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Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code

C. Ryan-Anderson, N. C. Brown, M. S. Allman, B. Arkin, G. Asa-Attuah, C. Baldwin, J. Berg, J. G. Bohnet, S. Braxton, N. Burdick, J. P. Campora, A. Chernoguzov, J. Esposito, B. Evans, D. Francois, J. P. Gaebler, T. M. Gatterman, J. Gerber, K. Gilmore, D. Gresh, A. Hall, A. Hankin, J. Hostetter, D. Lucchetti, K. Mayer, J. Myers, B. Neyenhuis, J. Santiago, J. Sedlacek, T. Skripka, A. Slattery, R. P. Stutz, J. Tait, R. Tobey, G. Vittorini, J. Walker, D. Hayes

arXiv:2208.01863v1quant-ph

TL;DR

The paper compares fault-tolerant logical entangling gates implemented with the five-qubit and color codes, addressing how different code and protocol choices perform in experiments. It implements the gates on trapped-ion processors with real-time classical co-processing and finds that the color-code logical sequence has higher average fidelity bounds than its physical-level analogue. The comparison is limited to the protocols and codes studied.

  • Problem

    The study compares how different quantum error-correction codes and implementations of fault-tolerant logical operations perform in similar experimental environments.

    Source: §V · p. 9

  • Method

    The experiments implement a pieceable non-transversal CNOT on two five-qubit codes and a transversal CNOT on two color codes, using real-time classical co-processing for fault-tolerant protocols.

    Source: §II · p. 3, §III · p. 6, §V · p. 9

  • Results

    The color-code state-preparation, transversal CNOT, and FT measure-out sequence had average fidelity bounds of [0.9957, 0.9963], versus [0.9850, 0.9903] for the analogous physical-level operations.

    Source: §III · p. 8

  • Takeaways & Limitations

    The results suggest that the relatively economical fault-tolerant circuitry of the color code may provide a better platform for computation than the qubit-efficient five-qubit code.

    Source: §V · p. 9

    The comparison is limited to the specific protocols and codes studied; other fault-tolerant protocols are left for future work.

    Source: §V · p. 9

Abstract

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We compare two different implementations of fault-tolerant entangling gates on logical qubits. In one instance, a twelve-qubit trapped-ion quantum computer is used to implement a non-transversal logical CNOT gate between two five qubit codes. The operation is evaluated with varying degrees of fault tolerance, which are provided by including quantum error correction circuit primitives known as flagging and pieceable fault tolerance. In the second instance, a twenty-qubit trapped-ion quantum computer is used to implement a transversal logical CNOT gate on two [[7,1,3]] color codes. The two codes were implemented on different but similar devices, and in both instances, all of the quantum error correction primitives, including the determination of corrections via decoding, are implemented during runtime using a classical compute environment that is tightly integrated with the quantum processor. For different combinations of the primitives, logical state fidelity measurements are made after applying the gate to different input states, providing bounds on the process fidelity. We find the highest fidelity operations with the color code, with the fault-tolerant SPAM operation achieving fidelities of 0.99939(15) and 0.99959(13) when preparing eigenstates of the logical X and Z operators, which is higher than the average physical qubit SPAM fidelities of 0.9968(2) and 0.9970(1) for the physical X and Z bases, respectively. When combined with a logical transversal CNOT gate, we find the color code to perform the sequence--state preparation, CNOT, measure out--with an average fidelity bounded by [0.9957,0.9963]. The logical fidelity bounds are higher than the analogous physical-level fidelity bounds, which we find to be [0.9850,0.9903], reflecting multiple physical noise sources such as SPAM errors for two qubits, several single-qubit gates, a two-qubit gate and some amount of memory error.

I. THE QUANTUM COMPUTERS

The five-qubit and color-code experiments ran on similar trapped-ion QCCD systems with different capacities and upgrades. A classical co-processor ran look-up-table decoders, with their outputs controlling quantum-program operations at runtime.

I. THE QUANTUM COMPUTERS

  • Both systems used surface-electrode traps, 171Yb+ qubits, reconfigurable QCCD registers, and ion transport for all-to-all connectivity.Each qubit ion was accompanied by a 138Ba+ ion for sympathetic cooling; gates and SPAM could run in parallel in dedicated zones.Source: §I · p. 2
  • H1-2 used 12 physical qubits and three active gate zones for two five-qubit code qubits; H1-1 supported up to 20 physical qubits and five active gate zones for color-code experiments.H1-1 also used spatial phase tracking to compensate for phase errors arising from position-dependent qubit frequencies.Source: §I · p. 2
  • The classical co-processor ran look-up-table decoders, and their returned data determined subsequent quantum-program control flow and operations.The decoders were partly written in Rust and compiled to WebAssembly, which could be called from QASM through a foreign-function interface.Source: §I · p. 3
  • Table I summarizes typical single- and two-qubit gate, SPAM, and mid-circuit measurement/reset crosstalk errors for H1-1 and H1-2.Single- and two-qubit gate error rates were measured using randomized benchmarking.Source: Table I · p. 3
  • WebAssembly functions could use high-level language features, subject to integer-only data passed through the interface and the control system’s execution-time limit.The text describes this support as allowing decoders to be implemented in languages such as Rust, C, and C++.Source: §I · p. 3

II. THE FIVE-QUBIT CODE

The five-qubit code’s non-transversal CNOT is implemented as a pieceable gate, with an intermediate error-correction cycle between gate pieces. Across the tested five-qubit constructions, the added circuitry intended to increase fault tolerance coincided with lower fidelity, while the non-fault-tolerant sequence had the highest reported fidelity.

II. THE FIVE-QUBIT CODE

  • Pieceable fault tolerance divides the non-transversal logical gate into individually fault-tolerant pieces, with QEC between pieces to limit fault spread.The five-qubit implementation adapts a round-robin construction to a logical CNOT using nine physical CNOT gates.Source: §II · p. 3
  • Fault-tolerant initialization began with a non-FT encoding to logical |−⟩L states, followed by verification and post-selection on successful preparations.The procedure was run once rather than as a repeat-until-success circuit; the reported five-qubit success probability was 0.909(1)%.Source: §II · p. 3
  • After six physical CNOTs mix the code-block stabilizers, an intermediate QEC cycle measures the resulting higher-weight stabilizers to maintain fault tolerance.Flagged measurements account for higher-weight hook errors; a triggered flag or non-trivial syndrome dynamically switches to unflagged measurements, whose syndromes are decoded to update the Pauli frame.Source: §II · pp. 3–4, Figure 2 · p. 4
  • FT SPAM fidelities were 0.967(4) and 0.952(5) for logical X and Z, below non-FT SPAM fidelities of 0.978(2) and 0.976(2), respectively.The authors attribute the difference to the FT SPAM circuit’s 30–40 two-qubit gates per code block, compared with five for non-FT SPAM.Source: §II · p. 4
  • The circuit layout holds encoding, local Clifford rotations, and both logical-gate pieces fixed while varying initialization, the two-qubit QEC cycle, and measure-out.The measure-out procedure rotates back to the canonical code and then uses a multi-step fault-tolerant measurement.Source: §II · p. 4, Figure 3 · p. 5
  • The five-qubit CNOT comparison tested three constructions: non-FT SPAM without QEC, FT SPAM with one unflagged syndrome round, and FT SPAM with a fully FT QEC cycle.The gate was evaluated on an informationally incomplete set of XX, ZZ, and XZ input states, yielding rigorous fidelity bounds rather than a process-fidelity estimate.Source: §II · pp. 4–6
  • Fidelity bounds centered around 0.926 made non-FT CNOT1f the highest-fidelity five-qubit state-preparation, CNOT, and measurement sequence; added FT circuitry was associated with lower fidelity.The paper reports fidelity negatively correlated with CNOT count and suggests the result may depend on its noise environment.Source: Figure 4 · p. 6, §II · p. 6

III. THE COLOR CODE

The color code’s transversal CNOT is naturally fault-tolerant, while added fault-tolerant SPAM circuitry improves preparation and measurement. The best tested preparation–CNOT–measurement sequence has higher average fidelity bounds logically than the analogous physical-level sequence.

III. THE COLOR CODE

  • The color code admits a transversal CNOT gate, whose natural fault tolerance prevents direct comparison with different instantiations of the five-qubit-code gate.Experiments used a single initialization round and were post-selected based on validation; successful FT initialization verification for the color code was 0.9834(3).Source: §III · p. 6
  • 0.99939(15) and 0.99959(13) are the fault-tolerant SPAM fidelities for logical X- and Z-eigenstates, respectively.The SPAM2c experiment adds the fault-tolerant check circuit.Source: §III · p. 6
  • Adding fault-tolerant circuitry to SPAM lowers error rates by roughly an order of magnitude, while adding QEC cycles for comparison increases the overall circuit error rate.The transversal logical CNOT is itself fault-tolerant, so the added SPAM circuitry is the additional circuitry required for end-to-end fault tolerance.Source: §III · p. 7
  • CNOT1c gives the highest fidelity among the tested CNOT sequences, despite having no QEC cycles, because its transversal gate lies between fault-tolerant initialization and measurement.Product-state outputs had higher fidelities than entangled-state outputs.Source: §III · p. 7
  • [0.9957, 0.9963] logical average fidelity bounds exceed the analogous physical-level bounds of [0.9850, 0.9903] for preparation–CNOT–measurement.These bounds include the logical SPAM contribution.Source: §III · p. 8, Table III · p. 8
  • The logical sequence’s higher fidelity does not establish arbitrary-algorithm superiority or break-even; QEC cycles and more careful measurements remain important for a fair comparison.The logical CNOT bounds are not SPAM-corrected because measurement-dominated SPAM is less justified at the logical level.Source: §III · p. 8

IV. SIMULATIONS

The simulations use characterized physical error sources and a fitted dephasing parameter to compare code behavior across error regimes. They indicate that the five-qubit code remains above physical error rates in the probed range, while a color-code CNOT with added fault-tolerant QEC could benefit at lower rates.

IV. SIMULATIONS

  • The emulator parameterizes gate, SPAM, crosstalk, and dephasing errors; its constant dephasing rate is a fitting parameter because spatially varying frequency errors and other noise details are not fully captured.The model uses coherent dephasing as a conservative estimate and fits effective rates to measured data.Source: §IV · p. 8
  • At physical-error scaling factors as low as 10^-3, corresponding to a two-qubit gate error of about 2 × 10^-6, five-qubit-code logical error rates still exceed physical error rates.The logical error scales nearly linearly over the probed range, indicating operation far from the pseudo-threshold.Source: §IV · p. 9
  • A color-code CNOT with an added fault-tolerant QEC cycle is predicted to outperform the standalone gate only at lower physical error rates than currently achieved; adding non-fault-tolerant QEC performed worse in the probed regimes.The study evaluates gating operations; the authors note that protocol, transport, and circuit optimization may remain possible.Source: §IV · p. 9

V. DISCUSSION AND CONCLUSION

The comparison favors the color code’s more economical fault-tolerant circuitry, while indicating a logical two-qubit-gate break-even milestone. The conclusions are limited to the protocols and codes studied.

V. DISCUSSION AND CONCLUSION

  • The color code’s relatively economical fault-tolerant circuitry suggests it may provide a better computational platform than the qubit-efficient five-qubit code.The authors caution that this conclusion is limited to the specific protocols studied.Source: §V · p. 9
  • The color code’s fault-tolerant SPAM circuit has an order of magnitude smaller error than its non-fault-tolerant encoding circuit, while the five-qubit code’s fault-tolerant SPAM performs worse.The codes perform similarly in the SPAM1f and SPAM1c experiments, which consist only of encoding circuits followed by measurements.Source: §V · p. 9
  • The authors caution that the protocol comparison is limited to these codes, although pieceable fault tolerance is a general concept with applications in other codes.They describe the results as one in a series of milestones toward beneficial logical quantum computation.Source: §V · pp. 9–10
  • The color-code transversal CNOT experiment found higher average fidelity bounds than the analogous physical-level experiment, indicating a logical two-qubit-gate break-even milestone.The authors attribute most of the increase to higher-fidelity logical SPAM; even after correcting for SPAM, the logical CNOT is higher fidelity than the physical operation.Source: §V · p. 9
  • Adding QEC cycles still yields error of approximately 1%, a further milestone to achieve before logical algorithms can be expected to outperform unencoded computations.The authors identify SPAM-independent measurement of logical gate fidelity as an important next step.Source: §V · p. 9
  • QCCD qubit mobility separates circuit geometry from hardware geometry, leaving open whether higher-dimensional codes could offer benefits over two-dimensional codes.The authors mention low-density parity-check codes as one example of recent progress toward lower physical resource overheads.Source: §V · p. 10

Appendix A: Process fidelity bounds from semidefinite programming

The appendix bounds process fidelity against an ideal unitary using state-fidelity measurements and semidefinite optimization. It also describes how physical-level measurement errors are incorporated into those bounds.

Appendix A: Process fidelity bounds from semidefinite programming

  • The procedure bounds quantum process fidelity relative to an ideal target unitary using state fidelities measured after applying the process to selected input states.The appendix describes the procedure as originally presented in Ref..Source: Appendix A · p. 11
  • A process is represented by its Choi matrix, defined by applying the process to one half of a maximally entangled state.The appendix states that complete positivity corresponds to a positive-semidefinite Choi matrix, while trace preservation imposes a partial-trace constraint.Source: Appendix A · p. 11
  • The minimum and maximum process fidelities consistent with the measured state-fidelity constraints are obtained by solving analogous semidefinite programs.The constraints use the selected input states and enforce the partial-trace condition for the Choi matrix.Source: Appendix A · p. 12
  • The experiment uses X-basis, Z-basis, and Bell-state inputs; the X- and Z-basis output fidelities are probabilities of measuring the correct state.Bell-state fidelities are obtained by measuring the logical qubits in the XX, YY, and ZZ bases.Source: Appendix A · p. 12
  • Physical-gate fidelity bounds account for measurement errors by estimating POVM elements from physical SPAM experiments while assuming perfect state preparation.The appendix reports average-fidelity bounds and describes noisy measurement operators used in the optimization.Source: Appendix A · pp. 11–12

Appendix B: Additional five-qubit code details

The appendix outlines how recovery operations are constructed for the five-qubit code and how the round-robin fault-tolerant gate changes the operators tracked during recovery.

Appendix B: Additional five-qubit code details

  • The appendix provides the five-qubit code’s logical, stabilizer, and destabilizer generators and refers to Fig. 2 for the stabilizers and their transformations under the pieceable round-robin gate.The supplied passage identifies these generator sets but does not enumerate their operators.Source: Appendix B · p. 12
  • Recovery operations for the five-qubit code are constructed from logical operators, stabilizers, and destabilizers.The appendix states that a Pauli recovery can be decomposed as an element of L × S × D.Source: Appendix B · p. 12
  • During the fault-tolerant round-robin construction, transformed destabilizers and the Pauli frame must be tracked to determine recovery and update later syndrome measurements and logical representatives.Source: Appendix B · p. 13

Appendix C: Dephasing simulations

The appendix calibrates a global coherent-dephasing parameter from experimental data, then examines how scaled dephasing affects simulated logical-circuit performance. The simulations show no significant fault-tolerance advantage overall, though reduced dephasing may change comparisons between circuit constructions.

Appendix C: Dephasing simulations

  • Using experimental data to tune a parameterized global coherent dephasing model by minimizing average experimental–simulation discrepancy yielded effective rates of 0.236 Hz for five-qubit code experiments on H1-2 and 0.172 Hz for color-code experiments on H1-1.Source: Appendix C · p. 13
  • Simulations show no significant fault-tolerance advantage overall, although reduced dephasing may let CNOT3f outperform CNOT2f; tighter bounds are needed to establish this, and CNOT2c is also reported to outperform a single-syndrome-round non-FT circuit.The passage repeats CNOT2c for both circuits in that final comparison.Source: Appendix C · p. 13
  • Deeper color-code circuits perform worse in the Z basis, while CNOT1c shows no significant difference between X- and Z-basis performance; five-qubit circuits perform similarly across bases overall.The passage attributes color-code Z-basis sensitivity to the weight-three logical Z operator and suggests depth contributes to CNOT1c's similar performance across bases.Source: Appendix C · p. 13
  • Figures 9–11 show simulated logical-circuit results as dephasing varies, including X- and Z-basis inputs and, in Figure 9, Bell-preparation inputs.The remaining error budget is held fixed during the scaling shown in Figures 9–11.Source: Figure 9 · p. 14, Figure 10 · p. 14, Figure 11 · p. 14

Appendix D: Additional circuit diagrams

The appendix provides circuit diagrams for five-qubit-code experiments. They document FT syndrome extraction and adaptive measure-out, alongside non-FT SPAM and round-robin gate circuits.

Appendix D: Additional circuit diagrams

  • The full FT QEC cycle measures flagged checks first and dynamically adds unflagged checks if a syndrome or flag is non-trivial; the non-FT cycle measures only unflagged checks.The resulting flagged and unflagged measurements provide enough syndrome information for FT decoding.Source: Figure 12 · p. 15
  • The FT measure-out scheme conditionally checks three flagged logical representatives, then uses decoding and a Pauli-frame update when needed to infer the corrected logical measurement.The representatives are related by stabilizer multiplication; agreement among their measurements determines when the scheme can return a result without decoding.Source: Figure 13 · p. 16
  • The non-FT SPAM circuit selects preparation and measurement bases with variable unitaries and measures raw logical output on qubits 1, 3, and 5..Source: Figure 14 · p. 17
  • The non-FT round-robin gate circuit uses unprimed variable unitaries to choose input states and primed unitaries to choose the measurement basis.The appendix presents these diagrams as additional circuitry used in the five-qubit-code experiments.Source: Appendix D · p. 15, Figure 15 · p. 17

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