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:2407.06068

Systematic time-coarse graining for driven quantum systems

Leon Bello, Fan, Wentao, Aditya Gandotra, Hakan E. Türeci

Published Jul 8, 2024• 0 citations
arXivHTMLPDF

Abstract

Many recent advancements in quantum computing leverage strong drives on nonlinear systems for state preparation, signal amplification, or gate operation. However, the interplay within such strongly driven system introduces multi-scale dynamics that affects the long-time behavior of the system in non-trivial ways that are very difficult to model. Therefore, the analysis of these systems often relies on effective Hamiltonian models that introduce additional nonlinear processes which approximate the long-time dynamics so that highly oscillatory terms may be ignored. However, the removal of such high frequency transitions can only be performed rigorously within a systematic framework of time-coarse graining, which is a fundamentally irreversible operation. This implies that standard approaches with unitary effective models cannot accurately capture the long-time behavior of strongly driven nonlinear quantum systems in general. We introduce a systematic perturbation theory for obtaining the complete non-unitary effective model of the time-coarse grained (TCG) dynamics of a driven quantum system to any order in the coupling strengths. We derive a closed-form analytical formula for both unitary and non-unitary contributions, in the form of an effective Hamiltonian and non-unitary (pseudo-)dissipators. Remarkably, even though the effective theory presumes unitary time evolution at the microscopic level, the time-coarse grained dynamics is found to follow a non-unitary time evolution in general. This occurs even when there is no open heat reservoir for the system to become entangled with or dissipate into. We demonstrate the effectiveness of the new method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.

DOI
10.48550/arxiv.2407.06068

Related papers

Systematic time-coarse-graining for driven quantum systems

Leon Bello, Wentao Fan, Aditya Gandotra +1 more

Real-world experiments on quantum systems are always performed with measurement apparatus whose interaction times with the systems are finite. This restricts the observable quantum states to the space of time-coarse-grained density matrices ${\overline{\ensuremath{\rho}}(t)}$, providing motivation for the time-coarse-graining (TCG) approach that solves for $\overline{\ensuremath{\rho}}(t)$ without ever referring to the unobservable $\ensuremath{\rho}(t)$ of infinite time resolution. Phenomenologically, this implies that coherent transitions far outside the bandwidth would be filtered out, leaving only their effective impacts on the ``slow'' dynamics resolvable by the finite time resolution of the measurements. Therefore, the TCG framework provides rigorous justification for many existing effective Hamiltonian methods in the literature that aim at capturing the unitary part of the long-time dynamics. However, since time-coarse-graining is fundamentally irreversible, the resulting effective model allows for secular loss of information and dissipation of energy in general, which cannot be captured by any unitary effective models and has to be treated with explicit time-coarse-graining. Such incoherent effects are particularly prominent in driven nonlinear quantum systems where exchange of information and energy with the drive gives rise to incoherent effective dynamics at all finite time resolutions in general. While rigorous in principle, existing TCG methods in the literature can be applied only to simple systems by one iteratively solving superoperator equations at low orders in the coupling strengths. The complexity of such methods prevents systematic study of the time-coarse-grained dynamics and limits analytical results to the IR (low-resolution) limit in most cases. We address these limitations in this paper, presenting a systematic time-coarse-graining method that overcomes the complexities and restrictions of current techniques, offering a comprehensive and accurate modeling framework for driven nonlinear quantum systems. We derive closed-form formulas as well as diagrammatic representations for both unitary and nonunitary contributions, in the form of an effective Hamiltonian and nonunitary dissipators at arbitrary order in the coupling strengths, and complement this with a computer-algebra software package. We demonstrate the effectiveness of the method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.

Physical Review Applied•May 15, 2025•2 citations

Model Order Reduction for Open Quantum Systems Based on Measurement-Adapted Time-Coarse Graining

Wentao Fan, Hakan E. Türeci

Model order reduction encompasses mathematical techniques aimed at reducing the complexity of mathematical models in simulations while retaining the essential characteristics and behaviors of the original model. This is particularly useful in the context of large-scale dynamical systems, which can be computationally expensive to analyze and simulate. Here, we present a model order reduction technique to reduce the time complexity of open quantum systems, grounded in the principle of measurement-adapted coarse graining. This method, governed by a coarse-graining timescale τ and the spectral band center ω 0 of the measurement channel, organizes corrections to the lowest-order model which aligns with the rotating-wave approximation Hamiltonian in certain limits and rigorously justifies the resulting effective quantum master equation (EQME). The focus on calculating to a high degree of accuracy what can only be resolved by the measurement introduces a principled regularization procedure to address singularities and generates low-stiffness models suitable for efficient long-time integration. The closed-form EQME parameters greatly enhance interpretability, while the predicted EQME structure provides a foundation for deriving stochastic master equations under continuous measurement and for constructing efficient gray-box models for system identification and control. As a demonstration, we derive the fourth-order EQME for a challenging problem related to the dynamics of a superconducting qubit under high-power dispersive readout in the presence of a continuum of dissipative waveguide modes. This derivation shows that the lowest-order terms align with previous results, while higher-order corrections suggest new phenomena.

arXiv:2410.23116•Physical Review X•Mar 12, 2026•1 citation

Effective formalism for open-quantum-system dynamics: Time-coarse-graining approach

Chang-Woo Lee, Changsuk Noh, Jaewan Kim

We formulate an effective-description framework for the dynamics of open quantum systems by extending the time-coarse-graining formalism to open systems. Our coarse-graining procedure efficiently removes high-frequency processes which are responsible for coherences between lower- and upper-manifold states and are irrelevant when considering only low-energy dynamics. We investigate the regime of validity of the resulting coarse-grained master equation by applying it to multilevel atoms driven by far-detuned lasers. Except for such high-frequency coherences, we find good agreement between the exact and coarse-grained dynamics unless the driving lasers are too strong or the initial high-frequency coherences are sizable.

arXiv:1709.00591•Physical Review A•Jan 8, 2018•6 citations

Time-dependent properties of solids : non-perturbative Floquet Hamiltonians and beyond

Michael Rudolf Vogl

In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.

•May 1, 2020•0 citations

Coarse-grained dynamics in quantum many-body systems using the maximum entropy principle

Adán Castillo, Carlos Pineda, Erick Sebastían Navarrete +1 more

Starting from a coarse-grained map of a quantum many-body system, we construct the inverse map that assigns a microscopic state to a coarse-grained state based on the maximum entropy principle. Assuming unitary evolution in the microscopic system, we examine the resulting dynamics in the coarse-grained system using the assignment map. We investigate both a two-qubit system, with \swap\ and controlled-\textsc{not}\ gates, and $n$-qubit systems, configured either in an Ising spin chain or with all-to-all interactions. We demonstrate that these dynamics exhibit atypical quantum behavior, such as nonlinearity and non-Markovianity. Furthermore, we find that these dynamics depend on the initial coarse-grained state and establish conditions for general microscopic dynamics under which linearity is preserved. As the effective dynamics induced by our coarse-grained description of many-body quantum systems diverge from conventional quantum behavior, we anticipate that this approach could aid in describing the quantum-to-classical transition and provide deeper insights into the effects of coarse graining on quantum systems.

arXiv:2407.11920•Physical Review A•Jul 22, 2025•1 citation

Coarse-Grained Effective Hamiltonian via the Magnus Expansion for a Three-Level System

Nicola Macrì, Luigi Giannelli, Elisabetta Paladino +1 more

Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control, they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties, hindering a systematic improvement of their accuracy in larger and larger systems. Here, we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.

arXiv:2212.08508•Entropy•Jan 27, 2023•3 citations
Many recent advancements in quantum computing leverage strong drives on nonlinear systems for state preparation, signal amplification, or gate operation. However, the interplay within such strongly driven system introduces multi-scale dynamics that affects the long-time behavior of the system in non-trivial ways that are very difficult to model. Therefore, the analysis of these systems often relies on effective Hamiltonian models that introduce additional nonlinear processes which approximate the long-time dynamics so that highly oscillatory terms may be ignored. However, the removal of such high frequency transitions can only be performed rigorously within a systematic framework of time-coarse graining, which is a fundamentally irreversible operation. This implies that standard approaches with unitary effective models cannot accurately capture the long-time behavior of strongly driven nonlinear quantum systems in general. We introduce a systematic perturbation theory for obtaining the complete non-unitary effective model of the time-coarse grained (TCG) dynamics of a driven quantum system to any order in the coupling strengths. We derive a closed-form analytical formula for both unitary and non-unitary contributions, in the form of an effective Hamiltonian and non-unitary (pseudo-)dissipators. Remarkably, even though the effective theory presumes unitary time evolution at the microscopic level, the time-coarse grained dynamics is found to follow a non-unitary time evolution in general. This occurs even when there is no open heat reservoir for the system to become entangled with or dissipate into. We demonstrate the effectiveness of the new method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.
Real-world experiments on quantum systems are always performed with measurement apparatus whose interaction times with the systems are finite. This restricts the observable quantum states to the space of time-coarse-grained density matrices ${\overline{\ensuremath{\rho}}(t)}$, providing motivation for the time-coarse-graining (TCG) approach that solves for $\overline{\ensuremath{\rho}}(t)$ without ever referring to the unobservable $\ensuremath{\rho}(t)$ of infinite time resolution. Phenomenologically, this implies that coherent transitions far outside the bandwidth would be filtered out, leaving only their effective impacts on the ``slow'' dynamics resolvable by the finite time resolution of the measurements. Therefore, the TCG framework provides rigorous justification for many existing effective Hamiltonian methods in the literature that aim at capturing the unitary part of the long-time dynamics. However, since time-coarse-graining is fundamentally irreversible, the resulting effective model allows for secular loss of information and dissipation of energy in general, which cannot be captured by any unitary effective models and has to be treated with explicit time-coarse-graining. Such incoherent effects are particularly prominent in driven nonlinear quantum systems where exchange of information and energy with the drive gives rise to incoherent effective dynamics at all finite time resolutions in general. While rigorous in principle, existing TCG methods in the literature can be applied only to simple systems by one iteratively solving superoperator equations at low orders in the coupling strengths. The complexity of such methods prevents systematic study of the time-coarse-grained dynamics and limits analytical results to the IR (low-resolution) limit in most cases. We address these limitations in this paper, presenting a systematic time-coarse-graining method that overcomes the complexities and restrictions of current techniques, offering a comprehensive and accurate modeling framework for driven nonlinear quantum systems. We derive closed-form formulas as well as diagrammatic representations for both unitary and nonunitary contributions, in the form of an effective Hamiltonian and nonunitary dissipators at arbitrary order in the coupling strengths, and complement this with a computer-algebra software package. We demonstrate the effectiveness of the method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.
Model order reduction encompasses mathematical techniques aimed at reducing the complexity of mathematical models in simulations while retaining the essential characteristics and behaviors of the original model. This is particularly useful in the context of large-scale dynamical systems, which can be computationally expensive to analyze and simulate. Here, we present a model order reduction technique to reduce the time complexity of open quantum systems, grounded in the principle of measurement-adapted coarse graining. This method, governed by a coarse-graining timescale τ and the spectral band center ω 0 of the measurement channel, organizes corrections to the lowest-order model which aligns with the rotating-wave approximation Hamiltonian in certain limits and rigorously justifies the resulting effective quantum master equation (EQME). The focus on calculating to a high degree of accuracy what can only be resolved by the measurement introduces a principled regularization procedure to address singularities and generates low-stiffness models suitable for efficient long-time integration. The closed-form EQME parameters greatly enhance interpretability, while the predicted EQME structure provides a foundation for deriving stochastic master equations under continuous measurement and for constructing efficient gray-box models for system identification and control. As a demonstration, we derive the fourth-order EQME for a challenging problem related to the dynamics of a superconducting qubit under high-power dispersive readout in the presence of a continuum of dissipative waveguide modes. This derivation shows that the lowest-order terms align with previous results, while higher-order corrections suggest new phenomena.
We formulate an effective-description framework for the dynamics of open quantum systems by extending the time-coarse-graining formalism to open systems. Our coarse-graining procedure efficiently removes high-frequency processes which are responsible for coherences between lower- and upper-manifold states and are irrelevant when considering only low-energy dynamics. We investigate the regime of validity of the resulting coarse-grained master equation by applying it to multilevel atoms driven by far-detuned lasers. Except for such high-frequency coherences, we find good agreement between the exact and coarse-grained dynamics unless the driving lasers are too strong or the initial high-frequency coherences are sizable.
In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.
Starting from a coarse-grained map of a quantum many-body system, we construct the inverse map that assigns a microscopic state to a coarse-grained state based on the maximum entropy principle. Assuming unitary evolution in the microscopic system, we examine the resulting dynamics in the coarse-grained system using the assignment map. We investigate both a two-qubit system, with \swap\ and controlled-\textsc{not}\ gates, and nnn-qubit systems, configured either in an Ising spin chain or with all-to-all interactions. We demonstrate that these dynamics exhibit atypical quantum behavior, such as nonlinearity and non-Markovianity. Furthermore, we find that these dynamics depend on the initial coarse-grained state and establish conditions for general microscopic dynamics under which linearity is preserved. As the effective dynamics induced by our coarse-grained description of many-body quantum systems diverge from conventional quantum behavior, we anticipate that this approach could aid in describing the quantum-to-classical transition and provide deeper insights into the effects of coarse graining on quantum systems.
Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control, they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties, hindering a systematic improvement of their accuracy in larger and larger systems. Here, we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.