, while its higher-weight stabilizers generally require more ancillas than toric-code stabilizers.Prior color-code decoders matched toric-code decoder performance assuming perfect syndrome measurement; extending competitiveness to circuit-level noise motivates this work.Source: §I · p. 1
  • The adapted Restriction Decoder addresses boundaries by finding connected syndrome components before local lifting, unlike a naive boundary adaptation that halves effective distance.For distances 5, 7, and 9, it corrects all errors of weight at most 2, 2, and 3, respectively; the authors expect correction up to about d/3.Source: §I · p. 2
  • The authors place data and ancilla qubits on a hexagonal lattice so stabilizer measurements use neighboring qubits and each qubit has degree-three connectivity.The additional ancillas can serve as flag qubits, whose information helps the decoder maintain the code's full effective distance; 1-flag circuits suffice.Source: §I · p. 2
  • 0.2% is the estimated storage threshold of the triangular color code under circuit-level depolarizing noise, comparable to the analyzed topological codes.Table I compares connectivity, qubit counts, transversal gates, and thresholds.Source: §I · p. 2, Table I · p. 2

A. Triangular color code

  • The triangular color code is defined on a two-dimensional lattice with three boundaries, with vertices incident to three edges except at the corners and faces colored in three colors.A qubit is placed at each vertex, and each face has X- and Z-type stabilizers supported on its qubits.Source: A. Triangular color code · p. 3

B. Color code decoding problem

The section frames decoding X errors as finding a correction with the measured syndrome, then adapts the Restriction Decoder to the triangular code’s three boundaries. Its adaptation pairs restricted syndromes, handles boundary-connected components, and locally lifts the pairings into a correction.

B. Color code decoding problem

  • Decoding X errors means finding an X-type correction whose syndrome matches the measured Z-type syndrome and whose combination with the error is a trivial logical operator.The discussion focuses on X errors because the two-dimensional color code is self-dual, making the X- and Z-correction procedures analogous.Source: B. Color code decoding problem · p. 3

C. Adaptation of the Restriction Decoder

  • For each color pair RG, RB, and GB, the decoder pairs the restricted syndrome, then combines the pairings and locally lifts them to obtain a correction.The lifting chooses incident faces whose local boundary matches the pairing or a connected component’s pairing.Source: C. Adaptation of the Restriction Decoder · p. 4
  • Unlike the original decoder, the boundary-aware adaptation finds syndrome components connected to vR before applying local lifting along their pairings.The original Restriction Decoder is described as lacking this boundary-specific step.Source: C. Adaptation of the Restriction Decoder · pp. 4–5
  • The adapted decoder is expected to correct errors of weight at most ∼d/3, with logical error rate scaling as pL ∼apd/3 below threshold.The leading-order coefficient is reported to be several orders of magnitude smaller than the next-order term.Source: C. Adaptation of the Restriction Decoder · p. 5

D. Incorporating measurement errors

To handle measurement errors, the decoder uses syndrome measurements repeated over time and a space-time matching graph. It pairs highlighted syndrome changes, processes boundary-connected components, then flattens and projects the pairings before local lifting.

D. Incorporating measurement errors

  • When measurements may be faulty, the syndrome is repeated T times, with T comparable to the code distance, and the observations over time inform the correction.The highlighted vertices mark space-time locations where outcomes change between consecutive rounds.Source: D. Incorporating measurement errors · p. 6
  • The space-time matching graph adds diagonal edges to represent space-time-correlated data errors from two-qubit gate failures, with edge weights depending on the circuits and flag outcomes.Flag edges in the matching graph capture possible weight-two data-qubit errors from a single circuit fault; weight-three errors from two faults do not require flag edges.Source: D. Incorporating measurement errors · p. 6
  • The decoder pairs restricted highlighted vertices, identifies components connected to space-time boundaries, then flattens and flag-projects the pairings before local lifting.The resulting adapted Restriction Decoder provides a correction for the triangular color code.Source: D. Incorporating measurement errors · p. 7

III. FAULT-TOLERANT IMPLEMENTATION OF THE TRIANGULAR COLOR CODE

This section presents a low-connectivity stabilizer-measurement implementation for the triangular color code and explains how flag qubits help preserve the Restriction Decoder’s effective distance.

III. FAULT-TOLERANT IMPLEMENTATION OF THE TRIANGULAR COLOR CODE

  • The stabilizer-measurement circuits are chosen to minimize each qubit’s connectivity degree.Source: §III · p. 7
  • Flag qubits help maintain the Restriction Decoder’s full effective distance under circuit-level noise.Source: §III · p. 7
  • The section also describes an alternative flag scheme to the one presented in the preceding sections.Source: §III · p. 7

A. Triangular color code stabilizer measurement circuits

The implementation places data and ancilla qubits on a hexagonal lattice with degree-three connectivity and schedules stabilizer measurements to limit circuit depth.

A. Triangular color code stabilizer measurement circuits

B. Use of flag qubits to correct high weight errors

The section uses flag outcomes to identify potentially high-weight data errors and establishes that 1-flag circuits suffice to preserve full code distance under circuit-level noise.

B. Use of flag qubits to correct high weight errors

C. Constructing the matching graph G

The matching graph adds flag edges to represent possible weight-two data errors, then adjusts edge weights according to flag outcomes and fault probabilities.

C. Constructing the matching graph G

D. Edge weight renormalization

E. Alternative flag scheme

The direct flag method uses flag outcomes to apply preliminary data corrections, while edge-weight renormalization offers an alternative when flag outcomes do not distinguish likely errors. The direct method preserves fault tolerance but can require more flags and may be suboptimal when measurement errors are frequent.

E. Alternative flag scheme

  • Flag outcomes can trigger preliminary Pauli corrections that leave at most one data-qubit error after a single stabilizer-measurement fault.For the weight-six Z stabilizer, the correction depends on the flag outcome; analogous rules apply to X and weight-four stabilizers.Source: E. Alternative flag scheme · p. 12
  • With the direct flag method, one fault leaves data error weight at most one, while compatible two-fault outcomes leave weight at most three and not entirely on a logical operator.The authors state that these properties satisfy the fault-tolerant definitions.Source: E. Alternative flag scheme · p. 12
  • A single flag on a weight-four stabilizer may not distinguish whether to correct with Z1 or Z3, so edge-weight renormalization can preserve effective distance in that case.The scheme renormalizes weights for edges corresponding to Z1, Z3, and Z1Z2 errors.Source: E. Alternative flag scheme · pp. 12–13
  • The direct method generally uses more flags than edge-weight renormalization and can apply unnecessary weight-one corrections when measurement errors are frequent.The numerical results therefore use edge-weight renormalization methods.Source: E. Alternative flag scheme · pp. 12–13

IV. NUMERICAL RESULTS

The numerical results estimate a 0.2% storage threshold under full circuit-level noise, alongside a 12.6% code-capacity threshold. The conclusion describes a degree-three triangular color-code implementation with flag information for preserving effective distance, and identifies further settings for study.

IV. NUMERICAL RESULTS

  • The code-capacity simulations consider distances d = 3 to d = 21, with X and Z errors corrected symmetrically.Each data qubit has X, Y, or Z error probability p/3; measurements, preparation, and gates are assumed perfect.Source: §IV · p. 13
  • 12.6% is the code-capacity threshold, matching the decoder-by-projection threshold; the Restriction Decoder achieves lower logical error rates.The lower logical error rates are attributed to its ability to correct a larger family of errors.Source: §IV · p. 13
  • The full-noise simulations use T = d + 1 rounds, with the last round introducing no errors to ensure projection to the codespace.Flag-edge placement and edge-weight renormalization are described as crucial to the reported logical-error scaling and threshold.Source: §IV · p. 13
  • Using α = 1.5 instead of α = 1 produced no significant difference in the logical-failure curves; a more detailed renormalization study remains future work.The tested edge probability was P′e = p^αmPe, where m counts the flagging stabilizer-measurement circuits.Source: §IV · p. 13
  • 0.2% is the threshold for both logical X and Z failure rates under the full circuit-level noise model.The simulations use the Restriction Decoder, flag methods, and edge-weight renormalization; Fig. 10 shows the logical error rates.Source: §IV · p. 13, Figure 10 · p. 14

V. CONCLUSION

  • The conclusion describes a degree-three layout, flag information that preserves the Restriction Decoder’s full effective distance, and transversal implementation of the full Clifford group.The authors report that the 0.2% circuit-level threshold is competitive with the surface code and other low-degree topological codes.Source: §V · p. 14
  • Future work includes applying flag schemes to other color-code families, studying low-degree layouts and biased noise, and exploring edge-weight renormalization parameters.The suggested families include 4.8.8 color codes and color codes with twist defects.Source: §V · p. 14

Appendix A: A naive adaptation of the Restriction Decoder

The naive triangular-code adaptation pairs restricted syndromes on two lattices and extends local lifting to boundary vertices. Its effective distance is reduced, while the adapted decoder is expected to correct errors of weight roughly d/3.

Appendix A: A naive adaptation of the Restriction Decoder

  • The naive adaptation is only guaranteed to correct errors of weight at most floor((d−3)/4), roughly half the effective distance of the color code.It pairs restricted syndromes on two lattices and allows pairing with boundary vertices before applying local lifting.Source: Appendix A · pp. 14–15
  • The adapted Restriction Decoder is expected to correct all errors of weight at most approximately d/3.The passage presents this as an expectation, not a proved guarantee.Source: Appendix A · p. 15

Appendix B: Edges and edge weight calculations for the space-time matching graphs eGRG, eGRB and eGGB

The appendix explains how circuit faults create correlated space-time edges for matching and how their weights are estimated from leading-order fault events. Diagonal edges capture CNOT-related changes across rounds, whereas measurement errors produce vertical edges.

Appendix B: Edges and edge weight calculations for the space-time matching graphs eGRG, eGRB and eGGB

  • CNOT failures can produce correlated space-time errors whose highlighted-vertex patterns require 3D diagonal edges in the matching graphs.For the described fault patterns, matching can select a diagonal edge that projects onto a 2D data edge after flattening.Source: Appendix B · p. 15
  • Measurement errors produce 3D vertical edges joining the same vertex across syndrome-measurement rounds.Diagonal edges instead connect different locations in the 2D lattice after flattening.Source: Appendix B · p. 15
  • The appendix considers CNOT scheduling at boundary locations as well as single faults projecting onto bulk edges e1 to e6.Edges sharing labels in the illustrated matching graphs have identical edge weights.Source: Appendix B · pp. 15–16
  • Edge weights are computed from fault-event probabilities, with the appendix giving leading-order polynomials based on single-fault events.The weight is defined as we = −log pe; the polynomials include only terms leading in p.Source: Appendix B · pp. 15–16

Appendix C: Proof of Theorem III.1

The proof establishes that 1-flag stabilizer measurement circuits preserve the triangular color code’s full distance. It reduces the fault analysis to a support-counting argument for purely Z-type logical operators, using the code’s structure to bound the required faults.

Appendix C: Proof of Theorem III.1

  • The triangular color code can be decoded with full distance when stabilizers are measured using 1-flag circuits.Source: Appendix C · p. 16
  • It suffices to show that at least d faults are needed to place a nontrivial logical error without any flag firing in one measurement round.Then fault sets of size at most ⌊(d −1)/2⌋ can be distinguished by flags or later syndrome rounds.Source: Appendix C · p. 17
  • By code symmetry, the proof reduces an arbitrary logical Pauli to a purely Z-type operator and lower-bounds the faults needed to place it by d.The proof defines Z(P) by retaining Z wherever P has Z or Y; X(P) = PZ(P).Source: Appendix C · p. 17
  • The proof assigns each face a set of errors, forming an over-partition of the logical operator’s support, and uses a sub-partition to count face sets by size.The support size is at most the sum of the sizes of the sets, with equality when they form a partition.Source: Appendix C · p. 17
  • By definition (Ref. [40]), if a 1-flag syndrome-extraction circuit results in two or more data errors, either multiple faults occurred in the circuit or a flag has flagged.Source: Appendix C · p. 17
  • A repeated lemma-based transformation removes size-three face sets while retaining a logical operator, yielding the bound needed to complete the distance proof.The construction uses complements on faces and distinguishes square from hexagonal faces when bounding support.Source: Appendix C · p. 20

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