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Temporal Coarse Graining for Classical Stochastic Noise in Quantum Systems

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Temporal Coarse Graining for Classical Stochastic Noise in Quantum Systems

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AbstractSimulations of quantum systems with Hamiltonian classical stochastic noise can be challenging when the noise exhibits temporal correlations over a multitude of time scales, such as for $1/f$ noise in solid-state quantum information processors. Here we present an approach for simulating Hamiltonian classical stochastic noise that performs temporal coarse-graining by effectively integrating out the high-frequency components of the noise. We focus on the case where the stochastic noise can be expressed as a sum of Ornstein-Uhlenbeck processes. Temporal coarse-graining is then achieved by conditioning the stochastic process on a coarse realization of the noise, expressing the conditioned stochastic process in terms of a sum of smooth, deterministic functions and bridge processes with boundaries fixed at zero, and performing the ensemble average over the bridge processes. For Ornstein-Uhlenbeck processes, the deterministic components capture all dependence on the coarse realization, and the stochastic bridge processes are not only independent but taken from the same distribution with correlators that can be expressed analytically, allowing the associated noise propagators to be precomputed once for all simulations. This combination of noise trajectories on a coarse time grid and ensemble averaging over bridge processes has practical advantages, such as a simple concatenation rule, that we highlight with numerical examples.Popular summaryClassical simulation of quantum systems is a valuable tool for understanding the role of different noise sources on a quantum system's behavior, as well as informing how different strategies can mitigate or eliminate the impact of such non-idealities. For this reason, the development of practical methods that allow a more faithful simulation of the noise experienced by a quantum system remains an important area of research. Many noise sources, such as charge noise in solid-state quantum technologies, exhibit temporal correlations over a multitude of time scales. This noise characteristic poses a challenge for classical simulations as it requires a sufficiently small time step to capture the short-time correlations (high-frequency components) of the noise but also long simulation times to capture the drift (low-frequency components) of the noise. In this work we present an approach for simulating Hamiltonian classical stochastic noise that performs temporal coarse-graining by effectively integrating out the high-frequency components. This is achieved by generating a realization of the noise on a coarse time grid and averaging over all noise realizations that connect the noise values on the coarse time grid. This approach then avoids the need for small time steps, allowing for long time simulations to be performed more effectively while still faithfully reproducing the temporal correlations of the noise.► BibTeX data@article{Albash2026temporalcoarse, doi = {10.22331/q-2026-06-15-2137}, url = {https://doi.org/10.22331/q-2026-06-15-2137}, title = {Temporal {C}oarse {G}raining for {C}lassical {S}tochastic {N}oise in {Q}uantum {S}ystems}, author = {Albash, Tameem and Young, Steve and Jacobson, N. Tobias}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2137}, month = jun, year = {2026} }► References [1] Joseph Emerson, Robert Alicki, and Karol Życzkowski. ``Scalable noise estimation with random unitary operators''. 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The Computer Journal 7, 308–313 (1965). https://doi.org/10.1093/comjnl/7.4.308Cited by[1] Tameem Albash and N. Tobias Jacobson, "Simulating the Open System Dynamics of Multiple Exchange-Only Qubits using Subspace Monte Carlo", arXiv:2603.15577, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-06-15 13:28:09). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-15 13:28:07: Could not fetch cited-by data for 10.22331/q-2026-06-15-2137 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractSimulations of quantum systems with Hamiltonian classical stochastic noise can be challenging when the noise exhibits temporal correlations over a multitude of time scales, such as for $1/f$ noise in solid-state quantum information processors. Here we present an approach for simulating Hamiltonian classical stochastic noise that performs temporal coarse-graining by effectively integrating out the high-frequency components of the noise. We focus on the case where the stochastic noise can be expressed as a sum of Ornstein-Uhlenbeck processes. Temporal coarse-graining is then achieved by conditioning the stochastic process on a coarse realization of the noise, expressing the conditioned stochastic process in terms of a sum of smooth, deterministic functions and bridge processes with boundaries fixed at zero, and performing the ensemble average over the bridge processes. For Ornstein-Uhlenbeck processes, the deterministic components capture all dependence on the coarse realization, and the stochastic bridge processes are not only independent but taken from the same distribution with correlators that can be expressed analytically, allowing the associated noise propagators to be precomputed once for all simulations. This combination of noise trajectories on a coarse time grid and ensemble averaging over bridge processes has practical advantages, such as a simple concatenation rule, that we highlight with numerical examples.Popular summaryClassical simulation of quantum systems is a valuable tool for understanding the role of different noise sources on a quantum system's behavior, as well as informing how different strategies can mitigate or eliminate the impact of such non-idealities. For this reason, the development of practical methods that allow a more faithful simulation of the noise experienced by a quantum system remains an important area of research. Many noise sources, such as charge noise in solid-state quantum technologies, exhibit temporal correlations over a multitude of time scales. This noise characteristic poses a challenge for classical simulations as it requires a sufficiently small time step to capture the short-time correlations (high-frequency components) of the noise but also long simulation times to capture the drift (low-frequency components) of the noise. In this work we present an approach for simulating Hamiltonian classical stochastic noise that performs temporal coarse-graining by effectively integrating out the high-frequency components. This is achieved by generating a realization of the noise on a coarse time grid and averaging over all noise realizations that connect the noise values on the coarse time grid. This approach then avoids the need for small time steps, allowing for long time simulations to be performed more effectively while still faithfully reproducing the temporal correlations of the noise.► BibTeX data@article{Albash2026temporalcoarse, doi = {10.22331/q-2026-06-15-2137}, url = {https://doi.org/10.22331/q-2026-06-15-2137}, title = {Temporal {C}oarse {G}raining for {C}lassical {S}tochastic {N}oise in {Q}uantum {S}ystems}, author = {Albash, Tameem and Young, Steve and Jacobson, N. Tobias}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2137}, month = jun, year = {2026} }► References [1] Joseph Emerson, Robert Alicki, and Karol Życzkowski. ``Scalable noise estimation with random unitary operators''. 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