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:2303.17308 · IEEE Conference on Decision and Control (CDC)

Heisenberg Formulation of Adiabatic Elimination for Open Quantum Systems with Two Timescales

François-Marie Le Régent, Pierre Rouchon

Published Dec 13, 2023• 4 citations
arXivHTMLPDF

Abstract

Consider an open quantum system governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from small decoherence and Hamiltonian dynamics. Usually adiabatic elimination is performed in the Schrödinger picture. We propose here a Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasi-equilibria subspace play a key role. Based on geometric singular perturbations, asymptotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from center-manifold and bifurcation theory. Second-order expansions are detailed and shown to ensure preservation, up to second-order terms, of the complete positivity for the slow propagator on a slow timescale. Such expansions can be exploited numerically to derive reduced-order dynamical models.

DOI
10.1109/cdc49753.2023.10384313

Related papers

Adiabatic elimination for composite open quantum systems: Reduced-model formulation and numerical simulations

François-Marie Le Régent, Pierre Rouchon

A numerical method is proposed for simulation of composite open quantum systems. It is based on Lindblad master equations and adiabatic elimination. Each subsystem is assumed to converge exponentially towards a stationary subspace, slightly impacted by some decoherence channels and weakly coupled to the other subsystems. This numerical method is based on a perturbation analysis with an asymptotic expansion. It exploits the formulation of the slow dynamics with reduced dimension. It relies on the invariant operators of the local and nominal dissipative dynamics attached to each subsystem. Second-order expansion can be computed only with local numerical calculations. It avoids computations on the tensor-product Hilbert space attached to the full system. This numerical method is particularly well suited for autonomous quantum error correction schemes. Simulations of such reduced models agree with complete full model simulations for typical gates acting on one and two cat qubits (z, zz, and cnot) when the mean photon number of each cat qubit is less than eight. For larger mean photon numbers and gates with three cat qubits (zzz and ccnot), full model simulations are almost impossible whereas reduced model simulations remain accessible. In particular, one observes numerically the simultaneous capture of the dominant phase-flip error rate and of the very small bit-flip error rate with its exponential suppression versus the mean photon number.

arXiv:2303.05089•Physical Review A•Mar 6, 2024•16 citations

Explicit formulas for adiabatic elimination with fast unitary dynamics

Angela Riva, Alain Sarlette, Pierre Rouchon

The so-called “adiabatic elimination” of fast decaying degrees of freedom in open quantum systems can be performed with a series expansion in the timescale separation. The associated computations are significantly more difficult when the remaining degrees of freedom (center manifold) follow fast unitary dynamics instead of just being slow. This paper highlights how a formulation with Sylvester’s equation and with adjoint dynamics leads to systematic, explicit expressions at high orders for settings of physical interest.

arXiv:2404.01802•IEEE Conference on Decision and Control (CDC)•Dec 16, 2024•0 citations

Complete positivity violation of the reduced dynamics in higher-order quantum adiabatic elimination

Masaaki Tokieda, Cyril Elouard, Alain Sarlette +1 more

This paper discusses quantum adiabatic elimination, which is a model reduction technique for a composite Lindblad system consisting of a fast decaying subsystem coupled to another subsystem with a much slower timescale. Such a system features an invariant manifold that is close to the slow subsystem. This invariant manifold is reached subsequent to the decay of the fast degrees of freedom, after which the slow dynamics follow on it. By parametrizing the invariant manifold, the slow dynamics can be simulated via a reduced model. To find the evolution of the reduced state, we perform an asymptotic expansion with respect to the timescale separation. So far, the second-order expansion has mostly been considered. It has then been revealed that the second-order expansion of the reduced dynamics is generally given by a Lindblad equation, which ensures complete positivity of the time evolution. In this paper, we present two examples where complete positivity of the reduced dynamics is violated with higher-order contributions. In the first example, the violation is detected for the evolution of the partial trace without truncation of the asymptotic expansion. The partial trace is not the only way to parametrize the slow dynamics. Concerning this nonuniqueness, it was conjectured in [Quantum Sci. Technol. 2, 044011 (2017)] that there exists a parameter choice ensuring complete positivity. With the second example, however, we refute this conjecture by showing that complete positivity cannot be restored in any choice of parametrization. We discuss these results in terms of the invariant slow manifold consisting of quantum correlated states.

arXiv:2303.04495•Physical Review A•Jun 3, 2024•7 citations

Adiabatic elimination and subspace evolution of open quantum systems

Daniel Finkelstein‐Shapiro, David Viennot, Ibrahim Saideh +3 more

Efficient descriptions of open quantum systems can be obtained by performing an adiabatic elimination of the fast degrees of freedom and formulating effective operators for the slow degrees of freedom in reduced dimensions. Here, we perform the construction of effective operators in frequency space, and using the final value theorem or alternatively the Keldysh theorem, we provide a correction for the trace of the density matrix which takes into account the non-trace-preserving character of the evolution. We illustrate our results with two different systems, one where the eliminated fast subspace is constituted by a continuous set of states and ones with discrete states. Furthermore, we show that the two models converge for very large dissipation and at coherent population trapping points. Our results also provide an intuitive picture of the correction to the trace of the density matrix as a detailed balance equation.

arXiv:1909.04211•Physical Review A•Apr 1, 2020•24 citations

Adiabatic Elimination for Multi-Partite Open Quantum Systems with Non-Trivial Zero-Order Dynamics

Paolo Forni, Alain Sarlette, Thibault Capelle +3 more

We provide model reduction formulas for open quantum systems consisting of a target component which weakly interacts with a strongly dissipative environment. The time-scale separation between the uncoupled dynamics and the interaction allows to employ tools from center manifold theory and geometric singular perturbation theory to eliminate the variables associated to the environment (adiabatic elimination) with high-order accuracy. An important specificity is to preserve the quantum structure: reduced dynamics in (positive) Lindblad form and coordinate mappings in Kraus form. We provide formulas of the reduced dynamics. The main contributions of this paper are (i) to show how the decomposition of the environment into $K$ components enables its efficient treatment, avoiding the quantum curse of dimension; and (ii) to extend the results to the case where the target component is subject to Hamiltonian evolution at the fast time-scale. We apply our theory to a microwave superconducting quantum resonator subject to material losses, and we show that our reduced-order model can explain the transmission spectrum observed in a recent pump probe experiment.

arXiv:1803.07810•IEEE Conference on Decision and Control (CDC)•Dec 1, 2018•10 citations

Approximate Reduced Lindblad Dynamics via Algebraic and Adiabatic Methods

Tommaso Grigoletto, Alain Sarlette, Francesco Ticozzi +1 more

We present an algebraic framework for approximate model reduction of Markovian open quantum dynamics that guarantees complete positivity and trace preservation by construction. First, we show that projecting a Lindblad generator on its center manifold -- the space spanned by eigenoperators with purely imaginary eigenvalue -- yields an asymptotically exact reduced quantum dynamical semigroup whose dynamics is unitary, with exponentially decaying transient error controlled by the generator's spectral gap. Second, for analytic perturbations of a Lindblad generator with a tractable center manifold, we propose a perturbative reduction that keeps the reduced space fixed at the unperturbed center manifold. The resulting generator is shown to remain a valid Lindbladian; explicit finite-time error bounds that quantify leakage from the unperturbed center sector are provided. We further clarify the connection to adiabatic elimination methods, by both showing how the algebraic reduction can be directly related to a first-order adiabatic-elimination and by providing sufficient conditions under which the latter method can be applied while preserving complete positivity. We showcase the usefulness of our techniques in dissipative many-body quantum systems exhibiting non-stationary long-time dynamics.

quant-pharXiv:2603.11982•Quantum Science and Technology•Mar 13, 2026•0 citations
Consider an open quantum system governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from small decoherence and Hamiltonian dynamics. Usually adiabatic elimination is performed in the Schrödinger picture. We propose here a Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasi-equilibria subspace play a key role. Based on geometric singular perturbations, asymptotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from center-manifold and bifurcation theory. Second-order expansions are detailed and shown to ensure preservation, up to second-order terms, of the complete positivity for the slow propagator on a slow timescale. Such expansions can be exploited numerically to derive reduced-order dynamical models.
A numerical method is proposed for simulation of composite open quantum systems. It is based on Lindblad master equations and adiabatic elimination. Each subsystem is assumed to converge exponentially towards a stationary subspace, slightly impacted by some decoherence channels and weakly coupled to the other subsystems. This numerical method is based on a perturbation analysis with an asymptotic expansion. It exploits the formulation of the slow dynamics with reduced dimension. It relies on the invariant operators of the local and nominal dissipative dynamics attached to each subsystem. Second-order expansion can be computed only with local numerical calculations. It avoids computations on the tensor-product Hilbert space attached to the full system. This numerical method is particularly well suited for autonomous quantum error correction schemes. Simulations of such reduced models agree with complete full model simulations for typical gates acting on one and two cat qubits (z, zz, and cnot) when the mean photon number of each cat qubit is less than eight. For larger mean photon numbers and gates with three cat qubits (zzz and ccnot), full model simulations are almost impossible whereas reduced model simulations remain accessible. In particular, one observes numerically the simultaneous capture of the dominant phase-flip error rate and of the very small bit-flip error rate with its exponential suppression versus the mean photon number.
The so-called “adiabatic elimination” of fast decaying degrees of freedom in open quantum systems can be performed with a series expansion in the timescale separation. The associated computations are significantly more difficult when the remaining degrees of freedom (center manifold) follow fast unitary dynamics instead of just being slow. This paper highlights how a formulation with Sylvester’s equation and with adjoint dynamics leads to systematic, explicit expressions at high orders for settings of physical interest.
This paper discusses quantum adiabatic elimination, which is a model reduction technique for a composite Lindblad system consisting of a fast decaying subsystem coupled to another subsystem with a much slower timescale. Such a system features an invariant manifold that is close to the slow subsystem. This invariant manifold is reached subsequent to the decay of the fast degrees of freedom, after which the slow dynamics follow on it. By parametrizing the invariant manifold, the slow dynamics can be simulated via a reduced model. To find the evolution of the reduced state, we perform an asymptotic expansion with respect to the timescale separation. So far, the second-order expansion has mostly been considered. It has then been revealed that the second-order expansion of the reduced dynamics is generally given by a Lindblad equation, which ensures complete positivity of the time evolution. In this paper, we present two examples where complete positivity of the reduced dynamics is violated with higher-order contributions. In the first example, the violation is detected for the evolution of the partial trace without truncation of the asymptotic expansion. The partial trace is not the only way to parametrize the slow dynamics. Concerning this nonuniqueness, it was conjectured in [Quantum Sci. Technol. 2, 044011 (2017)] that there exists a parameter choice ensuring complete positivity. With the second example, however, we refute this conjecture by showing that complete positivity cannot be restored in any choice of parametrization. We discuss these results in terms of the invariant slow manifold consisting of quantum correlated states.
Efficient descriptions of open quantum systems can be obtained by performing an adiabatic elimination of the fast degrees of freedom and formulating effective operators for the slow degrees of freedom in reduced dimensions. Here, we perform the construction of effective operators in frequency space, and using the final value theorem or alternatively the Keldysh theorem, we provide a correction for the trace of the density matrix which takes into account the non-trace-preserving character of the evolution. We illustrate our results with two different systems, one where the eliminated fast subspace is constituted by a continuous set of states and ones with discrete states. Furthermore, we show that the two models converge for very large dissipation and at coherent population trapping points. Our results also provide an intuitive picture of the correction to the trace of the density matrix as a detailed balance equation.
We provide model reduction formulas for open quantum systems consisting of a target component which weakly interacts with a strongly dissipative environment. The time-scale separation between the uncoupled dynamics and the interaction allows to employ tools from center manifold theory and geometric singular perturbation theory to eliminate the variables associated to the environment (adiabatic elimination) with high-order accuracy. An important specificity is to preserve the quantum structure: reduced dynamics in (positive) Lindblad form and coordinate mappings in Kraus form. We provide formulas of the reduced dynamics. The main contributions of this paper are (i) to show how the decomposition of the environment into KKK components enables its efficient treatment, avoiding the quantum curse of dimension; and (ii) to extend the results to the case where the target component is subject to Hamiltonian evolution at the fast time-scale. We apply our theory to a microwave superconducting quantum resonator subject to material losses, and we show that our reduced-order model can explain the transmission spectrum observed in a recent pump probe experiment.
We present an algebraic framework for approximate model reduction of Markovian open quantum dynamics that guarantees complete positivity and trace preservation by construction. First, we show that projecting a Lindblad generator on its center manifold -- the space spanned by eigenoperators with purely imaginary eigenvalue -- yields an asymptotically exact reduced quantum dynamical semigroup whose dynamics is unitary, with exponentially decaying transient error controlled by the generator's spectral gap. Second, for analytic perturbations of a Lindblad generator with a tractable center manifold, we propose a perturbative reduction that keeps the reduced space fixed at the unperturbed center manifold. The resulting generator is shown to remain a valid Lindbladian; explicit finite-time error bounds that quantify leakage from the unperturbed center sector are provided. We further clarify the connection to adiabatic elimination methods, by both showing how the algebraic reduction can be directly related to a first-order adiabatic-elimination and by providing sufficient conditions under which the latter method can be applied while preserving complete positivity. We showcase the usefulness of our techniques in dissipative many-body quantum systems exhibiting non-stationary long-time dynamics.