Self-Consistent Calibration of Quantum Gate Sets

• GSC fills a middle ground between RB and GST by distinguishing coherent errors with fewer resources than full process reconstruction.RB combines coherent and incoherent errors into one rate, whereas GST requires more sequences and a more involved fit.Source: §I · p. 1
• GSC constructs short sequences that extract all linearly independent coherent error generators, using fewer measurements than GST while providing more detail than RB.It can also be made robust to SPAM errors and moderately affected by decoherence.Source: §I · p. 1
• Automated tuning feeds extracted gate errors into an iterative solver, making moderate measurement requirements advantageous for scaling and handling cross-talk.Automatic tuning is presented as necessary for non-orthogonal parameters and multi-qubit cross-talk.Source: §I · p. 1
• 230 gate parameters can be calibrated in about 10 iterations, reaching fidelities beyond 99.6% including decoherence and leakage from initial CNOT systematic infidelities as large as 20%.The simulated gate set is predicted to be decoherence-limited near 99.8% fidelity.Source: §I · p. 2

II. CONSTRUCTION OF GATE SEQUENCES

• The algorithm enumerates gate sequences up to a chosen length, combining available gates into sequences whose responses are measured across initial states and observables.Each sequence is a time-ordered list of gates, and all state–sequence–measurement responses are collected experimentally.Source: §II · p. 2
• Each sequence is modeled with coherent error parameters in a Pauli basis, yielding one response vector for every sequence, initial state, and measurement combination.The parameter count is np = ng(d^2 − 1).Source: §II · p. 2
• The construction assumes independently applied gates, negligible simultaneous-gate cross-talk and concatenation errors, and sufficiently small gate errors for the linearized model.These assumptions define the regime in which the selected sequences provide first-order error extraction.Source: §II · p. 2
• If the response matrix is singular, additional gates, initial states, or measurements are required to make the error parameters identifiable.Invertibility depends on the available experimental resources and the condition number of the selected matrix.Source: §II · p. 2
• Rank-revealing QR decomposition selects np linearly independent, well-conditioned sequences so their responses can identify all coherent errors to first order.Rows with the largest diagonal factors are retained to improve extraction quality.Source: §II · pp. 2–3

III. TUNING ALGORITHM

• The solver iteratively minimizes deviations between measured and ideal sequence responses to adjust experimental control parameters.The control-to-error transformation is left unspecified, apart from assuming that some control setting removes coherent errors.Source: §III · p. 3
• The method assumes ideal reference responses at the target setting, while realistic SPAM and non-unitary dynamics can make measured responses differ from that ideal.These effects are treated as potential offsets in the calibration model.Source: §III · p. 3
• The procedure requires initialization sufficiently close to the desired solution because multiple solutions can exist.The paper ties this requirement to appropriate pre-characterization of the qubit system.Source: §III · p. 3
• Gradient-based Levenberg–Marquardt optimization is used in simulations because it handles vector-valued objectives, with finite-difference gradients feasible experimentally.Alternative Jacobian-estimation methods may improve efficiency but are outside the paper’s scope.Source: §III · p. 3

IV. CALIBRATION ERRORS

• Decoherence, leakage, SPAM errors, and faulty measurement calibration can add offsets to sequence measurements beyond the coherent gate errors targeted by GSC.GSC calibrates coherent gate errors self-consistently, but these additional effects alter the measured responses.Source: §IV · p. 3
• Modeling distortions as visibility factors and offsets means convergence can null the measured residual without guaranteeing the ideal control parameters are reached.The converged setting need not equal q^(0).Source: §IV · p. 3
• When decoherence is sufficiently low, GSC-tuned gate infidelity is not limited by remaining coherent errors.The section also presents sequences designed to be robust to offsets and coherent SPAM errors.Source: §IV · p. 3

A. Stochastic errors

• The analysis models stochastic errors with generalized Pauli channels and also finds linear-in-time diamond-distance scaling for single-qubit energy relaxation.The relaxation result supports using the same scaling assumption for the stochastic-error discussion.Source: A. Stochastic errors · pp. 4–5
• Residual coherent gate infidelity is bounded quadratically by the norm of sequence infidelities from stochastic errors.The bound is favorable when the sequence-infidelity norm is below one.Source: A. Stochastic errors · p. 4
• Residual coherent diamond distance is bounded linearly by the norm of sequence diamond distances from stochastic errors.For both metrics, residual coherent error is bounded by the total stochastic-error contribution, although infidelity has more favorable scaling.Source: A. Stochastic errors · p. 4
• For Markovian noise, sequence decoherence metrics are approximately l(r) times typical gate metrics, whereas correlated constructive interference can produce l(r)^2 scaling.The sequence metrics include gate errors and SPAM errors.Source: A. Stochastic errors · p. 4
• Longer repeated-gate sequences can reduce sensitivity to SPAM errors and increase readout signal, but they help only when decoherence grows sufficiently slowly.Such sequences can reduce the number of measurements without increasing the stated fidelity bounds.Source: A. Stochastic errors · pp. 4–5

B. Faulty readout calibration and coherent SPAM errors

• Coherent SPAM errors are represented as additional initialization and measurement gates, allowing them to be treated computationally like ordinary gate errors.Initialization errors are overparameterized across possible initial states.Source: B. Faulty readout calibration and… · p. 5
• Some coherent SPAM errors enter measurement deviations at first order, so they can remain relevant after calibration unless sequence design suppresses them.The protocol is first-order sensitive to gate errors, ensuring that not all SPAM terms can be eliminated automatically.Source: B. Faulty readout calibration and… · p. 5
• Sequence selection must retain full rank for gate errors while enforcing first-order insensitivity to selected SPAM parameters.The resulting accuracy improvement depends on which errors are present in the gate set.Source: B. Faulty readout calibration and… · p. 5
• Complementary sequence pairs cancel additive readout offsets when their sensitivity-matrix difference is invertible with a good condition number.At least n_m additional sequences may be needed, though one complementary sequence can pair with multiple primary sequences.Source: B. Faulty readout calibration and… · p. 5

C. Cross-talk

• Parallel single-qubit gate combinations can be modeled as two-qubit gates with additional coherent error parameters.Restricting allowed pairs can reduce the number of added parameters, but each pair introduces new systematic errors.Source: C. Cross-talk · p. 5
• Reducing cross-talk parameterization generally requires more calibration sequences when additional systematic errors are introduced.Source: C. Cross-talk · p. 5
• Unmodeled or unaddressable cross-talk produces measurement inconsistencies whose magnitude is set by the cross-talk strength.The resulting measurement errors can be propagated into fidelity and diamond-distance bounds.Source: C. Cross-talk · p. 5

V. TWO-QUBIT GATE SET

• The construction represents coherent errors with Pauli-product generators and measures P3⊗P0 or P0⊗P3 from the initial state |00⟩.The corresponding gate, initialization, and measurement error strengths are parameterized separately.Source: §V · p. 6
• The two-qubit sequences have |S^-1| between 5.1 and 8.1, while condition numbers range from 8.4 to 83.5 and can diverge if the system is not invertible.Careful sequence selection is therefore required for stable calibration.Source: §V · p. 6
• The two-qubit sequences are first-order insensitive to all coherent single-qubit initialization errors and all but one coherent single-qubit measurement error.Only the error proportional to 1⊗σ_x remains to be minimized for reducing SPAM errors.Source: §V · p. 6
• Adding 10 sequences for 12 new single-qubit error parameters is sufficient when single-qubit gates are tuned simultaneously.Source: §V · p. 6
• Additional sequences address decoherence, energy relaxation, leakage, and measurement offsets, while supplementary initial-state sequences prevent early optimization from drifting toward completely mixed-state statistics.Choosing those supplementary sequences requires knowledge of the relevant decoherence mechanisms.Source: §V · p. 6

Qubit model

• The simulated system comprises two S-T0 qubits in four adjacent quantum dots, with exchange-controlled gates, magnetic-field gradients, and leakage suppressed by choosing J23/|b23| ≪ 1.The model uses b23 = 7 ns^-1 and b12 = −b34 = 1 ns^-1, with pulses represented by 20–50 AWG samples and noise averaged over 100 traces.Source: §VI · pp. 6–7

Calibration sequences

• The calibration protocols use sequences that eliminate systematic errors, optionally extract decoherence and leakage, and can reduce sensitivity to initialization and measurement errors.Table I separates sequences 1–15 for systematic-error elimination from sequences 16–17 for decoherence and leakage; Table II excludes all listed SPAM errors except 1 ⊗ σx.Source: Table I · p. 7, Table II · p. 8, §VI · p. 8

Simulation results

• Coherent errors fall below 2 × 10^-4 in most runs, while tuning only the cnot gate reaches rates near 1 × 10^-8; incoherent errors remain unreduced.The gate-set simulations show different coherent-error removal across gates because the sequences are asymmetrically constructed.Source: §VI · p. 10
• A cnot gate with initial infidelity up to 20% can be reliably calibrated when single-qubit gates are pre-calibrated, whereas self-consistent tuning reaches about 50% success at 10% initial infidelity.Success means final cnot fidelity above 99.6%, close to the 99.8% optimum; the authors attribute the self-consistent difficulty to distance from the local minimum.Source: §VI · p. 11

Convergence

• Decoherence sequences slow convergence once coherent errors approach incoherent errors around iteration 4 because the noisy optimization has no exact solution.The authors suggest avoiding or removing those sequences after convergence slows unless a less sensitive optimizer is used.Source: §VI · pp. 10–11
• With decoherence sequences included, coherent errors fall below incoherent errors in around 10 iterations, and tuning the full gate set takes at most twice as many iterations despite 80 additional parameters.The study uses Levenberg-Marquardt optimization over experimentally bounded detunings and evaluates 2500 random starting points with initial infidelities up to 20%.Source: §VI · pp. 10–11

VII. CONCLUSION

• GSC automatically generates calibration sequences suited to different gate sets, experimental requirements, and error models.The protocol is presented as applicable to gate sets on one or several qubits.Source: §VII · p. 12
• Simulations indicate that two-qubit gate sets with 230 or more free parameters can converge rapidly, including calibration of S-T0 spin-qubit systems.The method is also proposed for calibrating classical and non-classical cross-talk in multi-qubit architectures.Source: §VII · p. 12
• Gate sequences can be designed to reduce sensitivity to some state-preparation, measurement, and decoherence errors.The work also gives sequences first-order sensitive to only one coherent single-qubit readout error.Source: §VII · p. 12
• For larger systems, future work should optimize measurement information, improve robustness to decoherence and SPAM errors, and estimate error bars on error syndromes.The current search for suitable sequences may be replaced by a more structured approach for longer sequences.Source: §VII · p. 12
• If initial infidelities cannot be reduced below 10^-20 %, other approaches may be needed despite GSC’s nearly ideal convergence.Suggested alternatives include coarse pre-calibration or global experimental search.Source: §VII · p. 12

ADDITIONAL GATE SEQUENCES

• Table IV provides CNOT-tuning sequences with cond(S) = 6.8 and supports offset cancellation by subtracting results from Table V.The paired construction is intended to make calibration insensitive to measurement offsets.Source: Table IV · p. 15
• Table V sequences share the corresponding final states with Table IV when coherent errors are absent, while subtraction yields cond(Sdiff) = 17.9.The measurement results from Table V are subtracted from those of Table IV.Source: Table V · p. 15
• Table VI supplies another CNOT-tuning sequence set with cond(S) = 11.4, paired with Table VII for measurement-offset subtraction.Table VII differs from Table VI by two additional gates and gives cond(Sdiff) = 23.0 after subtraction.Source: Table VI · p. 16, Table VII · p. 16