We propose a novel approach to numerically derive approximate reduced dynamical models for Markovian quantum open systems without perturbative iterations, projecting the evolution to the subspace associated to their slowest degrees of freedom. The two algorithms we develop are based on Oja's continuous-time principal component flow: the first returns the optimal reduction to the slowest decaying operator-subspace, and is extended to time-dependent dynamics, while the second one is designed to reduce the dynamics on a subspace of the system's Hilbert space, and thus preserve conditional complete positivity. The methods represent a non-perturbative alternative to well-established Adiabatic Elimination (AE) methods, and the second can be used to find noise-protected subspace codes for quantum information processing. Both are tested on a paradigmatic central spin model.
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.
Adiabatic elimination methods allow the reduction of the space dimension needed to describe systems whose dynamics exhibit separation of timescale. For open quantum systems, it consists in eliminating the fast part assuming it has almost instantaneously reached its steady state and obtaining an approximation of the evolution of the slow part. These methods can be applied to eliminate a linear subspace within the system Hilbert space or, alternatively, to eliminate a fast subsystem in a bipartite quantum system. In this work, we extend an adiabatic elimination method used for removing fast degrees of freedom within a system [Phys. Rev. A 101, 042102 (2020)] to eliminate a subsystem from an open bipartite quantum system. As an illustration, we apply our technique to a dispersively coupled two-qubit system and in the case of the open Rabi model.
We consider finite-dimensional many-body quantum systems described by time-independent Hamiltonians and Markovian master equations, and present a systematic method for constructing smaller-dimensional, reduced models that e x a c t l y reproduce the time evolution of a set of initial conditions or observables of interest. Our approach exploits Krylov operator spaces and their extension to operator algebras, and may be used to obtain reduced linear models of minimal dimension, well-suited for simulation on classical computers, or reduced quantum models that preserve the structural constraints of physically admissible quantum dynamics, as required for simulation on quantum computers. Notably, we prove that the reduced quantum-dynamical generator is still in Lindblad form. By introducing a new type of observable-dependent symmetries , we show that our method provides a non-trivial generalization of techniques that leverage symmetries, unlocking new reduction opportunities. We quantitatively benchmark our method on paradigmatic open many-body systems of relevance to condensed-matter and quantum-information physics. In particular, we demonstrate how our reduced models can quantitatively describe decoherence dynamics in central-spin systems coupled to structured environments, magnetization transport in boundary-driven dissipative spin chains, and unwanted error dynamics on information encoded in a noiseless quantum code.
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.
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.
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.