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:2506.14230

Norm Inequality for Perturbed Quantum Evolutions and Its Application to Grover's Algorithm

Kohei Kobayashi

Published Jun 17, 2025• 0 citations
arXivHTMLPDF

Abstract

We investigate the impact of coherent control errors on quantum state evolution by deriving a general norm inequality based on Gronwall's lemma. This inequality provides an explicit upper bound on the deviation between an ideal quantum state and one subject to arbitrary coherent perturbations, including both time-dependent and time-independent cases. The framework is broadly applicable, requiring no assumptions about the detailed structure of the perturbation and full dynamics of the quantum system. We apply this approach to analyze the robustness of Grover's search algorithm in the presence of coherent errors. Our results characterizes quantitative scaling relations between the error strength, algorithm runtime, and success probability, offering practical guidelines for the design of resilient quantum protocols. We further compare the effects of time-dependent and time-independent perturbations, showing the distinctive ways in which coherent errors accumulate in quantum dynamics.

DOI
10.48550/arxiv.2506.14230

Related papers

Robustness of quantum algorithms against coherent control errors

Julian Berberich, Daniel Fink, Christian Holm

Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.

arXiv:2303.00618•Physical Review A•Jan 10, 2024•20 citations

A Fundamental Bound for Robust Quantum Gate Control

Robert L. Kosut, Daniel A. Lidar, Herschel Rabitz

We derive a universal performance limit for coherent quantum control in the presence of modeled and unmodeled uncertainties. For any target unitary $W$ that is implementable in the absence of error, we prove that the worst-case (and hence the average) gate fidelity obeys the lower bound $F \ge \Flb\bigl(\tf \Omeff\bigr)$, where $\tf$ is the gate duration and $\Omeff$ is a single frequency-like measure that aggregates \emph{all} bounded uncertainty sources, e.g., coherent control imperfections, unknown couplings, and residual environment interactions, without assuming an initially factorizable system-bath state or a completely positive map. The bound is obtained by combining an interaction-picture averaging method with a Bellman-Gronwall inequality and holds for any finite-norm Hamiltonian decomposition. Hence it applies equally to qubits, multi-level qudits, and ancilla-assisted operations. Because $\Flb$ depends only on the dimensionless product $\tf\Omeff$, it yields a device-independent metric that certifies whether a given hardware platform can, in principle, reach a specified fault-tolerance threshold, and also sets a quantitative target for robust-control synthesis and system identification.

arXiv:2507.01215•Jul 1, 2025•0 citations

Exploring the Robustness of Stabilizing Controls for Stochastic Quantum Evolutions

Weichao Liang, Kentaro Ohki, Francesco Ticozzi

Abstract. In this work we analyze and bound the effect of modeling errors on the stabilization of pure states or subspaces for quantum stochastic evolutions. Different approaches are used for open-loop and feedback control protocols. For both, we highlight the key role of dynamical invariance of the target: if the perturbation preserves invariance, it is possible to prove that it also preserves its attractivity, under some additional assumptions. In addition, we prove boundedness in mean of the solutions of perturbed systems under open-loop protocols. For the feedback strategies, in the general case without assumptions on invariance, we provide bounds on the perturbation effect in expectation and in probability, as well as specific bounds for nondemolition nominal systems.

arXiv:2311.04428•SIAM Journal on Control and Optimization•Jan 10, 2025•7 citations

Robustness of quantum algorithms: Worst-case fidelity bounds and implications for design

Julian Berberich, Tobias Fellner, Robert L. Kosut +1 more

Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In this paper, we develop an alternative, algorithm-centered framework for understanding and improving the robustness against errors. For a given quantum algorithm and error model, we derive worst-case fidelity bounds which can be efficiently computed to certify the robustness. We consider general error models including coherent and (Markovian) incoherent errors and allowing for set-based error descriptions to address uncertainty or time-dependence in the errors. Our results give rise to guidelines for robust algorithm design and compilation by optimizing our theoretical robustness measure. We demonstrate the practicality of the framework with numerical results on algorithm analysis and robust optimization, including the robustness analysis of a 50-qubit modular adder circuit.

arXiv:2509.08481•Sep 10, 2025•0 citations

General Framework for Error Interference in Quantum Simulation

Boyang Chen, Jue Xu, Xiao Yuan +1 more

Quantum simulation is widely regarded as one of the most promising applications of quantum computing. A critical challenge in this domain is understanding and quantifying the accumulation of algorithmic errors over time, which is essential for designing more efficient simulation algorithms and for assessing the resources required to achieve quantum advantage. Conventional error analyses typically rely on the triangle inequality to bound the total simulation error, but such approaches tend to overestimate errors by ignoring error interference-a phenomenon in which errors from different simulation segments partially cancel. Here, we introduce a new framework for directly estimating long-time algorithmic errors in segmented quantum simulations. Our approach captures the full structure of error interference, enabling significantly tighter and more accurate error bounds. We identify both necessary and sufficient conditions for strict error interference and propose the notion of approximate error interference to account for realistic, imperfect cancellation. We demonstrate the broad applicability of our framework across a range of models and settings, including Heisenberg and Fermi-Hubbard systems, lattice Hamiltonians with power-law interactions, higher-order Trotter decompositions, and adiabatic evolution. By providing a unified and practical methodology for analyzing error interference, our Letter advances the theoretical understanding of quantum simulation and informs the design and benchmarking of algorithms for near-term and future quantum hardware.

arXiv:2411.03255•Physical Review Letters•Apr 1, 2026•0 citations

Unified error bounds for perturbations of non-Markovian open quantum systems in Gaussian environments

Zhen Huang, Yuanran Zhu, Gunhee Park +1 more

We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by a general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type inequality for unitary system-bath evolution. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.

arXiv:2411.08741•Nov 13, 2024•1 citation
We investigate the impact of coherent control errors on quantum state evolution by deriving a general norm inequality based on Gronwall's lemma. This inequality provides an explicit upper bound on the deviation between an ideal quantum state and one subject to arbitrary coherent perturbations, including both time-dependent and time-independent cases. The framework is broadly applicable, requiring no assumptions about the detailed structure of the perturbation and full dynamics of the quantum system. We apply this approach to analyze the robustness of Grover's search algorithm in the presence of coherent errors. Our results characterizes quantitative scaling relations between the error strength, algorithm runtime, and success probability, offering practical guidelines for the design of resilient quantum protocols. We further compare the effects of time-dependent and time-independent perturbations, showing the distinctive ways in which coherent errors accumulate in quantum dynamics.
Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.
We derive a universal performance limit for coherent quantum control in the presence of modeled and unmodeled uncertainties. For any target unitary WWW that is implementable in the absence of error, we prove that the worst-case (and hence the average) gate fidelity obeys the lower bound $F \ge \Flb\bigl(\tf \Omeff\bigr)$, where $\tf$ is the gate duration and $\Omeff$ is a single frequency-like measure that aggregates \emph{all} bounded uncertainty sources, e.g., coherent control imperfections, unknown couplings, and residual environment interactions, without assuming an initially factorizable system-bath state or a completely positive map. The bound is obtained by combining an interaction-picture averaging method with a Bellman-Gronwall inequality and holds for any finite-norm Hamiltonian decomposition. Hence it applies equally to qubits, multi-level qudits, and ancilla-assisted operations. Because $\Flb$ depends only on the dimensionless product $\tf\Omeff$, it yields a device-independent metric that certifies whether a given hardware platform can, in principle, reach a specified fault-tolerance threshold, and also sets a quantitative target for robust-control synthesis and system identification.
Abstract. In this work we analyze and bound the effect of modeling errors on the stabilization of pure states or subspaces for quantum stochastic evolutions. Different approaches are used for open-loop and feedback control protocols. For both, we highlight the key role of dynamical invariance of the target: if the perturbation preserves invariance, it is possible to prove that it also preserves its attractivity, under some additional assumptions. In addition, we prove boundedness in mean of the solutions of perturbed systems under open-loop protocols. For the feedback strategies, in the general case without assumptions on invariance, we provide bounds on the perturbation effect in expectation and in probability, as well as specific bounds for nondemolition nominal systems.
Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In this paper, we develop an alternative, algorithm-centered framework for understanding and improving the robustness against errors. For a given quantum algorithm and error model, we derive worst-case fidelity bounds which can be efficiently computed to certify the robustness. We consider general error models including coherent and (Markovian) incoherent errors and allowing for set-based error descriptions to address uncertainty or time-dependence in the errors. Our results give rise to guidelines for robust algorithm design and compilation by optimizing our theoretical robustness measure. We demonstrate the practicality of the framework with numerical results on algorithm analysis and robust optimization, including the robustness analysis of a 50-qubit modular adder circuit.
Quantum simulation is widely regarded as one of the most promising applications of quantum computing. A critical challenge in this domain is understanding and quantifying the accumulation of algorithmic errors over time, which is essential for designing more efficient simulation algorithms and for assessing the resources required to achieve quantum advantage. Conventional error analyses typically rely on the triangle inequality to bound the total simulation error, but such approaches tend to overestimate errors by ignoring error interference-a phenomenon in which errors from different simulation segments partially cancel. Here, we introduce a new framework for directly estimating long-time algorithmic errors in segmented quantum simulations. Our approach captures the full structure of error interference, enabling significantly tighter and more accurate error bounds. We identify both necessary and sufficient conditions for strict error interference and propose the notion of approximate error interference to account for realistic, imperfect cancellation. We demonstrate the broad applicability of our framework across a range of models and settings, including Heisenberg and Fermi-Hubbard systems, lattice Hamiltonians with power-law interactions, higher-order Trotter decompositions, and adiabatic evolution. By providing a unified and practical methodology for analyzing error interference, our Letter advances the theoretical understanding of quantum simulation and informs the design and benchmarking of algorithms for near-term and future quantum hardware.
We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by a general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type inequality for unitary system-bath evolution. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.