Driven-dissipative quantum devices are routinely operated with time-dependent drives and exposed to temporally correlated classical noise. When these perturbations vary on timescales comparable to the system's intrinsic dynamics, rotating-wave and Markovian approximations can fail, making effective static descriptions unreliable. As a result, accurately modeling systems in this regime often relies on simulating the full time-dependent master equation. This is particularly demanding for processes requiring long integration times, such as rare switching between metastable states, or for stochastic perturbations requiring averaging over many noise realizations. To address these challenges, we introduce Liouvillian spectral snapshots (LSS), a reduced-basis method that enables efficient simulation of such systems by projecting the master equation onto a low-dimensional subspace capturing their coarse-grained dynamics. LSS constructs this subspace by computing the slowly decaying, low-frequency eigenoperators of the Liouvillian at a few adaptively sampled perturbation values and compressing them into a common basis. This basis is built once and can be reused across noise realizations and perturbation parameters, reducing the cost of ensemble averages and parameter sweeps.
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Simulating driven open systems with temporally correlated classical noise can require long integrations and many realizations, particularly when the quantities of interest are rare switching events or logical relaxation rates. This paper introduces Liouvillian spectral snapshots (LSS), a projection-based reduction targeting these slow dynamics without replacing the perturbation by a static or Markovian model.
LSS computes slow, low-frequency eigenoperators of instantaneous Liouvillians at adaptively sampled perturbation values, combines them using singular-value decomposition, and propagates the master equation in the resulting fixed basis. A greedy residual test controls enrichment across the sampled parameter range. The basis can be reused across noise realizations and spectra; identity inclusion and conjugation symmetry enforce trace and Hermiticity preservation.
Benchmarks compare LSS with full-space evolution. For a bistable Kerr resonator, bases of roughly $100$ operators replace a $1024$-dimensional Liouville space. Floquet switching rates agree within $1.5\%$ at three tested detunings. Lorentzian detuning noise resonant with a long-lived oscillating mode enhances switching by about an order of magnitude; one resonant benchmark has a $3.4\%$ rate discrepancy. For an eleven-qubit continuous repetition-code model, a $37$-dimensional basis replaces $4^{11}\approx4.2\times10^6$ dimensions and closely reproduces tested logical trajectories.
Accuracy depends on spectral separation and perturbation-induced leakage into discarded modes. Tightening the residual tolerance does not remove spectral-window truncation errors, and the test can miss slow modes with negligible basis overlap. The paper derives a trace-distance bound from the integrated projection residual and a perturbative leakage estimate. The projected generator is not constrained to Lindblad form.
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Error found: The paper extracts a long-time switching rate from one finite-time noise-averaged propagator without establishing that the extracted rate has converged with integration time, although colored-noise averages generally lack the semigroup property.
The paper's general argument that discarded-mode dissipation limits long-time propagation errors overlooks feedback that can shift retained slow rates even when fast-sector population stays small; its first-order leakage estimate does not establish saturation of the total error.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Driven-dissipative quantum devices are routinely operated with time-dependent drives and exposed to temporally correlated classical noise. When these perturbations vary on timescales comparable to the system's intrinsic dynamics, rotating-wave and Markovian approximations can fail, making effective static descriptions unreliable. As a result, accurately modeling systems in this regime often relies on simulating the full time-dependent master equation. This is particularly demanding for processes requiring long integration times, such as rare switching between metastable states, or for stochastic perturbations requiring averaging over many noise realizations. To address these challenges, we introduce Liouvillian spectral snapshots (LSS), a reduced-basis method that enables efficient simulation of such systems by projecting the master equation onto a low-dimensional subspace capturing their coarse-grained dynamics. LSS constructs this subspace by computing the slowly decaying, low-frequency eigenoperators of the Liouvillian at a few adaptively sampled perturbation values and compressing them into a common basis. This basis is built once and can be reused across noise realizations and perturbation parameters, reducing the cost of ensemble averages and parameter sweeps.
Simulating driven open systems with temporally correlated classical noise can require long integrations and many realizations, particularly when the quantities of interest are rare switching events or logical relaxation rates. This paper introduces Liouvillian spectral snapshots (LSS), a projection-based reduction targeting these slow dynamics without replacing the perturbation by a static or Markovian model.
LSS computes slow, low-frequency eigenoperators of instantaneous Liouvillians at adaptively sampled perturbation values, combines them using singular-value decomposition, and propagates the master equation in the resulting fixed basis. A greedy residual test controls enrichment across the sampled parameter range. The basis can be reused across noise realizations and spectra; identity inclusion and conjugation symmetry enforce trace and Hermiticity preservation.
Benchmarks compare LSS with full-space evolution. For a bistable Kerr resonator, bases of roughly 100 operators replace a 1024-dimensional Liouville space. Floquet switching rates agree within 1.5% at three tested detunings. Lorentzian detuning noise resonant with a long-lived oscillating mode enhances switching by about an order of magnitude; one resonant benchmark has a 3.4% rate discrepancy. For an eleven-qubit continuous repetition-code model, a 37-dimensional basis replaces 411≈4.2×106 dimensions and closely reproduces tested logical trajectories.
Accuracy depends on spectral separation and perturbation-induced leakage into discarded modes. Tightening the residual tolerance does not remove spectral-window truncation errors, and the test can miss slow modes with negligible basis overlap. The paper derives a trace-distance bound from the integrated projection residual and a perturbative leakage estimate. The projected generator is not constrained to Lindblad form.
The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. We demonstrate the efficiency of our method on several examples of coupled bosonic systems.
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.
This thesis develops model reduction methods for open quantum system dynamics, yielding efficient models for bipartite systems, with direct application to circuit QED and broader applicability beyond this platform. First, we consider dissipative dynamics characterized by timescale separation, which translates to a gap on the real axis of the spectrum of the Liouvillian generating it. This allows to compute the reduced evolution for a target subsystem by adiabatically eliminating fast-decaying degrees of freedom. A Sylvester equation integral representation yields closed forms for the reduced generators and explicit reconstruction maps to arbitrary order without a preliminary interaction-picture averaging step, providing a linear center manifold description that separates long-lived from fast-decaying modes by the real parts of the spectrum. The reduced dynamics takes place on the center manifold, that is, the spectral subspace associated with eigenvalues whose real parts are small (long-lived modes), while placing no restriction on imaginary parts, thus allowing for fast unitary dynamics on the target subsystem. We further demonstrate the equivalence of such a geometric approach to adiabatic elimination with the time-convolutionless master equation framework, by showing that the maps defining the reduced model satisfy the same invariance equation in both frameworks. The time-convolutionless formulation of adiabatic elimination extends to cases where the target system undergoes fast unitary dynamics and retains explicit time dependence in the reduced model, hence capturing initial transients that are neglected in time-independent geometric reductions. Second, we apply a unitary chain mapping of microscopic system–environment Hamiltonians to a nearest-neighbor form, enabling efficient time evolution exploiting tensor network time evolution algorithms based on the time dependent variational principle. Accuracy of the tensor network representation, and thus the size of the reduced model, is controlled by entanglement entropy growth, which we quantify numerically. While the methods are general and apply beyond the specific platform considered, we focus on dispersive readout in circuit QED (a qubit dipolarly coupled to a driven, lossy cavity). We quantify the dependence of the relaxation time T1 of the qubit on intracavity photon number nˉ (and thus on the drive amplitude) and on the spectral structure of the environment, and compute the emission spectrum directly from environmental observables. Simulations including a Purcell notch filter predict a decrease of T1 with increasing drive power, a qualitatively different behavior with respect to Born–Markov–secular GKSL predictions but consistent with experimental trends, indicating that explicit filter modeling is required in this case. Together, the results specify when reduced Liouvillian models under a dissipative gap are reliable and when microscopic, non-perturbative simulations are required.
The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent , and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.
Open quantum systems with nearly degenerate energy levels have been shown to exhibit long-lived metastable states in the approach to equilibrium, even when modelled with certain Lindblad-form quantum master equations. This is a result of dramatic separation of timescales due to differences between Liouvillian eigenvalues. These metastable states often have nonzero coherences that die off only in the long-time limit once the system reaches thermal equilibrium. We examine two distinct situations that give rise to this effect: one in which dissipative dynamics couple together states only within a nearly degenerate subspace, and one in which they give rise to jumps over finite-energy splittings, between nearly degenerate subspaces. We find, in each case, that a change of basis can often lead to a representation that more naturally captures the impact of the system-bath interaction than does the energy eigenbasis, revealing that separate timescales are associated with separate processes (e.g., decoherence into a nonenergy eigenbasis, decay of population correlations to the initial state). This approach is paired with the inspection of quantum trajectories, which further provide intuition as to how open system evolution is characterized when coherent oscillations, thermal relaxation, and decoherence all occur simultaneously. Published by the American Physical Society 2024
The ubiquitous effects of the environment on quantum-mechanical systems generally cause temporally correlated fluctuations. This particularly holds for systems of interest for quantum computation where such effects lead to correlated errors. The Markovian approximation neglects these correlations and thus fails to accurately describe open-system dynamics where these correlations become relevant. In driven open systems, yet another approximation is persistently used, often unknowingly, in which one describes the decay effects independently from the time-dependent controlling fields acting on the system, thereby ignoring further temporally correlated effects. To overcome these shortcomings, we develop a quantum master equation for driven systems weakly coupled to quantum environments that avoids the aforementioned field-independent approximation, as well as the Markovian approximation. Our method makes it possible to track all occurring decay channels and their time-dependent generalized rates which we illustrate in the example of a generally driven two-level system. We also demonstrate that correlated and field-dependent dissipative effects can lead to an increase in the performance of single-qubit gate operations.