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arXiv:2401.11758 · Physical Review Research

Qubit fidelity distribution under stochastic Schrödinger equations driven by classical noise

Robert de Keijzer, Luke Visser, Oliver T.C. Tse, S. J. J. M. F. Kokkelmans

Published Apr 16, 2025• 7 citations
arXivHTMLPDF

Abstract

Environmental noise affecting controlled quantum systems is typically described by a dissipative Lindblad equation, which captures the system's average state through the density matrix ρ . One approach to deriving this equation involves a stochastic operator evolving under white noise in the Schrödinger equation; however, white noise fails to accurately depict real-world noise profiles, where lower frequencies often dominate. This study proposes a method to determine the of qubit fidelities in significant stochastic Schrödinger equation scenarios, with qubits evolving under more realistic noise profiles such as noise. This method enables the prediction of the mean, variance, and higher-order moments of qubit fidelities, offering insights crucial for assessing permissible noise levels in prospective quantum computing systems and guiding decisions about control systems procurement. Additionally, these methodologies are essential for optimizing qubit state control affected by classical control system noise.

DOI
10.1103/physrevresearch.7.023063

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Environmental noise affecting controlled quantum systems is typically described by a dissipative Lindblad equation, which captures the system's average state through the density matrix ρ . One approach to deriving this equation involves a stochastic operator evolving under white noise in the Schrödinger equation; however, white noise fails to accurately depict real-world noise profiles, where lower frequencies often dominate. This study proposes a method to determine the of qubit fidelities in significant stochastic Schrödinger equation scenarios, with qubits evolving under more realistic noise profiles such as noise. This method enables the prediction of the mean, variance, and higher-order moments of qubit fidelities, offering insights crucial for assessing permissible noise levels in prospective quantum computing systems and guiding decisions about control systems procurement. Additionally, these methodologies are essential for optimizing qubit state control affected by classical control system noise.
Creating robust quantum operations is a major challenge in the current noisy intermediate-scale quantum computing era. Recently, the importance of noise-resilient control methods has become more pronounced in the field. Ordinarily, noisy quantum systems are described by the Lindblad equation. However, minimizing noise susceptibility using this equation has proven challenging because of its irreversibility. In this study, we propose a method for creating robust pulses based on the stochastic Schr\"odinger equation. This equation describes individual noise realizations under any colored noise process, contrary to the Lindblad equation, which describes mean system behavior under white noise. Using stochastic optimal control techniques, our method, fidelity-enhanced variational quantum optimal control, is able to construct higher-fidelity paths than its nonstochastic counterpart. By accounting for both environmental noise sources as well as noise sources inherent to the control system, highly significant increases in fidelity are noted for both single- and multiple-qubit state preparations.
This paper investigates the dynamic suppression of quantum noise within the framework of quantum control theory. The core challenge in implementing precise quantum operations lies in the significant influence of environmental noise, which introduces uncertainties and degrades control fidelity. We propose a novel approach utilizing real-time feedback control, leveraging quantum sensors to monitor the noise levels within a quantum system. Based on this continuous assessment, control parameters are dynamically adjusted to mitigate the impact of noise. The effectiveness of this method is demonstrated through theoretical analysis and mathematical modeling. We aim to enhance the accuracy and stability of quantum control processes, ultimately contributing to the reliable execution of complex quantum operations. The system is described by the following Hamiltonian: H=H0+HIH = H_0 + H_IH=H0​+HI​, where H0H_0H0​ is the system Hamiltonian and HIH_IHI​ represents the interaction Hamiltonian. The time evolution of the system is governed by the equation: d∣ψ(t)>dt=iℏ[H,∣ψ(t)]\frac{d|ψ(t)>}{dt} = \frac{i}{\hbar} [H, |ψ(t)]dtd∣ψ(t)>​=ℏi​[H,∣ψ(t)]. Noise is modeled as a stochastic term, adding to the Hamiltonian: H=H0+HI+HNH = H_0 + H_I + H_NH=H0​+HI​+HN​, where HNH_NHN​ represents the noise Hamiltonian. The control pulse can be described as: U(t)=∑i=1Nαi e−iωitU(t) = \sum_{i=1}^N \alpha_i \, e^{-i\omega_i t}U(t)=∑i=1N​αi​e−iωi​t, where αi\alpha_iαi​ are the amplitudes and ωi\omega_iωi​ are the carrier frequencies. The fidelity of the quantum operation is evaluated using the overlap between the initial and final states: Fidelity=∣∣2Fidelity = ||^2Fidelity=∣∣2. The control parameters are optimized to maximize the fidelity. The quantum state evolution is described by the equation: ∣ψ(t+δt)>=U(t)∣ψ(t)>|ψ(t+\delta t)> = U(t) |ψ(t)>∣ψ(t+δt)>=U(t)∣ψ(t)>, where U(t) is the control pulse. The time-dependent Schrodinger equation is: i/ħ ∂/∂t |ψ(t)> = H|ψ(t)>. The noise introduces uncertainties which can be represented as: σ2=−2\sigma^2 = - ^2σ2=−2, where N is the noise operator.
We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler-Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.
Noise is a hindering factor for current-era quantum computers. In this study, we experimentally validate the theoretical relationships between amplitude noise of the control signal and qubit state fidelity. The experiment comprises a 10 × 10 site optical tweezer array stochastically loaded with single rubidium-85 atoms. A global microwave field is used to manipulate the state of the hyperfine qubits. With precise control of the time-dependent amplitude of the microwave drive, we apply control signals featuring artificial noise. We systematically analyze the impact of various noise profiles on the fidelity distribution of the quantum states. The measured fidelities are compared against theoretical predictions made using the stochastic Schrödinger equation. Our results show a good agreement between the experimentally measured and theoretically predicted results. This validation is consequential, as the model provides critical information on noise identification and optimal control protocols in noisy intermediate-scale quantum (NISQ)-era quantum systems.
We analytically investigate the fidelity dynamics of a Jaynes--Cummings qubit coupled to a single-mode cavity in the presence of stochastic fluctuations of the qubit--cavity coupling. Such fluctuations naturally arise in cavity- and circuit-QED platforms due to fabrication imperfections, control-field noise, mechanical vibrations, and environmental disturbances. Since the qubit--cavity coupling governs the rate of coherent energy and information exchange, understanding the effects of its fluctuations is essential for assessing the performance of quantum communication, quantum sensing, and quantum information-processing protocols. We consider both Gaussian white noise and Ornstein--Uhlenbeck noise with finite correlation time as models for the coupling fluctuations. Exploiting the closed dynamics within each excitation manifold, we derive exact analytical expressions for the noise-averaged density matrix and the fidelity relative to the ideal noiseless evolution. We show that the fidelity is determined by a single decoherence factor governed by the integrated noise correlation function, establishing a direct connection between microscopic coupling fluctuations and macroscopic coherence loss. In the white-noise limit, the fidelity decays exponentially, whereas finite noise correlation times produce nonexponential decay and slower coherence loss. We further show that the crossover between Markovian-like and non-Markovian regimes is controlled by the ratio of the noise correlation rate to the coherent coupling strength, and that decoherence increases with photon number because of the enhanced effective qubit--cavity coupling. These results provide a unified analytical framework for quantifying decoherence induced by stochastic coupling fluctuations and for evaluating the robustness of cavity-QED protocols under correlated noise.
Abstract We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler–Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.