Source-linked AI summary
A programmable two-qubit quantum processor in silicon
T. F. Watson, S. G. J. Philips, E. Kawakami, D. R. Ward, P. Scarlino, M. Veldhorst, D. E. Savage, M. G. Lagally, Mark Friesen, S. N. Coppersmith, M. A. Eriksson, L. M. K. Vandersypen
TL;DR
The paper examines noise, calibration, and crosstalk challenges in operating a programmable silicon spin-qubit processor. It characterizes gate performance and Bell-state tomography while using simulations to compare algorithmic behavior with experiment.
Problem
Noise, calibration errors, and crosstalk complicate consistent agreement between processor experiments and simulations.
Source: METHODS · pp. 9–10Method
The authors combine single-qubit calibration, Bell-state tomography, density-matrix estimation, and simulations of quantum algorithms on silicon spin qubits.
Source: METHODS · pp. 7, 9, S2. CALIBRATION OF SINGLE-QUBIT GATES · p. 20, S3. STATE TOMOGRAPHY OF BELL STATES · p. 21Results
Single-qubit Clifford gate fidelities were 98.8% for Q1 and 98.0% for Q2, while simulations reproduced many experimental algorithm features.
Source: METHODS · pp. 9, 14Takeaways & Limitations
The analyses identify residual exchange coupling and coupling-strength noise as relevant factors for interpreting algorithm experiments.
Source: METHODS · pp. 9–10The noise model omits calibration errors and non-detuning-axis charge noise, while tomography assumes perfect prerotation pulses.
Source: METHODS · p. 9, S3. STATE TOMOGRAPHY OF BELL STATES · p. 22
Abstract
from arXiv · showhide
With qubit measurement and control fidelities above the threshold of fault-tolerance, much attention is moving towards the daunting task of scaling up the number of physical qubits to the large numbers needed for fault tolerant quantum computing. Here, quantum dot based spin qubits may offer significant advantages due to their potential for high densities, all-electrical operation, and integration onto an industrial platform. In this system, the initialisation, readout, single- and two-qubit gates have been demonstrated in various qubit representations. However, as seen with other small scale quantum computer demonstrations, combining these elements leads to new challenges involving qubit crosstalk, state leakage, calibration, and control hardware which provide invaluable insight towards scaling up. Here we address these challenges and demonstrate a programmable two-qubit quantum processor in silicon by performing both the Deutsch-Josza and the Grover search algorithms. In addition, we characterise the entanglement in our processor through quantum state tomography of Bell states measuring state fidelities between 85-89% and concurrences between 73-80%. These results pave the way for larger scale quantum computers using spins confined to quantum dots.
METHODS
The processor’s measurements and simulations use calibrated gate models, readout-error correction, and noise models to analyze two-qubit algorithms. The methods quantify gate performance, characterize noise, and identify discrepancies between simulations and experiments.
Readout and tomography
- Readout errors are removed from measured two-spin probabilities using the readout fidelities of each qubit.The method models spin-down and spin-up readout fidelities separately for each qubit.Source: METHODS · p. 7
- State tomography reconstructs two-qubit density matrices from 16 measurement operators using maximum likelihood estimation.Measurements use combinations of I, X, and Y prerotations on both qubits, with 10,000 repetitions per measurement.Source: METHODS · p. 7
Noise modeling
- Simulations model charge, nuclear-spin, and coupling noise using quasistatic sampling and numerical time evolution.Each simulation averages 5,000 repetitions with independently sampled static-noise values.Source: METHODS · pp. 8–9
- Residual exchange coupling during single-qubit gates changes algorithm results by less than 2%, while omitted noise and calibration effects limit simulation agreement.The decoupled Grover simulation predicts a better outcome than experiment, especially for the longest sequence.Source: METHODS · pp. 9–10
Gate calibration
- 98.8% and 98.0% average Clifford gate fidelities are obtained for Q1 and Q2, respectively.The fidelities are estimated from randomized-benchmarking decay fits.Source: METHODS · p. 14
S1. FREQUENCY SHIFTS ON Q2 DUE TO OFF-RESONANT FREQUENCY PULSES
Off-resonant microwave pulses shift Q2’s resonance frequency, with the effect appearing rapidly and across charge regimes and microwave delivery paths. The shift is not explained by the AC Stark effect, spin coupling, or local heating.
- The shift persists in the (0,1) charge regime, eliminating coupling between the two electron spins as its cause.Source: S1. FREQUENCY SHIFTS ON Q2 DUE TO… · p. 19
- Q2’s resonance also shifts when off-resonant microwaves are applied through MW2 via gate P4, showing independence from the gate electrode or coaxial line.Source: S1. FREQUENCY SHIFTS ON Q2 DUE TO… · p. 19
- The shift occurs in less than 100 ns, ruling out local heating effects that would require time to dissipate.Source: S1. FREQUENCY SHIFTS ON Q2 DUE TO… · p. 19
- The observed shift is toward the off-resonant microwave frequency, unlike the predicted negligible AC Stark shift, and therefore rules out the AC Stark effect as its cause.Source: S1. FREQUENCY SHIFTS ON Q2 DUE TO… · p. 19
S2. CALIBRATION OF SINGLE-QUBIT GATES.
Accurate single-qubit gates require calibrating resonance frequencies and microwave powers, including an off-resonant compensation pulse that stabilizes Q2 during Q1 idle periods. Ramsey and AllXY sequences provide the frequency and power calibrations and expose associated errors.
- Three microwave-pulse parameters are calibrated: qubit resonance frequencies, π/2-gate powers, and Q2-shift compensation power during Q1 idle times.Source: S2. CALIBRATION OF SINGLE-QUBIT GATES · p. 19
- Ramsey measurements identify the resonance frequency as the setting that maximizes spin-up probability for each qubit.Source: S2. CALIBRATION OF SINGLE-QUBIT GATES · p. 20
- During the 300 ns wait, an off-resonant Q1 microwave pulse keeps Q2’s resonance frequency constant.Source: S2. CALIBRATION OF SINGLE-QUBIT GATES · p. 20
- AllXY applies every ordered pair of two single-qubit gates from {I, X, X2, Y, Y2}; ideal outcomes are spin-up probabilities of 0, 0.5, or 1.Source: S2. CALIBRATION OF SINGLE-QUBIT GATES · p. 20
- Frequency and power errors produce characteristic AllXY deviations, allowing both gate-power errors and compensation-pulse errors to be corrected.Source: S2. CALIBRATION OF SINGLE-QUBIT GATES · p. 20
S3. STATE TOMOGRAPHY OF BELL STATES.
Quantum state tomography reconstructs two-qubit density matrices from independent measurement operators and compares reconstruction choices for four Bell states. The analysis finds maximum-likelihood and linear-inversion results nearly identical, while prerotation choices affect the reported fidelities and concurrences.
- The two-qubit density matrix is expressed as a linear combination of 16 linearly independent measurement operators.Source: S3. STATE TOMOGRAPHY OF BELL STATES · p. 21
- Maximum-likelihood estimation ensures a physical density matrix that is Hermitian and positive semi-definite, unlike unconstrained linear inversion.Source: S3. STATE TOMOGRAPHY OF BELL STATES · p. 21
- For all measured states, density-matrix elements from linear inversion and maximum-likelihood estimation differ on average by ∼0.005.Source: S3. STATE TOMOGRAPHY OF BELL STATES · p. 21
- Using maximum-likelihood estimation, the four reconstructed Bell-state matrices contain distinct diagonal and off-diagonal populations consistent with their respective state structures.Source: S3. STATE TOMOGRAPHY OF BELL STATES · p. 21
- State fidelities and concurrences are 2−9% and 4−11% lower when using X, Y, X2 or I, X, Y, X2 than when using I, X, Y prerotations.Source: S3. STATE TOMOGRAPHY OF BELL STATES · p. 22
- The final density-matrix estimates use only I, X, Y prerotations because I is expected to estimate expectation values more accurately than X2 under decoherence and calibration errors.Source: S3. STATE TOMOGRAPHY OF BELL STATES · pp. 21–22