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Time-domain characterization and correction of on-chip distortion of control pulses in a quantum processor

M. A. Rol, L. Ciorciaro, F. K. Malinowski, B. M. Tarasinski, R. E. Sagastizabal, C. C. Bultink, Y. Salathe, N. Haandbaek, J. Sedivy, L. DiCarlo

arXiv:1907.04818v2quant-ph

TL;DR

Distortion in flux-control pulses can impair gate fidelity and repeatability through leakage and pulse-history dependence. Cryoscope uses Ramsey-style measurements to sample on-chip pulses and measure and correct the control-line step response. The authors report a reconstructed step response accurate to ∼0.1%, with corrections applied in real time.

  • Problem

    On-chip pulse distortions can impair gate fidelity and repeatability, causing leakage from the computational subspace and making pulse action depend on prior flux pulses.

    Source: p. 1

  • Method

    Cryoscope uses the qubit’s nonlinear frequency response and Ramsey-style experiments to sample on-chip flux pulses, measure the control-line step response, and apply real-time digital-filter predistortion.

    Source: pp. 1–3

  • Results

    The corrected, reconstructed step response was accurate to ∼0.1%.

    Source: p. 4

  • Takeaways & Limitations

    The demonstrated method characterizes and corrects on-chip flux-pulse distortion, with corrections implemented in real time.

    Source: p. 4

    In noiseless numerical analysis, reconstruction inaccuracy was small for a physically motivated step response but significant for a single-pole low-pass filter on its timescale.

    Source: §II · p. 6

Abstract

from arXiv · show

We introduce Cryoscope, a method for sampling on-chip baseband pulses used to dynamically control qubit frequency in a quantum processor. We specifically use Cryoscope to measure the step response of the dedicated flux control lines of two-junction transmon qubits in circuit QED processors with the temporal resolution of the room-temperature arbitrary waveform generator producing the control pulses. As a first application, we iteratively improve this step response using optimized real-time digital filters to counter the linear-dynamical distortion in the control line, as needed for high-fidelity, repeatable one- and two-qubit gates based on dynamical control of qubit frequency.

SUPPLEMENTAL MATERIALS

The supplemental materials outline experimental setup, Cryoscope details, signal-to-noise modeling, real-time filter implementations, and arbitrary-signal reconstruction.

SUPPLEMENTAL MATERIALS

SUPPLEMENTAL MATERIAL FOR “TIME-DOMAIN CHARACTERIZATION AND CORRECTION OF ON-CHIP DISTORTION OF CONTROL PULSES IN A QUANTUM PROCESSOR”

The supplement supports the main-text claims with experimental details and derivations, covering Cryoscope limitations, signal-to-noise modeling, real-time filters, and arbitrary-signal reconstruction.

SUPPLEMENTAL MATERIAL FOR “TIME-DOMAIN CHARACTERIZATION AND CORRECTION OF ON-CHIP DISTORTION OF CONTROL PULSES IN A QUANTUM PROCESSOR”

I. DEVICE AND EXPERIMENTAL SETUP

The experiments used two devices in separate dilution refrigerators and an AWG with real-time digital filters to generate flux pulses.

I. DEVICE AND EXPERIMENTAL SETUP

  • Data were acquired on two devices mounted in different dilution refrigerators, using a Zurich Instruments HDAWG with real-time digital filters.Source: §I · p. 6
  • The AWG output passed through a bias tee, while the flux line included 4 K attenuation and low-pass and eccosorb filters.Source: §I · p. 6
  • The two devices used different coaxial cable materials between 4 K and the mixing-chamber plate: superconducting NbTi in one and stainless steel in the other.Source: §I · p. 6
  • Distortions were corrected to within ∼0.1% in all cases.Source: §I · p. 6

II. LIMITATIONS OF THE CRYOSCOPE

The supplement evaluates reconstruction accuracy through noiseless numerical calculations of the acquired relative phase and considers physically motivated and single-pole-filter step responses.

II. LIMITATIONS OF THE CRYOSCOPE

  • For a physically motivated step response incorporating AWG bandwidth, bias tee, skin effect, and on-chip response, the modeled reconstruction inaccuracy is small using typical distortion parameters.Source: §II · p. 6
  • A single-pole low-pass filter produces significant reconstruction error on the timescale of that filter.The phase model uses square-pulse amplitude Φ_s, step response s(t), and a quadratic flux-to-detuning relation Δf_Q = aΦ^k with k = 2.Source: §II · p. 6
  • The analysis numerically calculates the acquired relative phase, yielding a noiseless Cryoscope measurement.Source: §II · p. 6

A. Cryoscope reconstruction of a typical step response

Cryoscope reconstructs a modeled, distorted step response by separating pulse and turn-off contributions, with the reconstructed flux matching the true flux to better than 1% after 3 ns.

A. Cryoscope reconstruction of a typical step response

  • Table S1 summarizes distortion models and coefficients, while the HDAWG response is derived from its measured step response and impulse response.The HDAWG step response is filtered to determine the derivative and impulse response.Source: A. Cryoscope reconstruction of a… · p. 6, Table S1 · p. 7, Figure S1 · p. 7
  • The reconstructed flux matches the true flux to better than 1% for t > 3 ns in the modeled response.The square pulses detune the qubit by 800 MHz; the difference between turn-off contributions is typically ≲8 MHz, about 1% of the detuning, except for the first few samples.Source: A. Cryoscope reconstruction of a… · p. 7
  • Individual turn-off transients can exceed the true-detuning contribution, but their difference is substantially smaller in the modeled case.The qubit takes multiple sampling intervals to return to the sweetspot after pulse turn-off.Source: A. Cryoscope reconstruction of a… · p. 7

B. Cryoscope reconstruction of a single-pole low-pass filter step response

For a single-pole low-pass response, Cryoscope can reconstruct the step response poorly; reconstruction accuracy improves with stronger nonlinear dependence of qubit frequency on flux.

B. Cryoscope reconstruction of a single-pole low-pass filter step response

  • The reconstructed single-pole low-pass step response differs from the true response by more than 1% for t ≲ 4τLP.This filter is analytically convenient but does not accurately represent the experimental setup.Source: B. Cryoscope reconstruction of a… · p. 7
  • For linear frequency dependence on flux, the reconstruction gives u(t) and misses the dynamics, whereas it matches the filter response as k → ∞.The comparison considers detuning forms Δf(Φ) = aΦ^k.Source: B. Cryoscope reconstruction of a… · pp. 7–8
  • Higher powers of flux dependence improve accuracy because they suppress turn-off contributions to the phase difference ε.With linear dependence, turn-off contributions are weighted as strongly as contributions during the pulse.Source: B. Cryoscope reconstruction of a… · p. 8

III. CRYOSCOPE SIGNAL-TO-NOISE RATIO

Cryoscope models the measured quadratures as noisy observations and uses amplitude and phase changes during a square pulse to reconstruct flux.

III. CRYOSCOPE SIGNAL-TO-NOISE RATIO

  • The measured values (x,y), given true values (x0,y0), are assumed normally distributed with standard deviation σ.The figure depicts σ as a grey circle.Source: §III · p. 9, Figure S3 · p. 9
  • For an ideal square pulse, qubit dephasing affects the measured amplitude r, while the phase accumulates according to the detuning.The phase is ϕτ = 2πτ∆fQ(ΦQ), with flux-dependent dephasing rate and α ∈ [1,2].Source: §III · p. 9
  • Cryoscope uses the phase difference between measurements at τ and τ+∆τ to reconstruct flux ΦR, with measurement error σ∆ϕ.The phase error σϕ depends on r and σ.Source: §III · p. 9

A. Error propagation

The error-propagation derivation rewrites the measurement distribution in polar form and simplifies the phase-error calculation under stated visibility conditions.

A. Error propagation

  • Setting ϕ0=0 also sets y0=0 before rewriting p(x,y) in spherical coordinates.The derivation then drops the subscript from rτ and ϕτ for brevity.Source: A. Error propagation · p. 9
  • Integrating the distribution gives the phase error, which simplifies when the Ramsey visibility x0 is much larger than σ.Source: A. Error propagation · p. 9
  • With ϕ0=0 and x0 much smaller than σ, the derivation additionally applies the small-angle approximation ϕ≪1 and concludes that the resulting distribution is normal.Source: A. Error propagation · p. 10

B. SNR formula

The SNR model combines flux-dependent detuning and dephasing with phase-measurement noise and filtering effects; the measured detuning dependence agrees with a quadratic model in the studied range.

B. SNR formula

  • Cryoscope SNR depends on flux-dependent qubit dephasing, the phase-acquisition rate proportional to ∆fQ(ΦQ), and filtering effects.Filtering affects SNR linearly.Source: B. SNR formula · p. 10
  • The phase-measurement noise contributes to the SNR, but its standard deviation σ cannot be distinguished from the filtering factor c′.The model absorbs their multiplicative effects.Source: B. SNR formula · p. 10
  • The quadratic detuning model matches the measured dependence up to 0.33 Φ0.The figure shows measured detuning versus applied SQUID-loop flux.Source: B. SNR formula · p. 10, Figure S4 · p. 11
  • The dephasing model uses Γ0 for flux-independent dephasing and Γ1 for the contribution from 1/f flux noise, with α set to 1.Source: B. SNR formula · p. 10
  • The joint fit uses Γ0 = 66.7 · 10^-3 s^-1, corresponding to T*2 = 15 µs, and yields Γ1 = 0.213×10^-3 Φ0 and √AΦ = 12 × 10^-6 Φ0.Source: B. SNR formula · p. 10

IV. REAL-TIME PREDISTORTION FILTERS

  • Real-time distortion correction uses an FIR filter for short-timescale (< 30 ns) distortions and a first-order IIR filter.Source: §IV · p. 11

A. FIR filter

The FIR filter convolves its input signal with a coefficient-defined impulse response, using a configurable set of coefficients.

A. FIR filter

  • The FIR filter implements convolution with an impulse response specified by coefficients bi.Source: A. FIR filter · p. 11
  • The filter’s input x[n] is the signal at time n, and its output y[n] is the signal at time n.Source: A. FIR filter · p. 11
  • Forty parameters set the first 72 coefficients, corresponding to a 30 ns filter at 2.4 GSa/s.The first eight parameters directly set eight coefficients; the remaining 32 set pairs.Source: A. FIR filter · p. 11

B. Exponential over- and under-shoot correction IIR filter

The section describes a first-order IIR correction for exponential step-response over- or undershoot, filter modeling, and Cryoscope’s reconstruction of arbitrary flux waveforms.

B. Exponential over- and under-shoot correction IIR filter

  • A first-order IIR filter is designed to correct exponential over- or undershoot in the step response, using coefficients set by the response parameters.The step-response model uses τIIR as its time constant and A as its amplitude; g is an ignored gain correction factor.Source: B. Exponential over- and under-shoot… · p. 11
  • Directly implementing the filter recursion is infeasible at 2.4 GSa/s, so hardware averages 16 samples and updates its state every eighth sample.The downsampled state is combined with the input at the full sampling rate, and these approximations are included in modeling the filter response.Source: B. Exponential over- and under-shoot… · p. 12

C. Modelling filters

  • SciPy’s lfilter predicts real-time predistortion effects while accounting for deviations between hardware-modified and ideal filter operation.The function applies filters defined by coefficient vectors b and a to an input signal.Source: C. Modelling filters · p. 12

V. USING CRYOSCOPE TO MEASURE ARBITRARY SHAPES

  • Cryoscope reconstructs the arbitrary canal-skyline flux waveform near perfectly, with no free parameters.Figure S5 shows the reconstruction.Source: §V · p. 12, Figure S5 · p. 12

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