Bias-preserving and error-detectable entangling operations in a superconducting dual-rail system

I. INTRODUCTION

• Bias-aware encodings can raise surface-code gate-error thresholds from 1% to more than 5%, while erasures can be corrected more readily than Pauli errors.The cited surface-code erasure threshold is almost 25% per round, and a given code distance can correct twice as many erasures as Pauli errors.Source: §I · p. 1
• Dual-rail cavity qubits have an idle-error hierarchy dominated by erasures, with dephasing 5–10 times lower and bit flips several orders of magnitude rarer.Source: §I · p. 1
• An entangling gate that preserves this favorable hierarchy was a missing ingredient for superconducting dual-rail cavity qubits.Source: §I · p. 1
• The SWAP-Wait-SWAP gate temporarily swaps a control-cavity excitation into a transmon coupler, uses its dispersive shift with the target, then swaps the excitation back for a roughly 500 ns CZ.Source: §I · p. 2
• Below 1% erasure, 0.1% or lower post-detection dephasing, and fewer than a few parts per million bit flips accompany the gate; target erasure and dephasing rates are 3–4 times lower.The asymmetry is between target and control qubits.Source: §I · p. 2

Gate construction

• The SWS gate swaps the control excitation into a coupler, accumulates a dispersive phase with the target, then swaps the excitation back.The gate takes about 500 ns and preserves total excitation number, keeping both dual-rails in the single-excitation manifold with the coupler included in the control rail during the gate.Source: §II · pp. 2–3, Figure 1 · p. 2

Gate benchmarking

• 0.53(2)% total erasure probability per CZ gate is extracted from the repeated-gate postselected fraction.A separate interleaved-RB estimate reports a 0.69% erasure probability per gate, which limits usable sequence depth to a few hundred gates.Source: §II · p. 4
• 0.1% error per CZ gate is estimated by interleaved randomized benchmarking, in reasonable agreement with repeated Bell-state tomography.The tomography estimate is 0.03% postselected infidelity per gate through 15 repeats; at larger depths, fidelity and purity decrease quadratically, possibly due to calibration drift or coupler-frequency fluctuations.Source: §II · p. 4, Figure 2 · p. 4

Error structure

• 0.400(4)% control and 0.096(4)% target erasures per CZ gate yield an erasure asymmetry of about four.The measured asymmetry is consistent with the relative T1 of the coupler and the target cavities.Source: §II · pp. 4–5
• 0.039(1)% control and 0.0112(9)% target Z errors per gate give a dephasing ratio of about 3.5.The proposed model attributes control dephasing to coupler Tϕ during the swap and target dephasing to dual-rail Tϕ while idling.Source: §II · pp. 4–5
• Bit-flip errors are bounded at a few parts per million per gate, while leakage to vacuum either leaves the target near identity or induces correlated target dephasing.For control preparations |00⟩ and |10⟩, the target process remains close to identity; loss from |01⟩ during the gate instead gives an incomplete, randomly timed CPHASE and a dephasing-like target channel.Source: §II · pp. 5–6

• Leakage before the CZ interaction switches the gate off, while leakage during it can produce either an identity process or a CZ error.The conditioned target process is close to identity for control preparations |00⟩ and |10⟩, and resembles a CZ error for |01⟩.Source: Figure 4 · p. 7
• The excitation-preserving SWS gate retains erasure-dominated errors, sub-microsecond timing, and strong suppression of bit flips.The dominant errors are erasures at approximately 0.5%, while bit flips are bounded below a few parts per million per gate.Source: §III · pp. 6–7
• The added entangling operation completes the universal gate set for dual-rail cavity qubits.The paper states this can enable high-performance short-depth circuits and new error-mitigation approaches.Source: §III · p. 7
• Erasure and dephasing rates are each reported at tenfold below their separate surface-code thresholds, though combined-error performance requires detailed numerical simulation.The authors describe the results as a realistic route to gains of 10 or more with increasing code distance.Source: §III · p. 7

• R.J.S. is a founder and shareholder of Quantum Circuits, Inc., and QCI-affiliated authors have a financial interest in the company.Source: §VIII · p. 8

• The supplied heading identifies this material as supplemental information for the paper.Source: Supplemental information for… · p. 11

1. 1Q gate characterization

• Single-qubit gates use parametric beamsplitter interactions, and randomized benchmarking samples 20 sequences per Clifford length.The benchmarking gateset generates the 24 single-qubit Cliffords.Source: §1 · p. 11
• Individual gate fidelity exceeds 99.98%; simultaneous operation decreases control and target fidelities by 0.001% and 0.002%, respectively.Source: §1 · p. 11, Figure 5 · p. 11

2. SPAM

• Repeated dual-rail measurements can exponentially suppress misassignments, at the cost of a first-order decrease in postselected fraction.The measurement detects photon loss and can flag errors arising during measurement.Source: §2 · pp. 11–12
• SPAM misassignment is 0.023% for target and 0.020% for control; SPAM erasure is 6.79% for target and 6.43% for control.These are the reported logical misassignment and overall SPAM erasure errors.Source: §2 · p. 12

Appendix B: Two-qubit gate calibration

• Swap amplitude is optimized for g_ac/κϕ, then pulse duration is set by minimizing cavity population after odd-numbered swaps.The calibration also selects the swap-back phase that minimizes erasure for the |01g10⟩ initial state.Source: Appendix B · p. 13
• The wait time is calibrated for entangling phase π, while Ramsey-like sequences measure the accumulated local Z phases.The swap-back phase is recalibrated as delay time changes.Source: Appendix B · p. 13

Appendix C: CZ gate on-off ratio

• χbc/2π is ∼1.51 MHz, χab/2π is ∼6.64 kHz, and the resulting CZ-gate on-off ratio is ∼230.The two interactions are inferred from target-cavity spectroscopy and Ramsey measurements, respectively.Source: Appendix C · pp. 13–14, Figure 7 · p. 15

Appendix D: Inferring gate fidelities

• The short-depth interleaved-RB analysis extracts the depolarizing parameter from a linear fit to survival-probability decay, with the estimate depending on fit amplitude and offset.The gate-error estimate is based on interleaved and reference RB slopes.Source: Appendix D · pp. 14–15

Appendix E: Evaluating the accuracy of IRB with postselection

• The postselected-fidelity model includes leakage and dephasing channels and renormalizes each input-state fidelity contribution by its postselected fraction.The model accounts for leakage-detection misassignments and compares simulated IRB with the modeled postselected fidelity.Source: Appendix E · pp. 15–16
• IRB systematically underestimates infidelity in the simulations, by 25% at the measured value, changing it from 0.1% to 0.13%.The underestimate persists with perfect simulated leakage detection and varies with the random gate sequence.Source: Figure 8 · p. 15, Appendix E · p. 16

1. Leakage before a CZ gate

• Leakage before the gate leaves the other qubit unchanged because the coupler-target interaction is not activated with only one excitation between the qubits.The leakage-propagation analysis considers leakage before and during the CZ gate.Source: Appendix F · p. 16, §1 · p. 17

2. Leakage during a CZ gate

• Leakage during the gate is modeled as a random partial entangling phase followed by leakage; after stabilizer measurement, preserving correlated error terms avoids bad hook errors.The effective channel corresponds to a 50-50 mixture of leakage before and after the gate.Source: §2 · p. 17

3. Analysis of experimental results

• The process-tomography analysis uses the χ-error matrix to quantify deviations from a reference process, with the ideal error process having only χI,I = 1 nonzero.The error process is represented in the Pauli basis.Source: §3 · p. 18

Appendix G: No-jump backaction in the SWS gate

• A single mid-sequence X pulse on the control cancels no-jump polarization, while interleaved rotations and measurements keep its effective error below 10^-5.No-jump backaction acts like a gradual Z-basis measurement favoring the less lossy |10⟩ state.Source: Appendix G · pp. 18–19

Appendix H: CZ induced coupler heating

• Unpostselected measurements find residual coupler population rising by 0.014% per CZ gate, whereas postselection on the control codespace finds no measurable increase after 64 gates.The excitation-preserving gate makes residual coupler excitation detectable through the control qubit’s vacuum outcome.Source: Appendix H · p. 19

Appendix I: Correlated leakage and coupler excitation in the SWS gate

• Target-photon loss midway through an SWS gate can leave both qubits in vacuum and an excitation in the coupler, producing a correlated double erasure.The mechanism is most pertinent when gac ∼ χbc, where the control photon may not fully swap into or back from the coupler.Source: Appendix I · p. 19
• The predicted double-erasure probability is 1.3 × 10^-5 per gate, while coupler dephasing is predicted to leak to detectable states with probability 6.5 × 10^-5 per gate.Both effects are suppressed quadratically with |χbc|/gac; the predicted total probability of ending with an excited coupler is 8.0 × 10^-5 per gate.Source: Appendix I · p. 20

Appendix J: Measuring bit-flip errors

• With the exception of the state that swaps an excitation into the coupler, measured bit-flip rates are indistinguishable from idling rates.The extracted rates may be apparent rates because finite leakage-discrimination fidelity combines with increasing erasures over time.Source: Appendix J · p. 20, Figure 10 · p. 20

Appendix K: Derivation of gate parameters

• The derived parameters make the gate locally equivalent to CZ on the dual-rail subspace, up to local Z rotations, under square, instantaneous-pulse assumptions.In the experiment, finite pulse ramps and terms such as cavity-cavity cross-Kerr make the calibrated values differ from theory.Source: Appendix K · pp. 20–21
• Fixing tSWAP = π/gac leaves twait and the second-SWAP phase ϕ as the parameters to derive and calibrate.For a given hardware parameter set, multiple pulse-parameter sets can enact CZ; this choice makes the experimental sweep involve a delay and a phase.Source: Appendix K · pp. 20–21

1. Deriving twait

• The wait time follows by summing the entangling phase from the wait and both SWAP segments, with the total phase set to π for CZ.The wait contributes ϕwait = χtwait; each SWAP segment also contributes phase because the detuned-Rabi trajectory is only partially completed.Source: §1 · p. 21

2. Deriving ϕSWAP

• The second-SWAP phase is derived geometrically from the XY-plane trajectory, relating ϕ3 to ϕSWAP and using ϕ1 + 2ϕ2 + ϕ3 = 2π.The trajectory angles are illustrated in Fig. 11.Source: §2 · pp. 21–22, Figure 11 · p. 22

Appendix L: QuTiP Master equation simulations and CZ gate error budget

• The simulations evolve all five modes under the three piecewise Hamiltonians, including measured-coherence collapse operators, white-noise dephasing, and excitation loss.They also capture no-jump backaction, expected Z-frame updates, and correlated errors.Source: Appendix L · p. 22
• The derived gate-parameter expressions are exact, and the simulated |0L1L⟩ state returns to its initial population with an entangling phase of π and a deterministic control Z-phase.The |1L0L⟩ and |1L1L⟩ states undergo identity evolution in the simulation; Fig. 12 depicts the basis-state trajectories.Source: Figure 12 · p. 22, Appendix L · p. 23
• The simulated error budget estimates probabilities for CZ error channels that are too small to measure directly, using measured hardware coherence times.The comparison uses the longer echo times as dephasing times; Table V gives the estimated single-gate budget and compares it with short-sequence measurements.Source: Appendix L · p. 23, Table V · p. 23

Appendix M: Fundamental limits to SWS gate performance

• The limit analysis assumes only the coupler has intrinsic loss and dephasing, with cavity modes otherwise taken to have infinite T1 and Tϕ.In this scenario, dispersively coupled cavities inherit coupler loss and dephasing through the inverse Purcell effect.Source: Appendix M · p. 23
• The estimated SWS error limits use gate duration τ = π/|χbc| and assume white-noise dephasing with hybridization G typically between 0.01 and 0.05.The analysis also assumes G is approximately equal to Ga and Gb and omits order-unity prefactors.Source: Appendix M · pp. 23–24
• Target-qubit Z errors may be several orders of magnitude below the transmon decoherence limit, while the maximum bias is set by 1/G^2.The passage describes this regime as promising for flag-qubit or stabilizer-measurement circuits.Source: Appendix M · p. 24
• Reducing extrinsic cavity dephasing, including dephasing inherited from ancilla transmons, is expected to further improve target-qubit Z errors.The analysis identifies these errors as a potential improvement target rather than a fixed limit.Source: Appendix M · p. 24