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Benchmarking Gate Fidelities in a Si/SiGe Two-Qubit Device

X. Xue, T. F. Watson, J. Helsen, D. R. Ward, D. E. Savage, M. G. Lagally, S. N. Coppersmith, M. A. Eriksson, S. Wehner, L. M. K. Vandersypen

arXiv:1811.04002v1quant-phcond-mat.mes-hall

TL;DR

Reliable two-qubit gate-fidelity characterization in silicon spin qubits is difficult with conventional randomized benchmarking, which requires many native gates per Clifford. This work combines standard benchmarking with character randomized benchmarking to characterize single- and two-qubit gates, cross-talk, and error correlations. It reports a CPhase fidelity around 91% and anticipates further improvement under specified device conditions.

  • Problem

    Conventional two-qubit benchmarking in semiconductor spin qubits has made gate-fidelity measurement difficult because Clifford sequences require many native gates and yield loose fidelity bounds.

    Source: INTRODUCTION · p. 1

  • Method

    The study combines standard randomized benchmarking with character randomized benchmarking, which uses a compact reference sequence to characterize two-qubit gates and separate decay channels without additional overhead.

    Source: INTRODUCTION · p. 1, CHARACTER RANDOMIZED BENCHMARKING · pp. 4–5

  • Results

    The comprehensive study reports independent single-qubit gate fidelities around 99%, simultaneous values of 98.8% for Q1 and 96.9% for Q2, and CPhase fidelity around 91%.

    Source: CONCLUSIONS · p. 6

  • Takeaways & Limitations

    The authors expect isotopically purified Si28/SiGe and operation at the symmetry point to make CPhase fidelity above the fault-tolerant threshold of 99% reachable.

    Source: CONCLUSIONS · p. 6

    Standard two-qubit Clifford benchmarking saturates within about eight operations, leaving large uncertainty, and the device cannot access the symmetry point where coupling is first-order insensitive to detuning.

    Source: TWO-QUBIT RANDOMIZED BENCHMARKING · p. 4, CHARACTER RANDOMIZED BENCHMARKING · p. 6

Abstract

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We report the first complete characterization of single-qubit and two-qubit gate fidelities in silicon-based spin qubits, including cross-talk and error correlations between the two qubits. To do so, we use a combination of standard randomized benchmarking and a recently introduced method called character randomized benchmarking, which allows for more reliable estimates of the two-qubit fidelity in this system. Interestingly, with character randomized benchmarking, the two-qubit CPhase gate fidelity can be obtained by studying the additional decay induced by interleaving the CPhase gate in a reference sequence of single-qubit gates only. This work sets the stage for further improvements in all the relevant gate fidelities in silicon spin qubits beyond the error threshold for fault-tolerant quantum computation.

INTRODUCTION

The paper addresses the difficulty of reliably measuring two-qubit gate fidelities in silicon spin qubits using conventional benchmarking. It combines standard randomized benchmarking with character randomized benchmarking to characterize gate fidelities, cross-talk, and correlated errors.

INTRODUCTION

  • Randomized benchmarking estimates gate fidelity with fewer resources than process tomography and is more robust to SPAM errors, including at low gate error rates.Process tomography scales exponentially with qubit number and cannot distinguish pure gate errors from SPAM errors.Source: INTRODUCTION · p. 1
  • Standard randomized benchmarking extracts average Clifford fidelity from return-probability decay, while interleaved benchmarking estimates a gate's fidelity by comparing decay with and without that gate.Simultaneous randomized benchmarking applies random Clifford operations to multiple qubits to characterize cross-talk.Source: INTRODUCTION · p. 1
  • Conventional two-qubit randomized benchmarking requires many native gates per Clifford, causing rapid decay and loose fidelity bounds that hinder measurement of silicon spin-qubit gates.Character randomized benchmarking can instead interleave a two-qubit gate in a reference sequence containing only a small number of single-qubit gates, while also resolving decay channels and error correlations.Source: INTRODUCTION · p. 1
  • The study combines standard and character randomized benchmarking to characterize single- and two-qubit fidelities, cross-talk, and correlated errors in silicon quantum dots.The authors note silicon spin qubits’ compact size, coherence times up to tens of milliseconds, and compatibility with existing semiconductor technology.Source: INTRODUCTION · p. 1

DEVICE AND QUBIT OPERATION

The device uses two electron spins in a Si/SiGe double quantum dot, with microwave-driven single-qubit control and an exchange-based CPhase gate. Gate pulsing and charge sensing support initialization, manipulation, and readout.

DEVICE AND QUBIT OPERATION

  • Two electron spins encoded in Zeeman-split states form qubits Q1 and Q2 in a double quantum dot in a 12 nm Si/SiGe quantum well, 37 nm below the surface.The device is cooled to about 20 mK and operated in a 617 mT in-plane magnetic field.Source: DEVICE AND QUBIT OPERATION · pp. 1–2
  • Electric-dipole spin resonance drives single-qubit rotations using a transverse magnetic-field gradient from three cobalt micromagnets.Microwave phase sets the rotation axis in the x-y plane; software changes to the rotating reference frame implement z rotations.Source: DEVICE AND QUBIT OPERATION · p. 2, Figure 1 · p. 2
  • The native two-qubit gate is CPhase, implemented by pulsing detuning to switch on exchange between the dots and produce an Ising interaction.The large difference in qubit energy splittings makes the exchange flip-flop terms ineffective; single-qubit z rotations are added to obtain the CPhase operator.Source: DEVICE AND QUBIT OPERATION · p. 2
  • Energy-selective tunnelling initializes and reads out Q2, while Q1 is initialized by fast spin relaxation and mapped onto Q2 for measurement.Plunger gates pulse the device for initialization, manipulation, readout, and detuning control of the two-qubit gate.Source: DEVICE AND QUBIT OPERATION · p. 2, Figure 1 · p. 2

INDIVIDUAL AND SIMULTANEOUS RANDOMIZED BENCHMARKING

Standard randomized benchmarking measures individual-qubit Clifford and average single-qubit gate fidelities, while simultaneous benchmarking tests cross-talk. The reported simultaneous fidelities are lower, with an asymmetric effect that is attributed to excitation rather than limited addressability.

INDIVIDUAL AND SIMULTANEOUS RANDOMIZED BENCHMARKING

  • Random Clifford operations twirl noise so the return probability decays exponentially with Clifford count, enabling average Clifford fidelity to be extracted from the decay parameter.The decay fit parameters A and B depend on state preparation and measurement, while d = 2^N for N qubits.Source: INDIVIDUAL AND SIMULTANEOUS RANDOMIZED… · p. 2
  • Individual randomized benchmarking yields average single-qubit gate fidelities of 99.20 ± 0.03% for Q1 and 98.79 ± 0.02% for Q2.The corresponding average Clifford fidelities are 98.50 ± 0.05% for Q1 and 97.72 ± 0.03% for Q2.Source: INDIVIDUAL AND SIMULTANEOUS RANDOMIZED… · p. 3
  • Simultaneous randomized benchmarking gives Clifford fidelities of 97.67 ± 0.04% for Q1 and 94.26 ± 0.10% for Q2, decreases of 0.8% and 3.5%, respectively.The frequency separation is 1.38 GHz, nearly three orders of magnitude larger than the approximately 2 MHz Rabi frequencies, so the cross-talk is not due to limited addressability.Source: INDIVIDUAL AND SIMULTANEOUS RANDOMIZED… · p. 3
  • The Q2 cross-talk persists when Q1 is driven off-resonantly, implicating excitation rather than twirling; its asymmetry suggests quantum-dot microscopic details also play a role.The authors report ruling out the AC Stark effect, heating, and residual qubit coupling as possible sources.Source: INDIVIDUAL AND SIMULTANEOUS RANDOMIZED… · p. 3

TWO-QUBIT RANDOMIZED BENCHMARKING

Standard two-qubit Clifford benchmarking produces a low-fidelity estimate with substantial uncertainty because each Clifford compiles to many native gates and the decay saturates quickly. A projected interleaved protocol and character randomized benchmarking provide routes to characterize the CPhase gate more directly.

TWO-QUBIT RANDOMIZED BENCHMARKING

  • 82.10 ± 2.75% is the measured average two-qubit Clifford fidelity from standard two-qubit randomized benchmarking.The Clifford group contains 11,520 elements, compiled from an average of 8.25 single-qubit rotations and 1.5 CPhase gates per Clifford.Source: TWO-QUBIT RANDOMIZED BENCHMARKING · pp. 3–4, Figure 3 · p. 4
  • The decay saturates within about eight two-qubit Cliffords, producing a large uncertainty and making gate-dependent errors difficult to assess.Because Clifford compilation mixes single-qubit and CPhase gates, it also cannot confidently yield the fidelity of an individual native gate.Source: TWO-QUBIT RANDOMIZED BENCHMARKING · p. 4
  • A simplified interleaved protocol measures CPhase fidelity projected into single-qubit subspaces by inserting CPhase into a reference sequence of single-qubit Cliffords.Depending on which qubit is control and its eigenstate, the target ideally receives either a z-axis π rotation or the identity; measured projected fidelities range from 91% to 95%.Source: TWO-QUBIT RANDOMIZED BENCHMARKING · p. 4

CHARACTER RANDOMIZED BENCHMARKING

Character randomized benchmarking (CRB) uses parallel single-qubit Clifford sequences to characterize two-qubit decay channels and extract the CPhase fidelity. Interleaving CPhase yields a two-qubit fidelity of 92.0 ± 0.5%, while the reference measurement indicates essentially independent errors.

CHARACTER RANDOMIZED BENCHMARKING

  • CRB uses a random character-group gate and character weighting to obtain single-exponential decays beyond the multi-qubit Clifford group while retaining SPAM resistance.The benchmark group is parallel single-qubit Cliffords, and the character group is the two-qubit Pauli group.Source: CHARACTER RANDOMIZED BENCHMARKING · pp. 4–5
  • The three CRB channels separately track each qubit’s decay and the two-qubit parity decay, allowing their rates to be extracted without fitting a sum of exponentials.CRB separates the decay rates in a procedure described as SPAM-immune and without additional overhead.Source: CHARACTER RANDOMIZED BENCHMARKING · pp. 5–6
  • The sequence initializes |00⟩, applies a random Pauli preparation and simultaneous single-qubit Clifford sequence, then applies recovery and measures both qubits.The Pauli is absorbed into the first Clifford; outcome probabilities are combined according to their character to obtain three fitting parameters.Source: CHARACTER RANDOMIZED BENCHMARKING · pp. 5–6
  • 91.9 ± 0.1% is the reference CRB fidelity for simultaneous single-qubit Clifford pairs, and the errors are essentially independent.The reported difference α12 − α1|2α2|1 is −0.0017 ± 0.0031.Source: CHARACTER RANDOMIZED BENCHMARKING · p. 6
  • 92.0 ± 0.5% is the extracted two-qubit CPhase gate fidelity, obtained by comparing interleaved and reference CRB decays.Interleaving CPhase adds decay and mixes the three subspaces; the comparison isolates the gate fidelity in two-qubit space.Source: Figure 4 · p. 5, CHARACTER RANDOMIZED BENCHMARKING · p. 6
  • The device could not access the coupling symmetry point, so charge noise directly affects the two-qubit coupling strength to first order.The authors identify nuclear spin noise and charge noise as dominant CPhase error sources.Source: CHARACTER RANDOMIZED BENCHMARKING · p. 6

CONCLUSIONS

The conclusions present CRB as a compact-reference method for characterizing multi-qubit behavior and extracting specific gate fidelities. They summarize measured gate fidelities, discuss possible routes to higher CPhase fidelity, and include a measurement-setup description.

CONCLUSIONS

  • CRB's reference-sequence decay can separate exponentials to characterize simultaneously applied gates, cross-talk, and noise correlations; interleaved CRB extracts specific gate fidelities.The conclusion describes CRB as a generic approach for multi-qubit characterization.Source: CONCLUSIONS · p. 6
  • Independent single-qubit gate fidelities are around 99%, falling to 98.8% for qubit 1 and 96.9% for qubit 2 under simultaneous twirling; CPhase fidelity is around 91%.The study characterizes single-qubit, simultaneous single-qubit, and two-qubit gate fidelities; CRB's compact reference sequence was essential for a reliable two-qubit estimate.Source: CONCLUSIONS · p. 6
  • CPhase fidelity above 99% is expected with isotopically purified Si28/SiGe and operation at the symmetry point; above 99.9% is described as in reach with further charge-noise improvements.Source: CONCLUSIONS · pp. 6–7

Appendix A: MEASUREMENT SETUP

  • The setup uses voltage pulses to control detuning and switch the controlled-phase gate, microwave sources for EDSR, and threshold detection of sensor-current traces for spin probabilities.The traces are sampled at approximately 60 kHz, converted to binary values, and averaged over repetitions.Source: Appendix A · p. 7

Appendix B: MATHEMATICAL BACKGROUND OF CRB

The appendix explains how CRB uses representation-theoretic averaging to isolate components of two-qubit noise independently of state preparation and measurement. It describes the chosen groups, the character-weighted projection, and possible uses beyond this experiment.

Appendix B: MATHEMATICAL BACKGROUND OF CRB

  • CRB uses the simultaneous single-qubit Clifford group as its benchmark group and the two-qubit Pauli group as its character group to isolate noise components.The character-group average is weighted by a character function; selecting a Pauli axis singles out a corresponding block of the average noisy operator.Source: Appendix B · p. 7
  • In the Pauli Transfer Matrix picture, CRB's character-group average projects onto a selected Pauli axis, allowing individual noise blocks such as α1|2 to be isolated independently of SPAM.The text identifies σz ⊗ I as the projection axis for α1|2.Source: Appendix B · p. 7
  • For the σz ⊗ I projector, multiplying the averaged operator by the projector yields α1|2^m times that projector.This links the selected noise parameter to the decay used in the CRB measurement.Source: Appendix B · p. 7
  • Projectors onto I ⊗ σz and σz ⊗ σz similarly provide estimates of α2|1 and α12, with corresponding character functions listed in Table I.The appendix also notes potential CRB use with native two-qubit gates, where conventional Clifford compilation is cumbersome and data can contain mixed exponential decays.Source: Appendix B · p. 8, Table I · p. 8

Appendix C: EXPERIMENTAL DETAILS FOR CRB

The experimental details explain how the single-qubit Clifford set is adapted to software-defined Z rotations. The compilation keeps Clifford durations equal, and Table II compares X/Y- and X/Z-based implementations.

Appendix C: EXPERIMENTAL DETAILS FOR CRB

  • Software reference-frame changes make Z rotations error-free, motivating a single-qubit Clifford generating set that uses X and Z rotations.Source: Appendix C · p. 8
  • Using X-Z compilation with a common Rabi frequency for X rotations keeps all 24 single-qubit Clifford operations the same duration, avoiding unnecessary idle time.An X(π) gate therefore lasts twice as long as an X(π/2) gate.Source: Appendix C · p. 8
  • Table II compares single-qubit Clifford compilations using X/Y rotations and X/Z rotations, with K(±π/2) and K(±π) denoting the listed rotation angles.Source: Table II · p. 8

Appendix D: Comparison of standard and character interleaved two-qubit RB

The appendix notes that interleaved-RB gate estimates are exact only for depolarizing noise and otherwise provide bounds. In a hypothetical comparison, CRB gives a tighter worst-case fidelity guarantee than standard two-qubit interleaved RB.

Appendix D: Comparison of standard and character interleaved two-qubit RB

  • An interleaved-gate fidelity estimate is exact only under exactly depolarizing qubit noise; with other noise types, it gives upper and lower bounds.Source: Appendix D · p. 8
  • Those bounds scale as O(√1 − αref) and depend strongly on reference-sequence fidelity, so higher-fidelity references tighten them.The appendix says CRB's single-qubit-only reference has an advantage over standard two-qubit interleaved RB on this point.Source: Appendix D · p. 8
  • With a hypothetical two-qubit reference fidelity of 82% and assumed interleaved CPhase fidelity of 92%, standard interleaved RB guarantees [0.58, 1], versus CRB's [0.69, 1].The comparison assumes multiplicative error scaling for the hypothetical standard experiment, described as optimistic given possible calibration errors.Source: Appendix D · p. 9
  • Assuming reference-gate noise is dominated by stochastic rather than coherent miscalibration errors can make the bounds significantly tighter.The authors argue that single-qubit reference gates typically suffer less from calibration errors than compiled two-qubit gates, favoring sharper interleaved-CRB bounds when more is known about the noise.Source: Appendix D · p. 9

Appendix E: Interleaved RB projected in single-qubit space

Appendix E reports CPhase fidelity projected into single-qubit space and presents results for four cases defined by target and control qubits. Figure 5 describes the interleaved randomized-benchmarking measurements used for this projection.

Appendix E: Interleaved RB projected in single-qubit space

  • The table covers CZ fidelities for different target qubits and control-qubit states.The accompanying text says the results are summarized for four possible cases.Source: Table III · p. 9, Appendix E · p. 9
  • Figure 5 plots outcome-0 probability against the number of single-qubit Clifford operations, with CPhase interleaved.Each data point averages 30 random sequences, each repeated 100 times; dashed curves are single-exponential fits.Source: Figure 5 · p. 9
  • With Q2 in |0⟩ or |1⟩, Q1 is expected to undergo identity or Z(π), respectively; panel (b) shows analogous data for Q2.The caption distinguishes the Q2 conditions with red circles and blue squares.Source: Figure 5 · p. 9

Appendix F: CROSSTALK

Appendix F describes microwave-drive cross-talk: driving Q1 shifts Q2’s frequency, and Q2 Rabi oscillations are measured with and without simultaneous Q1 drive.

Appendix F: CROSSTALK

  • Q2’s frequency shifts by about 2 MHz when Q1 is driven, with the shift depending on Q1 drive power.The text says the known shift is compensated by shifting Q2’s drive frequency.Source: Appendix F · p. 9
  • The Q2 Rabi measurement plots spin-up probability against resonant microwave-burst duration.A second panel shows the analogous measurement while Q1 is driven simultaneously.Source: Figure 6 · p. 10
  • The measurement setup compares Q2 Rabi oscillations with and without simultaneous Q1 drive.The caption identifies the two conditions but does not state their outcome.Source: Figure 6 · p. 10
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