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Software tools for quantum control: Improving quantum computer performance through noise and error suppression

Harrison Ball, Michael J. Biercuk, Andre Carvalho, Jiayin Chen, Michael Hush, Leonardo A. De Castro, Li Li, Per J. Liebermann, Harry J. Slatyer, Claire Edmunds, Virginia Frey, Cornelius Hempel, Alistair Milne

arXiv:2001.04060v2quant-ph

TL;DR

Noise and error in quantum hardware constrain useful computation, and existing frameworks leave a need for tools that integrate quantum control across hardware and software layers. This paper introduces software for developing and deploying quantum-control solutions, and reports experimental evidence of noise suppression, error homogenization, gate-fidelity stabilization, and noise spectroscopy on real quantum hardware.

  • Problem

    Noise and error in quantum hardware limit the range of useful computations, while existing software frameworks largely focus on higher-level algorithms and do not directly address this hardware challenge.

    Source: §I · p. 2

  • Method

    The paper introduces software tools that integrate quantum control into quantum-computing workflows, including optimization, noise and hardware characterization, and cloud and local integration.

    Source: §I · p. 3, §V · p. 37

  • Results

    Experimental evidence on real quantum hardware demonstrates suppression of noise susceptibility, error homogenization in multiqubit devices, gate-fidelity stabilization over time, and noise spectroscopy in multiqubit gates.

    Source: §V · p. 37

  • Takeaways & Limitations

    The toolset is designed to support research and application development for users ranging from students to quantum-hardware R&D teams.

    Source: §V · p. 37

Abstract

from arXiv · show

Manipulating quantum computing hardware in the presence of imperfect devices and control systems is a central challenge in realizing useful quantum computers. Susceptibility to noise limits the performance and capabilities of noisy intermediate-scale quantum (NISQ) devices, as well as any future quantum computing technologies. Fortunately quantum control enables efficient execution of quantum logic operations and algorithms with built-in robustness to errors, without the need for complex logical encoding. In this manuscript we introduce software tools for the application and integration of quantum control in quantum computing research, serving the needs of hardware R&D teams, algorithm developers, and end users. We provide an overview of a set of python-based classical software tools for creating and deploying optimized quantum control solutions at various layers of the quantum computing software stack. We describe a software architecture leveraging both high-performance distributed cloud computation and local custom integration into hardware systems, and explain how key functionality is integrable with other software packages and quantum programming languages. Our presentation includes a detailed mathematical overview of central product features including a flexible optimization toolkit, filter functions for analyzing noise susceptibility in high-dimensional Hilbert spaces, and new approaches to noise and hardware characterization. Pseudocode is presented in order to elucidate common programming workflows for these tasks, and performance benchmarking is reported for numerically intensive tasks, highlighting the benefits of the selected cloud-compute architecture. Finally, we present a series of case studies demonstrating the application of quantum control solutions using these tools in real experimental settings for both trapped-ion and superconducting quantum computer hardware.

I. INTRODUCTION

Quantum-hardware noise and error limit useful computation, while much existing software focuses on higher-level algorithms rather than directly addressing this hardware challenge. The paper presents quantum-control software and integrations spanning optimization, characterization, cloud computation, programming frameworks, and hardware.

I. INTRODUCTION

  • Hardware noise and error limit useful computation, while existing software frameworks largely focus on higher-level algorithm design rather than directly addressing this challenge.The introduction identifies a need for a unified framework that integrates quantum control across differing hardware systems and software abstractions.Source: §I · pp. 2–3

II. SOFTWARE ARCHITECTURE AND INTEGRATIONS

  • The suite targets hardware performance through error-robust control, flexible optimization, predictive error budgeting, and microscopic hardware characterization.Its objectives also include delivering more functionality from fixed qubits, gates, and runtime, and providing cloud resources for intensive computation and cross-workflow compatibility.Source: §I · p. 3, §II · p. 4

A. Cloud-compute architecture

  • A Python API connects locally installed client packages to cloud computation, with scalable worker pools and support for public, private, hybrid, or on-premises deployment.Processing resources scale with the computation, and specialized accelerators such as GPUs can be accessed for sufficiently complex tasks.Source: A. Cloud-compute architecture · p. 4

B. Package overview

  • BOULDER OPAL is the core Python toolkit for research and hardware experts, while BLACK OPAL uses guided web visualizations to build quantum-control intuition.BOULDER OPAL's local wrapper orchestrates calls to the web API, leaving intensive computations to the cloud engine.Source: B. Package overview · pp. 4–5
  • FIRE OPAL is designed to incorporate control robustness into algorithms, whereas Open Controls offers community-developed open-loop protocols for deployment on custom or cloud hardware.FIRE OPAL is described as forthcoming; Open Controls is an open-source Python package.Source: B. Package overview · p. 5

C. Compatibility with other programming languages

  • Python adaptors and convenience packages export control solutions to Qiskit, pyQuil, and Cirq while preserving sequence structure and timing within platform-specific limitations.Qiskit integration also supports OpenPulse export for analog-level microwave programming; pyQuil and Cirq exports are limited to timed standard control operations where analog-level access is absent.Source: C. Compatibility with other… · p. 6

D. Integration with quantum hardware

  • BOULDER OPAL controls can be formatted for hardware constraints, and a Quantum Machines integration supports real-time orchestration and hardware tuneup.The QUA-based integration demonstrated dephasing-robust single-qubit operations in a tunable transmon device.Source: D. Integration with quantum hardware · p. 6

4. Maximum-likelihood state estimation incorporat-

The section frames quantum control as designing Hamiltonian controls that implement target operations and resist noise, then describes corresponding software capabilities and hardware integration. Optimal control targets the desired operation; robust control adds a noise-susceptibility condition.

4. Maximum-likelihood state estimation incorporat-

  • BOULDER OPAL tuneup functions include IQ-nonlinearity calibration, impulse-response characterization, modulated spectroscopy, and Hamiltonian parameter estimation.The listed characterization tasks address hardware calibration and signal response.Source: §4 · p. 7
  • The Last-Mile Integration extension runs locally alongside experimental-control software to schedule calibration and characterization while intensive computations remain in the cloud.Scheduled noise sensing and reconstruction can detect changes in dominant noise power spectra.Source: §4 · pp. 7–8

A. General quantum-control setting

  • Quantum control separates designing Hctrl(t) to implement a target unitary over duration τ from designing it to resist noise over that interval.The paper calls these optimal control and robust control, respectively.Source: §III · p. 8, A. General quantum-control setting · p. 8

1. Optimal quantum control

  • Optimal control numerically seeks Hctrl(t) whose evolved unitary Uctrl(τ) matches Utarget; Frobenius-inner-product operator-distance fidelity is bounded from 0 to 1, with perfect implementation at 1.These measures describe whether the ideal Hamiltonian implements the target evolution and do not incorporate stochastic noise processes.Source: §1 · p. 8

2. Robust quantum control

  • Robust control combines target-operation fidelity with noise robustness, measured by the noise-averaged operator distance between the error action and identity.The robustness criterion is target-independent, and the conditions are solved numerically through multiobjective optimization.Source: §2 · p. 9

3. Controllability

The section relates controllability to device constraints and develops multi-dimensional filter functions for assessing how arbitrary controls respond to noise. Its frequency-domain formulation links noise spectra to control-dependent infidelity, providing a computational route to evaluating noise susceptibility.

3. Controllability

  • With unlimited bandwidth and power, the stated control set spans SU(n) using O(log2(n)^2/2) controls, fewer than O(n^2).For finite bandwidth, time, and power, controllability cannot be addressed analytically; realistic control requirements depend on targeted noise, available controls, and evolution-time limits.Source: §3 · pp. 9–10

B. Performance evaluation for arbitrary controls

  • Generalized multi-dimensional filter functions provide a frequency-resolved heuristic for noise susceptibility of arbitrary unitaries in high-dimensional Hilbert spaces.The framework covers noise from quasi-static regimes to fluctuations comparable to or faster than the control, assuming quasi-classical noise channels.Source: B. Performance evaluation for… · p. 10

1. Modelling noise and error in D-dimensional systems

  • The noise model assumes weak error fields and uses a leading-order Magnus expansion to approximate the error action for D-dimensional systems.Noise fields are modeled as classical, zero-mean, wide-sense stationary processes with associated power spectral densities.Source: §1 · p. 10

2. Multi-dimensional filter functions in the frequency domain

  • Each noise channel contributes an infidelity term from the frequency-domain overlap of its noise power spectrum and corresponding filter function.The overlap estimates operational fidelity; reducing filter weight where the noise spectrum is large improves fidelity, and predictive performance has been experimentally validated for single-qubit and higher-dimensional operations.Source: §2 · pp. 11–12
  • A matrix-element Fourier-transform form enables efficient filter-function calculations for single- and multi-qubit gates, high-dimensional systems, and multi-operation circuits.The implementation uses weighted squared magnitudes of Fourier-transformed traceless noise-operator elements, with weights from the projection matrix.Source: §2 · pp. 11–12

C. Flexible optimization tools for quantum control

The optimizer uses a flexible, TensorFlow-based framework to construct and optimize controls across gates, subcircuits, and algorithms. Its features incorporate hardware constraints and support additional estimation tasks through the same optimization engine.

C. Flexible optimization tools for quantum control

  • The toolkit creates high-fidelity unitary operations for optimal and robust control in high-dimensional systems, from individual gates to complete algorithms.The same system definition supports these different scales without changing the underlying toolkit.Source: C. Flexible optimization tools for… · p. 12

1. Flexible optimizer framework

  • TensorFlow supplies gradient calculation for a custom-built optimizer that maps parameterized controls and cost functions to standard gradient-based algorithms.The framework supports nonlinear gradients and combines distinct performance metrics through weighted cost terms.Source: §1 · pp. 12–13

2. Flexible optimizer features

  • Convenience methods model hardware constraints in cost functions, helping produce implementable controls while potentially improving optimization efficiency.Supported constraints include waveform smoothing for bandwidth limits and finite hardware response times; controls can also be discretized to match sampled hardware output.Source: §2 · pp. 13–14
  • The optimizer supports CRAB waveform parameterizations and can incorporate Schrödinger-equation integration using GPU-optimized matrix-exponentiation or Runge-Kutta routines.CRAB restricts the search to basis coefficients; integration routines are selected for piecewise-constant controls or smooth and larger systems where full exponentiation is infeasible.Source: §2 · p. 15

3. Optimizer performance benchmarking

The optimizer benchmarks show faster Q-CTRL local-instance performance than the compared tools, while cloud computation offers increasing runtime benefits for more complex problems. The simulation tools support realistic time-dependent noise and provide functions for generating, aligning, and propagating noisy dynamics.

3. Optimizer performance benchmarking

  • Greater than two orders-of-magnitude improvement over the internal NumPy optimizer and a typical 3–5× advantage over QuTiP were reported for the representative problems.The local-instance advantages grew with Hilbert-space dimension, number of controls, and number of time-segments.Source: §3 · p. 16
  • For three- to seven-qubit problems, cloud computation reduced optimization runtime by approximately 10×, while its roughly three-second overhead favored local instances for simple tasks.Beyond seven qubits, calculations route to a GPU, with extrapolated cloud benefits approaching two orders of magnitude relative to the local instance.Source: §3 · p. 16, Figure 4 · p. 16
  • Competing packages were not used for optimizations lasting beyond one hour.Source: §3 · p. 16

D. Time-domain simulation tools for realistic hardware error processes

  • Q-CTRL simulation tools model arbitrary quantum systems under broad classes of noise, including correlated and colored semiclassical processes, with user-defined noise power spectral densities.This extends beyond the quasi-static offsets or stochastic depolarizing models used by most available simulation tools.Source: D. Time-domain simulation tools for… · p. 17

1. Technical details of simulation functionality

  • The simulation package can generate time-domain signals from measured or user-specified noise spectra, interpolate series, and automatically align inputs with different time discretizations.The shared grid permits Schrödinger integration across inputs defined at different temporal resolutions.Source: §1 · pp. 17–18
  • The package computes unitary evolution at arbitrary sample times and can average propagated state vectors across noise realizations to produce a mean density matrix.Source: §1 · p. 19

2. Simulation example

The example simulates a noisy NOT gate on a superconducting qubit modeled as an anharmonic three-level system. In the presence of noise and leakage, state-transfer fidelity falls by approximately three orders of magnitude.

2. Simulation example

  • The example models a superconducting qubit as an anharmonic three-level system and applies a Gaussian Half-DRAG pulse to implement a NOT gate.It includes time-varying microwave-drive phase noise, microwave detuning, and an additive ambient dephasing field.Source: §2 · p. 19
  • In the noisy, leakage-inclusive simulation, state-transfer fidelity was reduced by approximately three orders of magnitude from the ideal case.Source: §2 · p. 19

E. Hardware characterization

The hardware-characterization tools use qubit controls and measurements to infer noise spectra and system parameters. For spectral estimation, the paper develops SVD and convex-optimization methods and demonstrates reconstruction of simulated spectra.

E. Hardware characterization

  • Characterization uses the qubit to probe local dynamics and signal distortion, with control choice constrained by noise type, device-supported controls, and classical hardware limits.Slepian waveforms are described as spectrally concentrated controls that restrict sensitivity to noise.Source: E. Hardware characterization · p. 19, §1 · p. 21

1. Noise spectral estimation

  • Noise spectra are inferred by probing selected frequency bands with controls, measuring responses, and combining the measurements using SVD or convex optimization.The controls’ filter functions determine spectral sensitivity; changing controls can shift the sensitivity peak for broader sampling.Source: §1 · pp. 19, 21, Figure 6 · p. 21
  • The two reconstruction techniques accept arbitrary measurement records and full filter functions, support parameter-free estimation, and generalize beyond single qubits.The full filter functions include harmonics and hardware-induced imperfections.Source: §1 · p. 22
  • SVD is numerically efficient for smoothly varying spectra, but narrow spectral spurs can cause oscillatory, unphysical estimates.The issue arises because the pseudoinverse lacks a strict positivity criterion; convex optimization enforces positivity at greater computational cost.Source: §1 · pp. 22–23
  • SVD and convex optimization both reasonably reconstruct smooth and complex simulated dephasing spectra, but SVD oscillates on complex spectra while convex optimization does not.The complex spectra include a 1/f background and discrete spurs; the figure compares each reconstruction with the true spectrum.Source: Figure 7 · p. 23, §1 · p. 24
  • Convex optimization incorporates strict positivity and configurable regularization to accommodate smooth backgrounds and sparse spectral features.The paper describes combining first-derivative and L1 regularization; the resulting objective and constraints are convex.Source: §1 · pp. 23–24

2. Hamiltonian parameter estimation

Hamiltonian parameter estimation fits model parameters to measurements from different experimental setups, using dynamics-based likelihood costs and optimization. A simulated single-qubit example illustrates the procedure and reports the best fit as a high-fidelity, efficient estimation demonstration.

2. Hamiltonian parameter estimation

  • The method reduces complex system dynamics to a finite set of proxy parameters and estimates them by optimizing a cost based on measurement results.The reduced model limits the characterization measurements but can cost some model accuracy.Source: §2 · p. 24
  • For independent Gaussian measurement results, the negative log-likelihood yields an optimization cost, whose Hessian provides the Fisher information used to bound parameter-estimate covariance.The paper uses the inverse Fisher information as the Cramér–Rao lower bound on covariance.Source: §2 · pp. 24–25
  • The single-qubit example identifies three Hamiltonian coefficients using preparations in three Pauli eigenstates and measurements in different bases after varied wait times.The simulated data comprise 20 points per initial state, with normally distributed measurement errors of standard deviation 0.01.Source: §2 · pp. 25–26
  • The lowest-cost result among 30 optimization runs is presented as a high-fidelity, efficient parameter-estimation demonstration.The runs used different random initial parameter values.Source: §2 · p. 26

A. Open-loop control benefits demonstrated in trapped-ion QCs

The case studies demonstrate error suppression and homogenization in trapped-ion hardware, simultaneous leakage and dephasing suppression in superconducting qubits, and optimized robust controls for parametrically driven entangling gates.

A. Open-loop control benefits demonstrated in trapped-ion QCs

  • Open-loop controls suppressed quasi-static amplitude and detuning errors and low-frequency time-varying noise, with measurements agreeing with filter-function predictions.Composite controls targeted individual error channels, while optimized controls demonstrated dual suppression of amplitude and detuning errors.Source: A. Open-loop control benefits… · p. 26, Figure 8 · p. 27
  • 10.2× lower standard deviation of randomized-benchmarking error rates across 10 qubits followed use of an amplitude-error-suppressing control solution.The best-performing qubit showed no net error-rate improvement beyond measurement uncertainty, with other error sources including laser-light leakage.Source: A. Open-loop control benefits… · pp. 26–27
  • ∼3–4× reduced susceptibility to system drifts was observed for error-suppressing two-qubit trapped-ion gates over repeated measurements.The study tracked gate populations over 3.5 hours; the robust gate’s measurement range was narrower and closer to the ideal case.Source: A. Open-loop control benefits… · p. 27, Figure 8 · p. 27
  • ∼5× shorter duration than the analytic approach was achieved for optimized trapped-ion Xπ gates with simultaneous amplitude- and detuning-error robustness.The solutions retained robustness with up to 10% miscalibrations in qubit frequency and Rabi rate.Source: A. Open-loop control benefits… · p. 28, Figure 9 · p. 28

B. Simultaneous leakage and noise-robust controls for superconducting circuits

  • Orders-of-magnitude reductions in leakage and dephasing sensitivity were achieved together by optimized controls at the same duration as Half-DRAG pulses.The controls also show low-frequency dephasing suppression and broad detuning plateaus with low infidelity.Source: B. Simultaneous leakage and… · p. 29, Figure 10 · p. 30

C. Robust control for parametrically-driven superconducting entangling gates

  • Three parametric-drive error channels manifest as effective dephasing, and optimized controls combining coupling-drive modulation with single-qubit control improved robustness under multiple constraints.An analytic solution had similar overall performance but required ∼18.5% more execution time than the fixed-amplitude constrained optimization; similar results were reported for CZ and non-parametric cross-resonance gates.Source: C. Robust control for… · pp. 31–32

D. Experimental noise characterization of multiqubit circuits on an IBM cloud QC

The IBM-device study reconstructs two-qubit dephasing noise, while a five-qutrit case study uses optimized circuit structure to suppress residual cross-talk. Together, the experiments demonstrate noise characterization and circuit-level control under hardware constraints.

D. Experimental noise characterization of multiqubit circuits on an IBM cloud QC

  • The probe sequence uses a fixed number of gates so entangling-gate execution errors do not vary between sequences and swamp the noise signal.The study used 66 gates of 110 ns each to keep the experiment within the device coherence time.Source: D. Experimental noise characterization… · p. 32
  • A probe protocol varies two SW-gate positions to construct filter functions and infer noise spectra from infidelity measurements using SVD reconstruction.The hardware-constrained measurements overlap in frequency, making standard matrix-inversion reconstruction impossible.Source: D. Experimental noise characterization… · p. 32
  • The reconstructed two-qubit dephasing spectrum contains a low-frequency component and a repeatable higher-frequency contribution at 150–200 kHz.Similar performance was observed across multiple qubit pairs, with variations in the quasi-static component.Source: D. Experimental noise characterization… · pp. 33–34
  • The observed spectral-feature breadth reflects Fourier limits of the individual measurements, as confirmed by numerical simulations.Source: D. Experimental noise characterization… · p. 33

E. Crosstalk-resistant circuit compilation

  • Generalized dynamic decoupling combines timed free evolution with specific unitary operations to cancel ZZ coupling while implementing a non-identity operation.The five-qutrit optimization searches timing, rotation, and phase variables under a coherence-time constraint.Source: E. Crosstalk-resistant circuit… · pp. 35–36
  • 99.4% target-unitary fidelity was achieved, up from approximately 2.2% without dynamic cross-talk suppression in the five-qutrit example.The optimized sequence combines driven rotations and their timing; compactifying it changes its cross-talk-suppressing behavior.Source: E. Crosstalk-resistant circuit… · p. 37

V. CONCLUSION AND OUTLOOK

The paper presents a quantum-control software toolset combining local packages with cloud computation. It reports theoretical and hardware demonstrations, and outlines future development in machine-learning analysis and adaptive calibration.

V. CONCLUSION AND OUTLOOK

  • The toolset combines locally installed packages with cloud-compute infrastructure, spanning interactive interfaces and Python toolkits for users from students to hardware R&D teams.Source: §V · p. 37
  • Case studies provide theoretical and real-hardware evidence of noise-susceptibility suppression, multiqubit error homogenization, gate-fidelity stabilization, and noise spectroscopy.Source: §V · p. 37
  • Future work includes machine-learning tools for time-series analysis and hardware characterization, plus adaptive tuneup and calibration beyond independent device calibration.Source: §V · p. 37

Appendix A: Technical definitions

The appendix establishes mathematical conventions for matrix operations, Fourier analysis, and noise spectra, then formalizes control Hamiltonians, control solutions, and segmented pulses.

1. Frobenius inner product and Frobenius norm

  • The appendix introduces the Frobenius inner product and the matrix norm it induces.Source: §1 · p. 39

2. Fourier transform

  • The paper uses a non-unitary angular-frequency Fourier-transform convention for scalar-, matrix-, and operator-valued time functions.The same symbol denotes the time- or frequency-domain function, distinguished by its argument.Source: §2 · p. 39

3. Power spectral density

  • For independent, zero-mean, wide-sense-stationary noise fields, the autocorrelation depends on time difference and relates to the power spectral density through Wiener–Khinchin.Source: §3 · p. 39

1. Generalized formalism

  • The generalized control Hamiltonian combines control operators with real- and complex-valued pulses, organized as drives, shifts, and drifts.Source: §1 · p. 41, Table IV · p. 41

2. Control solutions

  • A control solution is specified by its drive and shift pulse functions over the interval in which control is applied.Source: §2 · p. 41

3. Control segments

  • Control pulses can be specified locally by partitioning the time domain into segments on which a shift pulse is piecewise constant.The amplitude and duration of a segment determine its area.Source: §3 · p. 42, Figure 15 · p. 42

4. Generic shaped control segments

The appendix generalizes control pulses to shaped segments, relates complex drive coordinates to quadrature controls, and connects Magnus terms to error-cancellation order and filter functions.

4. Generic shaped control segments

  • Generic shaped control segments represent each control as time-dependent functions arranged by segment and control channel.Rows correspond to distinct segments and columns to distinct controls.Source: §4 · pp. 42–43

5. Control coordinates

  • A complex drive pulse can be represented in polar or Cartesian coordinates, with its modulus setting rotation rate and its phase setting rotation direction.The Cartesian components are the in-phase and in-quadrature controls.Source: §5 · pp. 43–44

1. Magnus expansion

  • The Magnus series defines error-cancellation order: controls achieve order α when Φ_k(τ) is approximately zero for every k ≤ α.The paper then formulates computation of these terms in the Fourier domain using filter functions.Source: §1 · p. 45

2. Leading order robustness infidelity in terms of filter functions

The appendix derives leading-order robustness infidelity in terms of filter functions and describes the setup used to benchmark the Q-CTRL optimization engine. The supplied passages identify the comparison tools, computing environments, and repeated-run protocol, but report no benchmark outcomes.

2. Leading order robustness infidelity in terms of filter functions

  • The derivation expands the leading-order error action and simplifies the robustness-fidelity expression using Hermiticity, a gauge property, and frequency-domain noise relations.The resulting term is substituted into the fidelity expression; these excerpts do not provide the final filter-function formula.Source: §2 · pp. 45–46

1. Optimization tools

  • Four optimizers were compared: NumPy with SciPy L-BFGS-B, QuTiP’s gradient-ascent pulse engineering, and local and cloud Q-CTRL implementations.The local Q-CTRL implementation also used SciPy L-BFGS-B; the cloud version used custom backend resource management.Source: §1 · p. 47

2. Software and hardware versions

  • Local optimizations ran on one 4-core Xeon machine, while cloud Q-CTRL used 20 such machines, each with an NVIDIA T4 GPU.The single machine had 16 GB RAM; each cloud machine also had 16 GB RAM.Source: §2 · p. 47

3. Physical systems

  • For each of two physical-system configurations, the authors ran 20 optimizations from randomly generated initial seeds and compared evaluation counts and final infidelities qualitatively.The comparisons were intended to check approximate consistency.Source: §3 · p. 47

a. Single controllable qubit in four-qubit space

The benchmark setups vary optimization complexity across a four-qubit space and a Rydberg-atom array. The experimental methods also compare pulse constructions under quasi-static and time-varying noise.

a. Single controllable qubit in four-qubit space

  • The four-qubit benchmark controls one qubit with three piecewise-constant controls; varying 10–500 segments changes the search space from 30 to 1,500 parameters.QuTiP used relaxed separate bounds on I and Q because it does not support the complex constraint used for the other tools.Source: a. Single controllable qubit in… · p. 48

b. Linear array of Rydberg atoms

  • The Rydberg-array benchmark varies atom count from 2 to 8 while optimizing global coupling and detuning to prepare a Greenberger-Horne-Zeilinger state.Results were omitted when 20 runs were projected to exceed roughly one hour: for NumPy at 7–8 atoms, and for QuTiP and local Q-CTRL at 8 atoms.Source: b. Linear array of Rydberg atoms · p. 48

1. Quasi-static error robustness

  • Four Xπ pulse constructions are compared by amplifying rotation-angle and detuning errors: Primitive, BB1, CORPSE, and CinBB.The sequences use 10 pulses; BB1 targets amplitude errors, CORPSE detuning errors, and CinBB both error types.Source: §1 · p. 49

2. Suppression of time-varying noise

  • Time-varying control-amplitude and dephasing filter functions are reconstructed by applying single-frequency disturbances and averaging responses across modulation phases.The phase-averaged modulation produces a delta-like noise spectrum, enabling extraction of the filter function at the disturbance frequency.Source: §2 · p. 49

3. Error homogenization characterized via 10-qubit parallel randomized benchmarking

The passages describe parallel randomized benchmarking across ten trapped ions and measurements of two phase-modulated Mølmer–Sørensen gate designs. They also describe interactive visualizations of circuit evolution, noise, and two-qubit entanglement.

3. Error homogenization characterized via 10-qubit parallel randomized benchmarking

  • Parallel randomized benchmarking measures spatially varying error rates across ten simultaneously illuminated ions, with each ion’s results extracted from spatially resolved camera images.The method uses sequences of up to 500 Clifford operations; camera-based SPAM errors are about 3–5%, roughly an order of magnitude above single-qubit RB.Source: §3 · p. 50

4. Mølmer–Sørensen drift measurements

  • Two phase-modulated Mølmer–Sørensen gates were alternated over several hours without recalibration; one uses four phase segments, while the other doubles that count for added low-frequency robustness.Both gates were set 2 kHz below mode k = 2, and the operation was repeated 500 times per measurement.Source: §4 · p. 50

Appendix F: Visualizations of noise and control in quantum circuits

  • The visualizer pairs Bloch-sphere state trajectories with correlation tetrahedra and concurrence to show two-qubit evolution and entanglement.For separable states the correlations are zero; entangled states move the indicators away from the tetrahedra’s origins, while noise can perturb the displayed evolution.Source: Appendix F · pp. 51–52
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