Source-linked AI summary
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.
Source: §III · p. 4Method
The paper controls microwave-drive phase to implement virtual Z-gates, benchmarks an HZ Clifford basis, and combines Z corrections with DRAG pulse shaping.
Source: §I · p. 3, §II · p. 4, §VI · p. 7Results
1.95(3)×10^-4 EPG and 3.1(6)×10^-6 LPG were measured for a 13.3 ns DRAGZ pulse.
Source: §IV · p. 6Takeaways & 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.
Source: §II · p. 4, §III · p. 4, §VI · p. 7
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.Source: §I · p. 2
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.Source: §I · p. 2
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.Source: §I · p. 3
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.Source: §I · p. 3
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.Source: §I · p. 3, Table I · p. 3
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.Source: §II · p. 3
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.Source: §II · p. 3
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.Source: §II · p. 4
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.Source: §II · p. 4
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].Source: §II · p. 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.Source: §III · pp. 4–5
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⟩.Source: §III · p. 4
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.Source: §III · p. 5
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.Source: §IV · p. 5
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.Source: §IV · p. 6
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.Source: §IV · p. 6
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.Source: §IV · p. 6
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.Source: §V · p. 7
- Cross-resonance gates require updating the CR-drive phase when a virtual Z-gate is applied to qubit 2.Source: §V · p. 7
- Parametric iSWAP gates require matching the iSWAP-drive phase to the difference between the relevant qubit phases.Source: §V · p. 7
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.Source: §V · p. 7
- For ZZ interactions, the single-qubit Z terms commute through the interaction and can be compensated by a subsequent virtual Z-gate.Source: §V · p. 7
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.Source: §VI · p. 7
- 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.Source: §VI · p. 7
- 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.Source: §VI · p. 7
- Further leakage improvement is limited by the qubit temperature.Source: §VI · p. 7