Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1101.3817 · Journal of Physics A Mathematical and Theoretical

Fidelity between unitary operators and the generation of robust gates against off-resonance perturbations

Renán Cabrera, Ofer M. Shir, Re-Bing Wu, HERSCHEL A. RABITZ

Published Feb 9, 2011• 12 citations
arXivHTMLPDF

Abstract

We perform a functional expansion of the fidelity between two unitary matrices in order to find necessary conditions for the robust implementation of a target gate. Comparison of these conditions with those obtained from the the Magnus expansion and Dyson series shows that they are equivalent in the first order. By exploiting techniques from robust design optimization , we account for issues of experimental feasibility by introducing an additional criterion to the search for control pulses. This search is accomplished by exploring the competition between the multiple objectives in the implementation of the NOT gate by means of evolutionary multi-objective optimization.

DOI
10.1088/1751-8113/44/9/095302

Related papers

Universally Robust Quantum Control

Pablo M. Poggi, Gabriele De Chiara, Steve Campbell +1 more

We study the robustness of the evolution of a quantum system against small uncontrolled variations in parameters in the Hamiltonian. We show that the fidelity susceptibility, which quantifies the perturbative error to leading order, can be expressed in superoperator form and use this to derive control pulses that are robust to any class of systematic unknown errors. The proposed optimal control protocol is equivalent to searching for a sequence of unitaries that mimics the first-order moments of the Haar distribution, i.e., it constitutes a 1-design. We highlight the power of our results for error-resistant single- and two-qubit gates.

arXiv:2309.14437•Physical Review Letters•May 7, 2024•36 citations

Exploring the tradeoff between fidelity and time optimal control of quantum unitary transformations

Katharine Moore Tibbetts, Constantin Brif, Matthew Grace +5 more

Generating a unitary transformation in the shortest possible time is of practical importance to quantum information processing because it helps to reduce decoherence effects and improve robustness to additive control field noise. Many analytical and numerical studies have identified the minimum time necessary to implement a variety of quantum gates on coupled-spin qubit systems. This work focuses on exploring the Pareto front that quantifies the tradeoff between the competitive objectives of maximizing the gate fidelity $\mathcal{F}$ and minimizing the control time $T$. In order to identify the critical time ${T}^{*}$ below which the target transformation is not reachable, as well as to determine the associated Pareto front, we introduce a numerical method of Pareto front tracking (PFT). We consider closed two- and multiqubit systems with constant interqubit coupling strengths and each individual qubit controlled by a separate time-dependent external field. Our analysis demonstrates that unit fidelity (to a desired numerical accuracy) can be achieved at any $T\ensuremath{\ge}{T}^{*}$ in most cases. However, the optimization search effort rises superexponentially as $T$ decreases and approaches ${T}^{*}$. Furthermore, a small decrease in control time incurs a significant penalty in fidelity for $T<{T}^{*}$, indicating that it is generally undesirable to operate below the critical time. We investigate the dependence of the critical time ${T}^{*}$ on the coupling strength between qubits and the target gate transformation. Practical consequences of these findings for laboratory implementation of quantum gates are discussed.

arXiv:1112.0333•Physical Review A•Dec 12, 2012•70 citations

Protecting quantum gates from control noise

Constantin Brif

External controls are necessary to enact quantum logic operations, and the inevitable control noise will result in gate errors in a realistic quantum circuit. We investigate the robustness of quantum gates to the random noise in an optimal control field, by utilizing properties of the quantum control landscape that relates the physical objective (in the present case, the quantum gate fidelity) to the applied controls. An approximate result obtained for the statistical expectation value of the gate fidelity in the weak noise regime is shown to be in excellent agreement with direct Monte Carlo sampling over noise process realizations for fidelity values relevant for practical quantum information processing. Using this approximate result, we demonstrate that maximizing the robustness to additive/multiplicative white noise is equivalent to minimizing the total control time/fluence. Also, a genetic optimization algorithm is used to identify controls with improved robustness to a colored noise characterized by its autocorrelation function.

•Jan 1, 2011•2 citations

Minimal time robust control for two superconducting qubits

Niril George, Joseph L. Allen, Robert L. Kosut +1 more

High-fidelity quantum gates are crucial for achieving fault-tolerant quantum computing; however, decoherence significantly reduces gate fidelities during long operation times. Although optimal control techniques can theoretically minimize these operation times, they often neglect realistic uncertainties in system parameters. In this work, we demonstrate that by using robust optimal control strategies, the cross-resonance gate in superconducting systems can be operated within 64 ns, achieving fidelities of [Formula: see text] while maintaining robustness against up to 10% uncertainty in a single parameter. Alternatively, by extending the control time to 71 ns, we achieve fidelities of [Formula: see text] with robustness against up to 3% uncertainty. Our results identify the minimal control times attainable with experimentally feasible pulses and system parameters, as well as the maximum allowable static parameter error for high-fidelity operations. Furthermore, we demonstrate simultaneous robustness against both static and time-dependent errors by generating 100 ns control pulses ([Formula: see text]) that maintain robustness against 10% static parameter error and time-dependent parameter fluctuations two orders of magnitude stronger than typical experimental noise. These findings demonstrate a viable open-loop strategy for implementing fast, high-fidelity quantum gates in the presence of realistic system uncertainties that would otherwise degrade conventional control pulses.

arXiv:1902.08056•Scientific Reports•Dec 24, 2025•5 citations

Sixth-order robust gates for quantum control

D. Mc Hugh, Jason Twamley

Composite pulse sequences designed for nuclear magnetic resonance experiments are currently being applied in many quantum information processing technologies. We present an analysis of a family of composite pulse sequences used to address systematic pulse-length errors in the execution of quantum gates. It has been demonstrated by Cummins et al. [Phys. Rev. A 67, 042308 (2003)] that for this family of composite pulse sequences, the fidelity of the resulting unitary operation compared with the ideal unitary operation is $1\ensuremath{-}C{ϵ}^{6}$, where $ϵ$ is the fractional error in the length of the pulse. We derive an exact expression for the sixth-order coefficient $C$ and from this deduce conditions under which this sixth-order dependence is observed. We also present pulse sequences which achieve the same fidelity.

arXiv:quant-ph/0404127•Physical Review A•Jan 19, 2005•13 citations

Comparing and correcting robustness metrics for quantum optimal control

Andrew T. Kamen, Samuel Fine, Bikrant Bhattacharyya +2 more

Control pulses that nominally optimize fidelity are sensitive to routine hardware drift and modeling errors. Robust quantum optimal control seeks error-insensitive control pulses that maintain fidelity thresholds and obey hardware constraints. Distinct numerical approximations to the first-order error susceptibility include adjoint end-point and toggling-frame approaches. Although theoretically equivalent, we provide a novel, systematic study demonstrating important numerical differences between these two approaches. We also introduce a critical discretization correction to the widely-used toggling-frame robustness estimator, measurably improving its estimate of first-order error susceptibility. We accomplish our study by positioning robustness as a first-class objective within direct, constrained optimal control. Our approach uniquely handles control and fidelity constraints while cleanly isolating robustness for dedicated optimization. In both single- and two-qubit examples under realistic constraints, our approach provides an analytic edge for obtaining precise, physics-informed robustness.

quant-pharXiv:2602.10349•Feb 12, 2026•0 citations
We perform a functional expansion of the fidelity between two unitary matrices in order to find necessary conditions for the robust implementation of a target gate. Comparison of these conditions with those obtained from the the Magnus expansion and Dyson series shows that they are equivalent in the first order. By exploiting techniques from robust design optimization , we account for issues of experimental feasibility by introducing an additional criterion to the search for control pulses. This search is accomplished by exploring the competition between the multiple objectives in the implementation of the NOT gate by means of evolutionary multi-objective optimization.
We study the robustness of the evolution of a quantum system against small uncontrolled variations in parameters in the Hamiltonian. We show that the fidelity susceptibility, which quantifies the perturbative error to leading order, can be expressed in superoperator form and use this to derive control pulses that are robust to any class of systematic unknown errors. The proposed optimal control protocol is equivalent to searching for a sequence of unitaries that mimics the first-order moments of the Haar distribution, i.e., it constitutes a 1-design. We highlight the power of our results for error-resistant single- and two-qubit gates.
Generating a unitary transformation in the shortest possible time is of practical importance to quantum information processing because it helps to reduce decoherence effects and improve robustness to additive control field noise. Many analytical and numerical studies have identified the minimum time necessary to implement a variety of quantum gates on coupled-spin qubit systems. This work focuses on exploring the Pareto front that quantifies the tradeoff between the competitive objectives of maximizing the gate fidelity F\mathcal{F}F and minimizing the control time TTT. In order to identify the critical time T∗{T}^{*}T∗ below which the target transformation is not reachable, as well as to determine the associated Pareto front, we introduce a numerical method of Pareto front tracking (PFT). We consider closed two- and multiqubit systems with constant interqubit coupling strengths and each individual qubit controlled by a separate time-dependent external field. Our analysis demonstrates that unit fidelity (to a desired numerical accuracy) can be achieved at any $T\ensuremath{\ge}{T}^{*}$ in most cases. However, the optimization search effort rises superexponentially as TTT decreases and approaches T∗{T}^{*}T∗. Furthermore, a small decrease in control time incurs a significant penalty in fidelity for T<T∗T<{T}^{*}T<T∗, indicating that it is generally undesirable to operate below the critical time. We investigate the dependence of the critical time T∗{T}^{*}T∗ on the coupling strength between qubits and the target gate transformation. Practical consequences of these findings for laboratory implementation of quantum gates are discussed.
External controls are necessary to enact quantum logic operations, and the inevitable control noise will result in gate errors in a realistic quantum circuit. We investigate the robustness of quantum gates to the random noise in an optimal control field, by utilizing properties of the quantum control landscape that relates the physical objective (in the present case, the quantum gate fidelity) to the applied controls. An approximate result obtained for the statistical expectation value of the gate fidelity in the weak noise regime is shown to be in excellent agreement with direct Monte Carlo sampling over noise process realizations for fidelity values relevant for practical quantum information processing. Using this approximate result, we demonstrate that maximizing the robustness to additive/multiplicative white noise is equivalent to minimizing the total control time/fluence. Also, a genetic optimization algorithm is used to identify controls with improved robustness to a colored noise characterized by its autocorrelation function.
High-fidelity quantum gates are crucial for achieving fault-tolerant quantum computing; however, decoherence significantly reduces gate fidelities during long operation times. Although optimal control techniques can theoretically minimize these operation times, they often neglect realistic uncertainties in system parameters. In this work, we demonstrate that by using robust optimal control strategies, the cross-resonance gate in superconducting systems can be operated within 64 ns, achieving fidelities of [Formula: see text] while maintaining robustness against up to 10% uncertainty in a single parameter. Alternatively, by extending the control time to 71 ns, we achieve fidelities of [Formula: see text] with robustness against up to 3% uncertainty. Our results identify the minimal control times attainable with experimentally feasible pulses and system parameters, as well as the maximum allowable static parameter error for high-fidelity operations. Furthermore, we demonstrate simultaneous robustness against both static and time-dependent errors by generating 100 ns control pulses ([Formula: see text]) that maintain robustness against 10% static parameter error and time-dependent parameter fluctuations two orders of magnitude stronger than typical experimental noise. These findings demonstrate a viable open-loop strategy for implementing fast, high-fidelity quantum gates in the presence of realistic system uncertainties that would otherwise degrade conventional control pulses.
Composite pulse sequences designed for nuclear magnetic resonance experiments are currently being applied in many quantum information processing technologies. We present an analysis of a family of composite pulse sequences used to address systematic pulse-length errors in the execution of quantum gates. It has been demonstrated by Cummins et al. [Phys. Rev. A 67, 042308 (2003)] that for this family of composite pulse sequences, the fidelity of the resulting unitary operation compared with the ideal unitary operation is $1\ensuremath{-}C{ϵ}^{6}$, where ϵϵϵ is the fractional error in the length of the pulse. We derive an exact expression for the sixth-order coefficient CCC and from this deduce conditions under which this sixth-order dependence is observed. We also present pulse sequences which achieve the same fidelity.
Control pulses that nominally optimize fidelity are sensitive to routine hardware drift and modeling errors. Robust quantum optimal control seeks error-insensitive control pulses that maintain fidelity thresholds and obey hardware constraints. Distinct numerical approximations to the first-order error susceptibility include adjoint end-point and toggling-frame approaches. Although theoretically equivalent, we provide a novel, systematic study demonstrating important numerical differences between these two approaches. We also introduce a critical discretization correction to the widely-used toggling-frame robustness estimator, measurably improving its estimate of first-order error susceptibility. We accomplish our study by positioning robustness as a first-class objective within direct, constrained optimal control. Our approach uniquely handles control and fidelity constraints while cleanly isolating robustness for dedicated optimization. In both single- and two-qubit examples under realistic constraints, our approach provides an analytic edge for obtaining precise, physics-informed robustness.