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Fast universal quantum control above the fault-tolerance threshold in silicon
Akito Noiri, Kenta Takeda, Takashi Nakajima, Takashi Kobayashi, Amir Sammak, Giordano Scappucci, Seigo Tarucha
TL;DR
Reliable high-fidelity quantum control requires identifying operating conditions that support reproducible gates. The paper models controlled rotations and noise, finding that Fp can reproducibly exceed 99% but drops at large detuning.
Problem
The study addresses the need to identify conditions for reproducibly achieving high-fidelity gates under detuning.
Method
The paper models electrically controlled rotations using a theoretical description and measured resonance-frequency fluctuations under quasi-static noise.
Results
Fp higher than 99% is reproducibly obtained, while fidelity sharply drops at large positive and negative detuning.
Takeaways & Limitations
The results identify a detuning-dependent operating regime for reproducible high-fidelity control.
Takeaways & Limitations
The analysis assumes noise is quasi-static during each 100 µs measurement and changes between measurements.
Abstract
from arXiv · showhide
Fault-tolerant quantum computers which can solve hard problems rely on quantum error correction. One of the most promising error correction codes is the surface code, which requires universal gate fidelities exceeding the error correction threshold of 99 per cent. Among many qubit platforms, only superconducting circuits, trapped ions, and nitrogen-vacancy centers in diamond have delivered those requirements. Electron spin qubits in silicon are particularly promising for a large-scale quantum computer due to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent due to the slow operation.Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the condition of qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate the universal gate fidelity beyond the fault-tolerance threshold and pave the way for scalable silicon quantum computers.
Measurement setup.
The measurement setup used a dilution refrigerator, filtered dc gate control, synchronized waveform generators, and I/Q-modulated microwave pulses. Microwave sideband modulation reduced leakage and enabled rapid frequency switching, while modulation amplitudes controlled single-qubit Rabi frequencies.
- Cryogenic environment: ∼20 mK base temperature and ∼60 mK electron temperature were achieved in a dry dilution refrigerator.The refrigerator was an Oxford Instruments Triton.
- Microwave control: EDSR microwave pulses were generated by an I/Q-modulated Anapico APMS20G and Marki MLIQ-0218 mixer, then applied to the bottom screening gate.I/Q modulation signals came from a Tektronix AWG70002A synchronized with the gate-pulse generator.
- Microwave control: Microwave sideband modulation from -240 to 180 MHz avoided leakage-induced spin rotation and enabled rapid microwave-frequency switching.Typical isolation was ∼50 dBc after calibration, with additional pulse modulation during initialization and measurement.
- Single-qubit control: Single-qubit Rabi frequencies were controlled by the amplitudes of the I/Q modulation signals.This provided direct control over the rotation rates of single-qubit gates.
Sample fabrication.
The quantum dots are fabricated in an isotopically enriched silicon quantum well 50 nm below the wafer surface, using overlapping aluminum gates and an integrated titanium–cobalt micromagnet. The gates and micromagnet are separated by thin aluminum-oxide insulating layers.
- Sample fabrication.: Quantum dots are defined in an isotopically enriched silicon quantum well with residual 29silicon concentration of 800 parts per million, 50 nm below the wafer surface.The quantum well is located beneath the wafer surface at the stated depth.
- Sample fabrication.: Three layers of overlapping aluminum gates are fabricated using electron-beam lithography and lift-off processes.Each gate layer is insulated by thin native aluminum oxide.
- Sample fabrication.: A micromagnet made from titanium and cobalt films is placed above the overlapping gates.The titanium and cobalt films have thicknesses of 5 and 250 nm, respectively.
- Sample fabrication.: The micromagnet is separated from the gates by a 30 nm thick aluminum-oxide insulating layer grown by atomic layer deposition.The micromagnet design is similar to those in previous reports.
Sequence fidelity and gate fidelity extraction in randomized benchmarking.
Randomized benchmarking extracts single- and two-qubit sequence fidelities from depolarizing-parameter fits, then converts Clifford fidelities into primitive-gate fidelities using average gate counts. CNOT fidelity is obtained by comparing reference and CNOT-interleaved benchmarking, with uncertainties estimated by Monte Carlo sampling.
- Single-qubit randomized benchmarking: Single-qubit Clifford fidelity is calculated as F_C,m = (1 + p_s)/2 for qubit m = 1 or 2.The depolarizing parameter p_s is obtained from randomized-benchmarking sequence measurements.
- Single-qubit randomized benchmarking: Primitive single-qubit fidelity is extracted as F_p,m = 1 − (1 − F_C,m)/1.875 because each Clifford contains 1.875 primitive gates on average.This conversion applies separately to qubits 1 and 2.
- Two-qubit randomized benchmarking: Two-qubit Clifford fidelity is obtained as F_C = (1 + 3p_t)/4, and the primitive fidelity uses an average of 2.57 primitive gates per Clifford.The reported primitive conversion is F_p = 1 − (1 − F_C)/2.57.
- CNOT fidelity extraction: CNOT fidelity is extracted by comparing depolarizing parameters from reference Clifford sequences and sequences with a CNOT inserted between random Clifford gates.The resulting fidelity is calculated from the reference and CNOT-interleaved depolarizing parameters.
- Uncertainty estimation: Gate-fidelity errors are estimated with a Monte Carlo method by fitting the resulting fidelity distribution to a Gaussian and extracting its standard deviation.This procedure provides the reported uncertainty measure.
Estimation of resonance frequencies fluctuations.
Repeated Ramsey-fringe measurements with Bayesian estimation tracked fluctuations in four spin-dependent resonance frequencies. The analysis indicates that dephasing is mainly limited by single-qubit frequency noise rather than fluctuations in J.
- Estimation of resonance frequencies fluctuations.: Four resonance frequencies were estimated from sequential Ramsey-fringe measurements using Bayesian estimation.The measurements varied evolution time from 0.04 µs to 4.0 µs in 0.04 µs steps.
- Estimation of resonance frequencies fluctuations.: 1.706 s was required for one measurement cycle, repeated over 10,000 cycles to extract time-dependent resonance-frequency fluctuations.The tracked frequencies were f1,↓, f1,↑, f2,↓, and f2,↑.
- Estimation of resonance frequencies fluctuations.: The analysis quantified fluctuations in J/2 and in the single-qubit frequencies of Q1 and Q2 from the spin-dependent resonance-frequency shifts.The reported relations are ΔJ/2 = (Δf1,↑−Δf1,↓)/2, Δf1 = (Δf1,↑+Δf1,↓)/2, and Δf2 = (Δf2,↑+Δf2,↓)/2.
- Estimation of resonance frequencies fluctuations.: Dephasing times are mostly limited by single-qubit frequency noise because Δfm is larger than ΔJ/2.Accordingly, J does not significantly affect the dephasing times.
Theoretical description of controlled-rotation.
The theoretical description models EDSR-controlled time evolution and explains how selecting resonance frequencies produces controlled rotations. It also identifies off-resonant effects and a Rabi-frequency condition for mitigating them.
- Theoretical description of controlled-rotation.: EDSR control is analyzed through the system’s time evolution.The passage frames the description as a simpler way to understand dynamics under EDSR control.
- Theoretical description of controlled-rotation.: Selecting one of f1,↓, f1,↑, f2,↓, or f2,↑ from the off-diagonal terms produces a CROT gate.The gate is generated by choosing one of the four resonance frequencies.
- Theoretical description of controlled-rotation.: During the π/2 CROT time thp = 1/(4fR), terms oscillating much faster than fR are averaged out.The averaging occurs over the specified π/2 CROT interval.
- Theoretical description of controlled-rotation.: When J is comparable to fR, oscillations at frequency J cause unwanted off-resonant rotation of the target qubit.This identifies the coupling-to-Rabi-frequency regime associated with the undesired rotation.
- Theoretical description of controlled-rotation.: The prescribed Rabi frequency is fR = J/√(16k^2 − 1), where k is an integer.The passage states this condition for selecting the Rabi frequency.
Simulation of two-qubit gate infidelity by quasi-static noise in resonance frequencies.
The study simulates two-qubit primitive-gate infidelity from quasi-static resonance-frequency noise using measured parameter variations and randomized Clifford sequences. With J fixed at √15f_R, the simulation identifies f_R = 4–5 MHz as the optimal gate condition, where infidelity is minimized.
- Noise model: The noise is assumed quasi-static during each 100 µs measurement and variable between measurements.This assumption defines how resonance-frequency fluctuations are sampled in the simulation.
- Noise model: The simulation uses measured time dependence of ΔJ, ΔdẼZ = Δf2 − Δf1, and ΔEZ = (Δf1 + Δf2)/2.These measured variations enter the simulated resonance-frequency-noise model.
- Gate simulation: π/2 CROT operators are calculated by time-evolving the Hamiltonian with ΔHR over N = 1000 time steps.The perturbation is represented by ΔHR, with Δt = thp/N.
- Infidelity extraction: The ideal final-state probability is averaged over 60 random Clifford sequences, each repeated 100 times with different noise parameters.The resulting decay is used to extract the two-qubit primitive-gate infidelity.
- Simulation result: With J fixed at √15f_R, the simulated infidelity is minimized around the optimal gate condition f_R = 4–5 MHz.The f_R dependence of infidelity is shown in Extended Data Fig. 5b.
Additional information
Additional measurements characterize relaxation, Rabi decay, gate performance, noise, and algorithmic output states. Two-qubit fidelity exceeds 99% near the charge-symmetry point but drops at large detuning because of fast Rabi decay.
- Qubit characterization: Spin relaxation is negligible for both qubits, as the measured spin-up probabilities show no decay with wait time.Measurements use J = 18.85 MHz and fR = 4.867 MHz.
- Rabi characterization: Rabi oscillations are measured by varying microwave burst time t_MW from 0.01 µs to 0.41 µs in 0.01-µs increments.Longer-burst traces are also measured, and decay fits extract the Rabi decay during a π/2 CROT.
- Single-qubit gates: Single-qubit gate fidelities depend on f_R, with the best values obtained at f_R = 2-5 MHz under fixed J = 32.0 MHz.This measurement assesses the impact of f_R without involving the effect of J.
- Noise analysis: Single-qubit frequency noises Δf_1 and Δf_2 are larger than exchange noise ΔJ/2, identifying single-qubit frequency fluctuations as the larger measured noise contribution.The comparison is made with J fixed at 18.85 MHz.