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Efficient Z-Gates for Quantum Computing
David C. McKay, Christopher J. Wood, Sarah Sheldon, Jerry M. Chow, Jay M. Gambetta
TL;DR
The paper addresses how to exploit virtual Z-gates for efficient superconducting-qubit control while correcting pulse errors and leakage. It combines phase-controlled Z rotations with benchmarking and DRAG-based pulse strategies. The approach yields lower Clifford error and a 13.3 ns DRAGZ Xπ/2 gate with 1.95(3)×10^-4 EPG and 3.1(6)×10^-6 LPG.
Problem
The paper studies how virtual Z-gates can improve circuit fidelity and correct unitary rotation errors and leakage in weakly anharmonic superconducting qubits.
Method
The paper controls microwave-drive phase to implement virtual Z-gates, benchmarks an HZ Clifford basis, and combines Z corrections with DRAG pulse shaping.
Results
1.95(3)×10^-4 EPG and 3.1(6)×10^-6 LPG were measured for a 13.3 ns DRAGZ pulse.
Takeaways & Limitations
Virtual Z-gates provide a low-overhead way to construct arbitrary SU(2) gates, reduce Clifford error, and correct rotation errors without adding gate duration.
Abstract
from arXiv · showhide
For superconducting qubits, microwave pulses drive rotations around the Bloch sphere. The phase of these drives can be used to generate zero-duration arbitrary "virtual" Z-gates which, combined with two $X_{π/2}$ gates, can generate any SU(2) gate. Here we show how to best utilize these virtual Z-gates to both improve algorithms and correct pulse errors. We perform randomized benchmarking using a Clifford set of Hadamard and Z-gates and show that the error per Clifford is reduced versus a set consisting of standard finite-duration X and Y gates. Z-gates can correct unitary rotation errors for weakly anharmonic qubits as an alternative to pulse shaping techniques such as DRAG. We investigate leakage and show that a combination of DRAG pulse shaping to minimize leakage and Z-gates to correct rotation errors (DRAGZ) realizes a 13.3~ns $X_{π/2}$ gate characterized by low error ($1.95[3]\times 10^{-4}$) and low leakage ($3.1[6]\times 10^{-6}$). Ultimately leakage is limited by the finite temperature of the qubit, but this limit is two orders-of-magnitude smaller than pulse errors due to decoherence.
I. THEORY
The paper explains how microwave phase control implements zero-duration virtual Z-gates and uses them with Xπ/2 gates to construct arbitrary SU(2) operations. It also connects this construction to the physical control chain from waveform generation through IQ mixing.
- Physical implementation: The AWG programs I/Q waveforms, and the IQ mixer combines them with a local oscillator to produce the microwave drive.The drive envelope and frequency are set through AWG voltages and single-sideband modulation.
- Error sources: For transmon qubits, higher-level coupling produces leakage and unitary errors that arise during shaped microwave driving.The oscillator’s |0⟩ and |1⟩ levels define the qubit, while higher levels generate the discussed errors.
- Virtual Z-gates: Virtual Z-gates change subsequent drive phases, implementing Z rotations with zero duration through software-defined axis redefinition.The phase offset applies to later single- and two-qubit drives, while the gate is effectively perfect in the stated control model.
- Universal single-qubit control: Two Xπ/2 gates combined with virtual Z-gates can construct any arbitrary SU(2) gate.The paper gives U(θ, φ, λ) = Zφ−π/2 · Xπ/2 · Zπ−θ · Xπ/2 · Zλ−π/2 as one decomposition.
- Algorithmic relevance: Efficient Z-gate synthesis is presented as essential for universal quantum algorithms.Common SU(2) gates are tabulated in a Z-gate notation.
II. RANDOMIZED BENCHMARKING OF THE Z-GATE
The randomized-benchmarking experiment compares Clifford constructions using finite-duration X/Y pulses with a Hadamard-plus-Z basis. The latter reduces error per Clifford because most of its gates are near-perfect virtual Z-gates rather than finite-duration rotations.
- Basis sets: The XY π/2 basis uses four finite-duration X/Y rotations, while the HZ basis uses a finite-duration Hadamard gate plus virtual Z-gates.The average Clifford requires 2.25 gates from XY π/2 and 2.4583 gates from HZ, but only one HZ gate per Clifford is finite-duration.
- Interpretation: The lower HZ error per Clifford is attributed to requiring fewer finite-duration gates in each Clifford representation.The experiment uses DRAG pulses for each finite-duration gate.
- Pulse calibration: The DRAG parameter β optimizes fidelity by cancelling Stark-shift errors, but the β that minimizes leakage need not maximize fidelity.This separates pulse-fidelity optimization from leakage minimization.
- Randomized benchmarking: 3.0(1)×10^-4 EPC was measured for the HZ set, versus 5.6(1)×10^-4 for the XY π/2 set.The corresponding EPG values were 1.22(4)×10^-4 and 2.48(5)×10^-4, respectively.
- Virtual-Z characterization: Interleaved benchmarking of the S gate measured an error of -1.7(1.0)×10^-5, consistent with zero error within systematic bounds.The stated systematic-error bounds are [0,6×10^-4].
III. CORRECTING ERRORS WITH VZ-GATES
Virtual Z-gates correct off-resonance rotation errors and reduce randomized-benchmarking error by replacing finite-duration rotations with phase-based corrections.
- Off-resonance rotation correction: VZ-gates correct most off-resonance-rotation errors by applying an appropriate phase-based Z correction.The correction is implemented as an axes rotation without adding finite-duration pulse errors.
- Correction boundary: A valid VZ correction exists only when the detuned rotation can reach the plane containing the desired final state.A sufficiently detuned π pulse cannot be compensated because a qubit starting in |0⟩ does not complete the rotation to |1⟩.
- Pulse implementation: GZ pulses use Gaussian pulses plus VZ corrections to compensate ORR errors and remain viable across pulse-shape variations and sideband frequencies.The calibrated phase compensates passive DRAG shaping, while pulse shaping remains available for other errors such as leakage.
IV. LEAKAGE
Leakage to higher transmon levels is measured alongside gate error, and combining leakage-optimized DRAG shaping with VZ correction yields short, low-error pulses.
- Leakage mechanism: Leakage is driven mainly by pulse-frequency components near the |1⟩→|2⟩ transition, while pulse shaping can mitigate it.The transition frequency is ω12 = ω01 + α.
- Measurement: Leakage measurements simultaneously estimate |0⟩, |1⟩, and |2⟩ populations using thresholded single-shot readout and randomized-benchmarking fits.The leakage-per-gate rate is extracted from the |2⟩ population decay, with long-sequence |2⟩ population serving as a calibration proxy.
- Pulse comparison: For pulses shorter than 20 ns, DRAGZ and FILTZ show nearly an order-of-magnitude lower leakage-per-gate rates than the other tested pulse types.DRAG and GZ otherwise show similar leakage trends, generally decreasing for longer pulses because of reduced Fourier broadening.
- Design trade-offs: Replacing DRAG with GZ preserves EPG while allowing DRAG pulse shaping to target leakage specifically in DRAGZ pulses.FILTZ obtains the lowest LPG but works only for specific sideband frequencies; leakage is ultimately limited by finite temperature.
V. THE VZ-GATE IN MULTIQUBIT SYSTEMS
Virtual-Z phase updates can be incorporated into microwave-activated two-qubit gates, but their compatibility depends on the interaction implementation. Flux-tunable interactions require special treatment, whereas ZZ interactions permit compensation with a subsequent virtual Z-gate.
- Microwave-activated gates: Microwave-activated two-qubit interactions inherit the single-qubit drive phase, so virtual-Z updates must be applied consistently to subsequent interactions.For cross-resonance, the CR-drive phase must be updated; for iSWAP, the drive phase is matched to the difference of the qubit-drive phases.
- Microwave-activated gates: Cross-resonance gates require updating the CR-drive phase when a virtual Z-gate is applied to qubit 2.
- Microwave-activated gates: Parametric iSWAP gates require matching the iSWAP-drive phase to the difference between the relevant qubit phases.
- Flux-tunable interactions: Flux-tunable interactions are harder to combine with virtual-Z gates because time-dependent single-qubit Z terms may not commute with the interaction.
- Flux-tunable interactions: For ZZ interactions, the single-qubit Z terms commute through the interaction and can be compensated by a subsequent virtual Z-gate.
VI. CONCLUSIONS
The paper investigates near-perfect virtual Z-gates implemented through microwave-drive phase control. It uses them to reduce circuit and pulse errors while combining pulse shaping for leakage reduction, achieving a hybrid gate whose leakage is temperature-limited and fidelity is coherence-limited.
- Virtual Z-gates are implemented by controlling the phase of microwave drives used for X and Y rotations.
- Virtual Z-gates can improve circuits with many single-qubit gates, correct typical gate errors, and implement arbitrary SU(2) gates with a calibrated Xπ/2 gate.
- Combining pulse shaping for leakage reduction with virtual Z-gates for rotation-error correction produces a hybrid pulse with temperature-limited leakage and coherence-limited gate fidelity.
- Further leakage improvement is limited by the qubit temperature.