High-fidelity operation and algorithmic initialisation of spin qubits above one kelvin

A. Device and two-qubit operation

• The SiMOS double-dot processor encodes qubits in unpaired-electron spins and uses PSB parity readout, microwave single-qubit gates and decoupled controlled-phase two-qubit gates.Three electrons in at least one dot avoid the narrow PSB range associated with valley excitation for a single electron.Source: A. Device and two-qubit operation · pp. 2–3
• At 1 K and 0.79 T, coherent single-qubit Rabi and two-qubit exchange oscillations are stable without feedback on gate voltages, resonance frequencies or microwave amplitudes.RFSET feedback is retained to maintain the readout signal level over time.Source: A. Device and two-qubit operation · p. 3

B. Initialisation and readout

• The three-stage protocol filters odd-parity states by PSB readout, then uses zCNOT to convert residual |↑↑⟩ into a filterable odd-parity state.The final ESR spectra yield an initialization fidelity of 99.6%.Source: B. Initialisation and readout · p. 3
• Initialization remains above 90% at 35 mT despite kBT exceeding 20hfqubit, and is estimated at 99.6% for |↓↓⟩ at 0.79 T.The overall initialization and readout fidelities are estimated at 99.6% for |↓↓⟩ and at least 95.0% for |↓↑⟩ and |↑↓⟩.Source: B. Initialisation and readout · p. 3
• Above 1 K, T1-induced errors and charge-readout limits affect readout, while PSB relaxation falls to 0.45 ms at 1.4 K.Above 0.5 K, PSB relaxation time scales as T^-2.8; overall SPAM remains workable through at least 1.4 K.Source: B. Initialisation and readout · p. 3
• With 20 readout cycles, Pinit,even is 99.34 ± 0.27% and Pread,even is 99.34 ± 0.08%.Machine learning on cumulative PSB readout outcomes estimates initialization, readout and state-change probabilities.Source: B. Initialisation and readout · p. 4

C. Single-qubit performance

• Above 1 K, T1 scales as T^-3.1; T Hahn 2 in different configurations falls between T^-1 and T^-1.1, while T*2 scales uniformly as T^-0.2.Thermal equilibrium shifts from |↓↓⟩ when kBT ≪ hfqubit to a mixed state when kBT ≥ hfqubit.Source: C. Single-qubit performance · p. 5
• In the white-noise regime at frequencies ≥200 kHz, noise power spectral density increases tenfold from 0.14 K to 1.2 K.The study uses CPMG noise spectroscopy; higher-frequency apparent noise is characterized as blue noise and may reflect microwave pulse miscalibration or high-power driving.Source: C. Single-qubit performance · p. 5, Figure 3 · p. 5
• 99.85 ± 0.01% is the optimized single-qubit Clifford fidelity measured by randomized benchmarking at 1 K.Fidelity drops by less than 1% from 0.14 K to 1 K.Source: C. Single-qubit performance · p. 5
• Single-qubit fidelity falls at low B0, mainly because crosstalk limits maximum Rabi frequency, and near high-field excited-state degeneracy, where dephasing increases.At B0 = 0.1 T, the one-electron qubit operates at 99.18 ± 0.03% fidelity; qubits remain operable down to 25 mT.Source: C. Single-qubit performance · p. 5
• The SMART dressing protocol shows coherence optima at periodic microwave-modulation conditions at 1 K.The authors present this as evidence of the potential for continuous global driving of many spin qubits above 1 K.Source: Figure 3 · p. 5, C. Single-qubit performance · p. 6

D. Two-qubit performance

• The DCZ inserts an individual-qubit X(π) pulse midway through the CZ gate to extend coherence and cancel Stark shift-induced phase errors.Source: D. Two-qubit performance · p. 6
• Above 1 K, the DCZ oscillation quality factor remains at least 50, with coherence reduction similar to that in T Hahn2.Source: D. Two-qubit performance · p. 6
• 98.92 ± 0.67% DCZ fidelity was measured by FBT at T = 1 K, while the interleaved-RB ratio was 97.7±1.5%.The authors note that the interleaved-RB ratio mainly reflects dephasing during exchange and echoing in the DCZ gate, rather than the true DCZ fidelity.Source: D. Two-qubit performance · p. 6
• At both 0.1 K and 1 K, exchange noise appears among the main noise sources, but stochastic errors contribute most to infidelity.The analysis identifies Heisenberg and antisymmetric exchange terms, while the main infidelity contributions take the form of stochastic errors.Source: D. Two-qubit performance · p. 7
• No single stochastic error source dominates DCZ infidelity, and inconsistent coupling strength across executions may require advanced pulse shaping to address non-Markovian noise.Source: D. Two-qubit performance · p. 7
• Q1 and Q2 show different error types despite a symmetric operation point; local environmental factors may contribute to this asymmetry.Source: D. Two-qubit performance · p. 7
• The error rates are biased toward dephasing by generally more than 100:1 up to at least 1.5 K, a bias that may permit increased fault-tolerant thresholds.Fully exploiting this bias requires syndrome-extraction circuits that avoid injecting spin-flip errors from ancilla SPAM into data qubits.Source: D. Two-qubit performance · p. 7

E. Outlook

• Algorithmic initialization and high-fidelity universal logic bring SiMOS spin qubits above 1 K into the realm of fault tolerance.The authors also report robustness to charge configuration and magnetic field above 1 K, and suggest weaker qubit-variability effects from similar T1 and T2 temperature dependence across configurations.Source: E. Outlook · p. 7
• Future campaigns could implement SPAM analysis and FBT in an FPGA for real-time calibration and automate algorithmic-initialization tune-up.Source: E. Outlook · p. 7
• SPAM and control fidelities must rise far above 99% for truly fault-tolerant operation, and control may inject errors into spin readout.The authors identify improved Si/SiO2 interfaces, lower experimental noise, and faster readout as potential improvement directions.Source: E. Outlook · p. 7
• Scalable spin-qubit operation will rely on scalable control such as multi-qubit SMART, which continuously drives qubits with a modulated microwave field.The passage says the driving pulses decouple qubits from noise and eliminate free precession, during which they are most sensitive to decoherence.Source: E. Outlook · pp. 7–8
• CMOS-compatible silicon operation at temperatures suitable for integrating classical control circuits makes a scalable semiconductor quantum processor plausible.Source: E. Outlook · p. 8

A. Measurement setup

• The device is measured in a dilution refrigerator, with a heater used up to 1 K and reduced helium-mixture cooling above 1 K; temperature control becomes nonviable above 1.5 K.The RFSET charge sensor uses a single-island SET connected to a tank circuit for reflectometry.Source: A. Measurement setup · p. 8

B. Device tune-up

• In the PSB region, even-parity states are blockaded and produce a lower RF signal; single-shot readout thresholds the two signal levels to estimate blockade probability.The initial spin proportions depend on ramp speed and exchange through transition adiabaticity, though increased thermalization above 1 K reduces this bias.Source: B. Device tune-up · p. 8
• Above VJ = 1.175 V, significant exchange produces four ESR transitions corresponding to CROT operations; latch readout is excluded from algorithmic initialization because it involves reservoir tunneling and may not scale.Source: B. Device tune-up · p. 9

C. Algorithmic initialisation

• Algorithmic initialization enters the double-dot state space, rejects odd-parity outcomes, calibrates the CZ operation, then uses a zCNOT to convert |↑↑⟩ into |↑↓⟩ while leaving |↓↓⟩ unchanged before another parity filter.The initial load can produce any of the four two-qubit states, with ground-state probability decreasing as temperature rises.Source: C. Algorithmic initialisation · p. 9
• The initialization protocol restarts on an unblockaded outcome and completes on a blockaded |↓↓⟩ outcome, with the if conditions implemented in FPGA real-time logic.The protocol can also prepare other parity-basis states by applying microwave π pulses or replacing zCNOT with CNOT.Source: C. Algorithmic initialisation · p. 9
• At low magnetic field, the protocol remains effective at producing an even-parity state, but low-field control and readout contribute to reduced ESR transition amplitude.Source: C. Algorithmic initialisation · p. 9
• At 85 mT and 1 K, initialization takes about three iterations, totaling around 150 µs; readout integration dominates the time cost.Iteration count also increases with temperature, while high-field excited-state crossings enhance spin randomization after loading.Source: C. Algorithmic initialisation · p. 9

D. SPAM error analysis with repeated readout

• Repeated parity measurements produce binary outcomes, and machine learning uses increased statistics from multiple measurements for SPAM error analysis.The initialisation-and-measurement sequence is performed for 1,000 shots.Source: D. SPAM error analysis with repeated… · pp. 9–10
• The HMM represents hidden spin states as a Markov chain, with start probabilities, state-transition probabilities, and measurement probabilities conditioned on the current state.These are encoded by Π, A, and Θ, respectively.Source: D. SPAM error analysis with repeated… · p. 10
• Expectation maximisation fits the HMM by maximising the marginal likelihood, with Baum–Welch providing iterative updates without gradient backpropagation.Source: D. SPAM error analysis with repeated… · p. 10
• The Cramér–Rao bound quantifies uncertainty in the fitted parameters, using inverse Fisher-information diagonal elements as lower bounds.The Forward–Backward algorithm computes the marginal likelihood needed for the Fisher information matrix.Source: D. SPAM error analysis with repeated… · p. 10
• The Viterbi algorithm identifies the most likely sequence of true spin states underlying the measurements for a given model.Source: D. SPAM error analysis with repeated… · p. 10

E. Crosstalk correction

• For N = 4, 8, 12, ... the cancellation criterion makes each target-qubit π/2 pulse produce a full 2πN off-resonance rotation on the ancilla.The criterion relates the Zeeman frequency difference ΔE_Z to the target Rabi frequency f_Rabi.Source: E. Crosstalk correction · p. 10
• Two-qubit sequences also require AC Stark-shift correction, which is distinct from cancelling off-resonance driving.Source: E. Crosstalk correction · p. 10
• At ultra-low B0, diminishing ΔE_Z makes higher-order crosstalk significant and individual-qubit control unmanageable; SMART addresses all qubits simultaneously.The correction does not prevent crosstalk-induced perturbation from causing ancilla decoherence.Source: E. Crosstalk correction · p. 10

F. Randomised benchmarking

• Single-qubit RB sequences use elementary π/2 and π gates plus identity, with one physical elementary gate per Clifford on average, excluding virtual Z gates.Source: F. Randomised benchmarking · p. 10
• Two-qubit RB Cliffords average 1.8 single-qubit and 1.5 DCZ elementary gates, executed sequentially; the sequences also support FBT and interleaved RB inserts DCZ gates.Source: F. Randomised benchmarking · p. 11
• The fitted decay exponent gives single-qubit Clifford fidelity as 1 − 0.5b and two-qubit Clifford fidelity as 1 − 0.75b.The exponent c reflects error Markovianity; a depends on readout fidelity and d is close to 0.5.Source: F. Randomised benchmarking · p. 11
• At T = 1 K, the longest RB sequences average 1,400 elementary gates and take up to several hundred microseconds, within the measured T1.Spin relaxation, excitation, incorrect rotations, or slow charge-readout drifts can obscure long runs.Source: F. Randomised benchmarking · p. 11

G. Fast Bayesian tomography

• FBT uses variable-length two-qubit RB sequences and their outcomes to reconstruct gate-set process matrices, feeding native parity readout directly into the analysis.The experiments ran 32,000 sequences at T = 0.1 K and 26,000 at T = 1 K.Source: G. Fast Bayesian tomography · p. 11
• FBT is bootstrapped with guessed fidelity values to help the analysis converge with a finite number of experiments.Source: G. Fast Bayesian tomography · p. 11
• Each noisy gate is modelled as a noise channel times an ideal gate, with the channel linearised around identity; output receives CPTP projection and gauge optimisation.The projection and optimisation address the fact that FBT does not guarantee physical reconstructed channels or flag gauge ambiguity.Source: G. Fast Bayesian tomography · p. 11

H. Error taxonomy with pyGSTi

• The error generator is obtained from the estimated noisy gate and its ideal counterpart, and its coefficients are used to estimate entanglement infidelity.The approximation is validated by Hamiltonian errors dominating stochastic errors in magnitude; average gate fidelity is then related to entanglement fidelity for a four-dimensional two-qubit system.Source: H. Error taxonomy with pyGSTi · p. 11
• pyGSTi projects each gate’s error generator into Hamiltonian and stochastic components, with coefficients represented in the Pauli basis.The analysis covers DCZ, X1(π/2) and X2(π/2) at 0.1 K and 1 K.Source: H. Error taxonomy with pyGSTi · p. 11

CORRESPONDING AUTHORS

• Correspondence is directed to J.Y.H., A.S.D. or C.H.Y.Source: CORRESPONDING AUTHORS · p. 12

• A.S.D. is Diraq Pty Ltd’s CEO and a director; several authors declare equity interests, and J.Y.H., A.S. and C.H.Y. are inventors on a related provisional patent filed by Diraq.The filing is Australian provisional application 2023902138, with a priority date of 3 July 2023.Source: COMPETING INTERESTS · p. 12

CODE AVAILABILITY

• The analysis codes supporting the study’s findings are available from the corresponding authors on reasonable request.Source: CODE AVAILABILITY · p. 12

• At 1 K, the measured T1 decay amplitude decreases as thermal energy approaches or exceeds qubit energy, with readout degradation becoming dominant at higher temperatures.The B0-dependent measurements also report a T1 drop at low B0 and near the excited-state-crossing hot spot, where readout is degraded.Source: Figure 4 · p. 16, Figure 6 · p. 18
• The reported finding is that the quality factor of the Rabi oscillation improves with faster driving; the cited passage gives no numerical improvement magnitude or DCZ threshold.Source: CODE AVAILABILITY · p. 18
• The two-qubit benchmarking and tomography figure reports IRB results and DCZ Pauli transfer matrices at 0.79 T and 0.1 K and 1 K.The random sequences include Clifford gates followed by a recovery gate and parity projections.Source: Figure 9 · p. 21

• Algorithmic |↓↓⟩ initialization fidelity is 90.77% at 35 mT and 99.56% at 0.79 T.The estimate uses ESR transition amplitudes measured after the mixed even-parity and |↓↓⟩ initialization stages.Source: Figure 1 · p. 22
• The initialization spectra cover 1 K conditions where thermal energy is 20 times, 8 times or near the qubit energies.The respective fields are 35 mT, 85 mT and 0.79 T.Source: Figure 2 · p. 23
• Machine-learning SPAM analysis reports inferred probabilities for algorithmic |↓↓⟩ and |↑↓⟩ initialization at 0.79 T and 1 K.Source: Figure 3 · p. 24
• Single-qubit RB fidelities are 99.8484 ± 0.0021%, 99.8547 ± 0.0016% and 99.8519 ± 0.0013% for the three stated projection analyses.The projections yield decays at about the same rate toward an equilibrium of 0.5; supplementary figures also show residual PTMs and error-channel magnitudes for DCZ and X(π/2) gates.Source: Figure 5 · p. 25, Figure 6 · p. 25, Figure 7 · p. 26