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The impact of classical control electronics on qubit fidelity

Jeroen P. G. van Dijk, Erika Kawakami, Raymond N. Schouten, Menno Veldhorst, Lieven M. K. Vandersypen, Masoud Babaie, Edoardo Charbon, Fabio Sebastiano

arXiv:1803.06176v1quant-ph

TL;DR

As quantum processors improve, classical-controller non-idealities may limit overall performance, and prior analyses addressed only specific signal effects. This paper analyzes control errors across quantum operations to derive fidelity-targeted electronics specifications, including a 99.9% case study and an assessment of room-temperature equipment.

  • Problem

    Prior work examined specific controller signal effects, but a comprehensive analysis is needed to specify electronics as quantum processors improve and scale.

    Source: §I · pp. 1–2

  • Method

    The paper analyzes static and dynamic controller errors across single-qubit gates, two-qubit gates, and read-out to derive electrical specifications for a target fidelity.

    Source: §I · p. 2, §VIII · p. 20

  • Results

    The analysis provides electrical specifications for targeting 99.9% fidelity; the discussed room-temperature instruments can support that level but are bulky, consume several Watts, and cannot operate cryogenically.

    Source: §VII · p. 19, §VIII · p. 20

  • Takeaways & Limitations

    The specifications support designing tailor-made controllers optimized for performance, power, cost, and size to improve quantum-computer scalability.

    Source: §VIII · p. 20

    Power estimates use room-temperature CMOS as a reference and assume 50-Ω loads, unlike a fully integrated controller; the estimates likely overstate required power.

    Source: §VII · p. 19

Abstract

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Quantum processors rely on classical electronic controllers to manipulate and read out the quantum state. As the performance of the quantum processor improves, non-idealities in the classical controller can become the performance bottleneck for the whole quantum computer. To prevent such limitation, this paper presents a systematic study of the impact of the classical electrical signals on the qubit fidelity. All operations, i.e. single-qubit rotations, two-qubit gates and read-out, are considered, in the presence of errors in the control electronics, such as static, dynamic, systematic and random errors. Although the presented study could be extended to any qubit technology, it currently focuses on single-electron spin qubits, because of several advantages, such as purely electrical control and long coherence times, and for their potential for large-scale integration. As a result of this study, detailed electrical specifications for the classical control electronics for a given qubit fidelity can be derived, as demonstrated with specific case studies. We also discuss the effect on qubit fidelity of the performance of the general-purpose room-temperature equipment that is typically employed to control the few qubits available today. Ultimately, we show that tailor-made electronic controllers can achieve significantly lower power, cost and size, as required to support the scaling up of quantum computers.

I. INTRODUCTION

As quantum processors improve and scale, classical-controller imperfections may become a performance bottleneck, while electronics specifications require a comprehensive fidelity analysis. The paper frames that analysis around spin qubits and the signals used for control and read-out.

I. INTRODUCTION

  • Earlier analyses addressed specific signal effects, while this work examines controller non-idealities across single-qubit gates, two-qubit gates, and read-out.The framework covers frequency-, voltage-, and time-domain inaccuracies, including static, dynamic, systematic, and random errors.Source: §I · p. 2

A. The quantum processor

  • The modeled processor encodes qubits in electron spins in quantum dots, with plunger gates controlling dot potential and barrier gates tuning interdot tunneling.The analysis assumes negligible or compensated crosstalk; qubits may have distinct Larmor frequencies.Source: A. The quantum processor · pp. 2–3

B. The classical electronic controller

  • The controller biases gates, generates microwave drives using a local oscillator and waveform envelope, and processes charge-sensor signals for read-out.Separate waveform generators are assumed for two-qubit gates and read-out because their specifications can differ.Source: B. The classical electronic controller · pp. 3–4

III. THE METHOD OF ANALYZING THE EFFECTS OF SIGNAL NON-IDEALITIES

  • The analysis computes qubit evolution from signal-dependent Hamiltonians, approximates dynamic drives with short time-independent steps, and compares process fidelity with an ideal controller.The time step must be small enough to provide the required approximation accuracy.Source: §III · p. 4

IV. SIGNAL SPECIFICATIONS FOR SINGLE-QUBIT OPERATIONS

  • For single-qubit rotations, microwave frequency matching and phase set the rotation axis, while signal amplitude and duration determine the rotation angle.The rotating wave approximation requires the Rabi frequency to be sufficiently below the Larmor frequency.Source: §IV · p. 4, Figure 4 · p. 5

A. Specifications for the microwave carrier

This section relates microwave-carrier inaccuracies and oscillator noise to qubit fidelity to derive carrier specifications. It treats frequency and phase errors alongside phase noise and wideband additive noise.

A. Specifications for the microwave carrier

  • 3% relative frequency accuracy achieves 99.9% fidelity for a π-rotation, corresponding to 3 kHz at 100 kHz Rabi frequency or 0.3 MHz at 10 MHz.For a given rotation angle, larger Rabi frequency improves fidelity.Source: A. Specifications for the microwave… · p. 5
  • 0.03 rad (1.8°) phase accuracy is required for 99.9% fidelity in a π-rotation, and achieving a given phase accuracy is harder at higher LO frequencies.Higher LO frequencies may be required when increasing the Rabi frequency to satisfy the RWA requirement.Source: A. Specifications for the microwave… · p. 5
  • Larmor-frequency drift can act as a random static error, while oscillator phase noise requires a model that also accounts for wideband additive noise.Low-frequency flicker noise produces an f^-1 frequency-noise roll-off; dynamical decoupling can add high-pass filtering.Source: A. Specifications for the microwave… · pp. 5–6
  • The frequency-noise filter has a low-pass response; its ENBW grows with Rabi frequency, so faster operations expose the qubit to noise over a wider band.For white noise, the analysis favors the highest possible Rabi frequency; for flicker noise, it likewise favors a higher minimum frequency.Source: A. Specifications for the microwave… · p. 5, Figure 5 · p. 6
  • At high offsets, the phase-noise analyzer trace can include additive noise rather than actual phase noise; the example shows a noise floor near -150 dBc/Hz.The actual phase-noise component is bandwidth-limited by the qubit filter.Source: Figure 6 · p. 6, A. Specifications for the microwave… · p. 6
  • Wideband additive noise is evaluated separately and filtered by a low-pass response whose noise spectrum is centered around the qubit frequency.Without additional band-pass filtering, the high-offset additive-noise contribution can make the fidelity integral diverge.Source: A. Specifications for the microwave… · p. 6, Figure 7 · p. 7

B. Specifications for the microwave envelope

The section derives how microwave-envelope errors and qubit frequency multiplexing affect gate fidelity. It identifies amplitude and timing tolerances, noise filtering, and crosstalk constraints, including correction requirements for simultaneous operations.

B. Specifications for the microwave envelope

  • For a π-rotation at 99.9% fidelity, the relative amplitude or duration error must be < 2%.The rotation angle depends on both signal amplitude and duration, so errors in either cause under- or over-rotation.Source: B. Specifications for the microwave… · p. 7
  • Amplitude noise is filtered by an intrinsic low-pass qubit filter, with relevant bandwidth proportional to the nominal Rabi frequency.For white noise, the required signal-to-noise ratio in the qubit’s sensitivity band is fixed for a given fidelity.Source: B. Specifications for the microwave… · p. 7
  • Phase noise is high-pass filtered for duration jitter, with the filter corner set by pulse duration; the resulting jitter contributes to infidelity.The duration variation is estimated using a Gaussian-jitter assumption.Source: B. Specifications for the microwave… · p. 7

C. Requirements for qubit frequency multiplexing

  • Gaussian envelopes do not produce a much faster crosstalk roll-off than rectangular envelopes, and higher drive amplitudes broaden pulse bandwidth and extend crosstalk to farther-separated qubits.FDMA can support simultaneous gates on one line, but requires engineering the applied pulses to compensate for off-resonant Z-rotations during the operations.Source: C. Requirements for qubit frequency… · pp. 8–9
  • Unaddressed-qubit fidelity is mainly limited by unintended Z-rotations, which can be corrected by updating the microwave phase for subsequent operations.The correction is a software update, with fidelity limited by microwave-source phase accuracy.Source: C. Requirements for qubit frequency… · p. 8
  • A target fidelity for an unaddressed qubit sets a minimum frequency separation or, for parasitic drive coupling, a required tone attenuation.For a rectangular envelope, the fidelity lower bound varies with rotation angle.Source: C. Requirements for qubit frequency… · p. 9

D. Specifications for the idle operation

Idle periods can degrade qubit fidelity through frequency mismatch, residual driving, and thermal noise, while the case study translates fidelity targets into electronics specifications. The two-qubit analysis uses exchange interaction controlled by tunnel coupling and detuning to set gate speed and type.

D. Specifications for the idle operation

E. Case study of the specifications for a single-qubit operation

  • The single-qubit case study allocates electronics error budgets for 99.9% fidelity at a 1 MHz π-rotation and for preserving the idle state over an equal-duration interval.It assumes simple rectangular pulses without echo; specifications are stated at the gate, so wiring attenuation and filtering may need to be included when referring them to the electronics.Source: E. Case study of the specifications… · pp. 9–10

V. SIGNAL SPECIFICATIONS FOR TWO-QUBIT OPERATIONS

  • For two neighboring qubits, finite tunnel coupling produces an avoided crossing at |ϵ| = U, and the resulting energy-level changes form the basis of two-qubit operations.Source: §V · pp. 10–11, Figure 12 · p. 11
  • Increasing tunnel coupling or detuning raises the exchange interaction; its value sets two-qubit operation speed, while adiabatic or diabatic control yields C-phase, exchange, or mixed gates.Operation at zero detuning is possible by controlling tunnel coupling alone.Source: §V · pp. 10, 12

The C-Phase Gate

The paper derives how exchange and C-phase gates respond to control errors, then specifies accuracy and noise requirements for gate and idle operation. The requirements depend on operating point, with charge-symmetry operation especially tolerant of detuning error.

The Exchange Gate

  • An exchange gate uses a rapid, diabatic control change and requires δω0 ≪ ωop; θJ = π gives a SWAP, limiting FDMA use for single-qubit operations.The gate depends on the qubits’ Larmor frequencies, tunnel coupling, charging energy, detuning, and operation duration.Source: The Exchange Gate · pp. 12–13

The C-Phase Gate

A. Specifications for an exchange and C-phase gate

  • For a 99.9% π gate, Δϵ/U must be <0.023% at ϵ≈0.99U, <0.25% at ϵ≈0.9U, or <15% at charge symmetry; timing and tunnel-coupling accuracy are ~2.3% and ~1.2%.Detuning-error infidelity is fourth-order at ϵ = 0, but quadratic otherwise; low-frequency Gaussian variations follow the static-error formulas with standard deviation replacing inaccuracy.Source: A. Specifications for an exchange and… · p. 13, Table II · p. 14
  • At high frequencies, signal noise near allowed transitions can affect the state; exchange gates permit only limited filtering because their control change is diabatic.Simulations find sensitivity to noise above ωop only within a limited band set by operation duration.Source: A. Specifications for an exchange and… · p. 13

B. Specifications for the idle operation

  • For a 99.9%-fidelity identity lasting ten nominal π-gate times, interaction strength must fall by ~430×, achievable by reducing tunnel coupling ~21× at unchanged detuning.Operating at finite detuning can instead turn the interaction off at zero detuning, but requires operation at least at 99.9% of U, near the avoided crossing where detuning tolerance is reduced.Source: B. Specifications for the idle operation · pp. 13–14

C. Case study of the specifications for a two-qubit operation

The two-qubit case studies compare electronics specifications for C-phase gates at zero and finite detuning. The read-out analysis separates spin-to-charge conversion, charge sensing, and signal discrimination, and examines how conversion depends on operating conditions.

C. Case study of the specifications for a two-qubit operation

  • Both C-phase examples target 99.9% fidelity; finite detuning raises the example operation speed from 2 MHz to 20 MHz but makes detuning-error specifications more than 100 times stricter and noise-power requirements almost five orders of magnitude different.The finite-detuning example uses detuning at 95% of U; moving closer to the avoided crossing increases speed but makes specifications increasingly challenging.Source: C. Case study of the specifications… · p. 14
  • In the Table IV example, detuning control can use an AWG at 1 GS/s for maximum timing inaccuracy of 0.5 ns; covering −U to U requires 10-bit resolution to meet detuning-energy accuracy.Source: C. Case study of the specifications… · p. 14

VI. SIGNAL SPECIFICATIONS FOR QUBIT READ-OUT

  • Read-out fidelity combines correct spin-to-charge projection, charge sensing, and circuit discrimination; the analysis excludes T1 relaxation and does not further quantify sensor-dependent Psense errors.The assumed setup measures the Pauli spin-blockade charge state with a charge sensor.Source: §VI · p. 15

A. Specifications for the electronics controlling the spin to charge conversion

B. Specifications for the electronics processing the read-out signal

The paper derives read-out electronics requirements and uses case studies to assess both existing instruments and cryogenic CMOS controllers against a 99.9% fidelity target. It finds cryogenic CMOS appears feasible, while power trade-offs matter for scaling.

B. Specifications for the electronics processing the read-out signal

  • For 99.9% correct discrimination, the read-out signal should be about 6 times the noise standard deviation under Gaussian noise and matched-filter detection.The matched filter integrates the signal current over Tread and compares it with a threshold; for white noise, ENBW = 1/(2Tread).Source: B. Specifications for the electronics… · p. 17

C. Case study of the specifications for a qubit read-out

  • In the direct-QPC example, Tread = 0.6 µs gives SNR 46 and Pdetect = 99.967% for F = 99.9%.The case assumes Is = 400 pA, circuit noise about half the QPC shot-noise level, and a typical EST of 50 µeV; the optimum detuning is 83.2 mV. The specifications also require an approximately 18× tunnel-coupling tuning range and a 9-bit detuning AWG.Source: C. Case study of the specifications… · pp. 17–18

VII. DISCUSSION

  • The compared general-purpose instruments can support 99.9% fidelity, but their bulk, multi-watt power use, and inability to operate cryogenically hinder scalability.The cited AWG also meets the stated microwave-envelope, two-qubit detuning, and read-out specifications.Source: §VII · p. 19
  • A 10-bit 500 MS/s DAC meets the microwave-envelope specification at 24 mW, while a 12-bit 1.6 GS/s DAC meets detuning requirements at 40 mW.A CMOS PLL covering 9.2–12.7 GHz is estimated at about 13 mW, with phase noise slightly worse than required at room temperature.Source: §VII · p. 19
  • Without multiplexing, the estimated controller power is 120 mW/qubit; sharing a 53 mW fast DAC and PLL across 64 qubits brings that component below 1 mW/qubit.The estimates use room-temperature CMOS references and assume 50-Ω loads; the authors note cryogenic CMOS may require less power and higher-impedance gates could reduce DAC consumption.Source: §VII · p. 19
  • A cryogenic CMOS controller for a large-scale processor appears feasible at a 99.9% target fidelity, but minimizing power requires systematic trade-off analysis.The analysis provides a basis for optimizing fidelity against controller power.Source: §VII · p. 19, §VIII · p. 20

Appendix A: General Derivations

The appendix sets out how process fidelity is derived and averaged over random errors, then applies those methods to numerical simulations and single-qubit control inaccuracies.

Appendix A: General Derivations

  • Process fidelity is obtained from unitary operations using its relation to average gate fidelity, and random-parameter fidelity is averaged over the parameter’s probability density.For zero-mean Gaussian errors, the appendix gives a simple expected-fidelity solution for certain quadratic forms such as F(x) = 1 − cx^4.Source: Appendix A · p. 20

1. Numerical Simulations

  • Gaussian-envelope microwave pulses are simulated by finite time steps, with 10× carrier-frequency oversampling reported as accurate.The time step is constant and chosen sufficiently small not to affect simulation results.Source: §1 · p. 21

Appendix B: Derivations for Single-Qubit Operation

  • For single-qubit rotations, the ideal unitary is derived from the rotating-frame Hamiltonian and used as the reference for process fidelity.The rotation angle is θ = ωR T, and the rotation axis is φ.Source: Appendix B · p. 21

1. Inaccuracies

  • Frequency, phase, amplitude, and duration inaccuracies are each related to fidelity loss through relative-error expressions and Taylor expansions.The duration error is relative to Tideal; phase error is ∆φ, while frequency, amplitude, and duration errors are normalized to their respective control parameters.Source: §1 · p. 21

2. Quasi-static Noise

This subsection gives expected-fidelity treatments for quasi-static Gaussian variations in microwave phase, amplitude, and timing, building on fidelity relations for random errors.

2. Quasi-static Noise

  • For Gaussian-distributed inaccuracies with fidelity form F(x) = 1 − cx^2, expected fidelity can be evaluated from the Gaussian-noise result; exact formulas are more accurate for large noise where available.The exact formula is unavailable for microwave frequency inaccuracy in the cases described.Source: §2 · p. 22
  • The appendix specifies expected-fidelity expressions for Gaussian microwave phase noise, with α defined as the phase standard deviation σφ.Source: §2 · p. 22
  • Expected-fidelity expressions are also given for Gaussian amplitude and timing variations, with α defined respectively as relative amplitude deviation and σT/Tideal.Source: §2 · p. 22

3. Noise Filtering

The noise-filtering derivation represents control noise with a generalized noise Hamiltonian, relates noise spectra to fidelity through control-dependent filter functions, and treats additive wideband noise separately.

3. Noise Filtering

  • In first-order approximation, expected process fidelity is expressed using a generalized noise Hamiltonian and control matrices that depend on the control propagator.The total Hamiltonian is written as a noise-free control term plus βx(t), βy(t), and βz(t) noise terms.Source: §3 · p. 22
  • For single-qubit operations, microwave-frequency and envelope fluctuations are treated as statistically independent noise sources with separate power spectral densities and filter responses.The resulting expressions evaluate expected fidelity for amplitude and microwave-frequency noise.Source: §3 · pp. 22–23
  • Wideband additive noise maps to transverse noise terms modulated by the Larmor-frequency sine and cosine, which shape its spectrum before expected fidelity is evaluated.Source: §3 · p. 23

4. Frequency Multiplexing

The section analyzes unintended effects of drive signals on other qubits and on qubits during idle periods. It relates these errors to drive conditions and noise filtering.

4. Frequency Multiplexing

  • Driving one qubit can also affect another; the residual infidelity includes X, Y, and Z-rotation contributions, with Z rotations removable by a software reference-frame update.For the other qubit, the residual infidelity resembles the power spectrum of a rectangular microwave envelope.Source: §4 · pp. 23–24
  • The other-qubit drive amplitude may differ from the intended qubit’s, and the stated approximation applies when α^2 ≫ β^2, including sufficient qubit spacing and ωR,other ≤ ωR.The intended rotation is θ = ωRT over duration T.Source: §4 · pp. 23–24

5. Idle Gate

  • Residual noise on the drive line can affect idle qubits, and its response is described as a sinc-shaped band-pass filter centered at the qubit frequency.At zero frequency, the response is described as a low-pass filter with effective noise bandwidth ωRπ/|θ| and DC gain θ^2/2.Source: §5 · p. 24
  • During an idle operation, frequency inaccuracy causes qubit rotation, while an unintended residual spurious tone also reduces identity-operation fidelity.The passages give fidelity expressions and Taylor-series expansions for both effects.Source: §5 · p. 24

Appendix C: Derivations for Two-Qubit Operation

Appendix C derives two-qubit operation results from a Hamiltonian and analyzes its relevant eigenenergies. The eigenenergy analysis compares exact and approximate solutions under stated assumptions.

1. Hamiltonian Eigenenergies

  • The Hamiltonian has six eigenvalues, but the analysis excludes the two single-dot singlet states and studies the remaining four across Larmor-frequency differences δω0.The appendix identifies its two-qubit Hamiltonian as one used in the main text.Source: Appendix C · p. 24, §1 · p. 25
  • For δω0 = 0 and small tunnel coupling, the eigenenergies are derived using a second-order Taylor expansion around t0 = 0.The derivation also treats nonzero Larmor-frequency differences with a second-order expansion.Source: §1 · p. 25
  • The sum ωλ2 + ωλ3 is independent of δω0, while the fraction ω′λ3 changes with δω0.The passage states that the generally valid solutions for ωλ2 and ωλ3 are obtained by solving Eq. C10 and can be simplified under stated assumptions.Source: §1 · p. 25
  • Figure 19 compares Hamiltonian eigenvalues with approximations from Eqs. C8 and C9 across interdot tunnel coupling and detuning, using nominal tunnel coupling t0n = 1 GHz.Its colors indicate different δω0 values.Source: §1 · p. 25

2. The C-Phase Gate

The C-phase derivation models an adiabatic excursion through a desired operating point, while the exchange derivation treats a diabatic control change. Both derive gate operations and fidelity expressions for control inaccuracies.

2. The C-Phase Gate

  • For the C-phase gate, an adiabatic excursion to and from the operating point yields a diagonal operation depending on the Hamiltonian eigenenergies.The ideal operation preserves eigenstates during the adiabatic control change.Source: §2 · pp. 25–26
  • The C-phase infidelity formulas cover duration, tunnel-coupling, and detuning inaccuracies, using second-order expansions except at ϵ = 0, where a fourth-order expansion is used.Figure 19 plots two-qubit operation speeds against interdot tunnel coupling and detuning for nominal tunnel coupling t0n = 1 GHz.Source: Figure 19 · p. 26, §2 · p. 27

3. The Exchange Gate

  • For a diabatic control change, the exchange-gate operation is described by the derived unitary in the rotating frame.The operation’s rotation angle θJ changes when control parameters are inaccurate.Source: §3 · p. 27
  • Exchange-gate infidelity formulas cover duration, tunnel-coupling, and detuning inaccuracies, using second-order expansions except at ϵ = 0, where a fourth-order expansion is used.The fidelity is expressed in terms of the ideal and inaccurate acquired rotation angles.Source: §3 · p. 27

4. Idle Gate

The idle-gate derivation evaluates two-qubit operation fidelity against identity. The read-out derivation combines sensing and detection contributions and specifies assumptions for estimating detection error.

4. Idle Gate

  • For idle-gate analysis, fidelity of the C-phase and exchange operations is evaluated relative to an identity operation, with expansions in θcz and θJ.The passage identifies the corresponding expansions as second order.Source: §4 · p. 27

Appendix D: Qubit Read-out

  • The read-out fidelity includes sensing and detection errors, which may jointly produce the correct outcome when their probabilities are assumed uncorrelated.For targeted higher fidelities and therefore smaller tolerable errors, the passage gives a simplified approximation to Eq. D1.Source: Appendix D · p. 27
  • Detection error is estimated assuming Gaussian noise and threshold discrimination halfway between the two charge-configuration signals; the read-out filter transfer function is specified for integration time Tread.The threshold is It = Is/2.Source: Appendix D · p. 28
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