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arXiv:2411.13166 · Communications Physics

Making quantum collision models exact

Thibaut Lacroix, Dario Cilluffo, Susana F. Huelga, Martin Bodo Plenio

Published Jul 1, 2025• 5 citations
arXivHTMLPDF

Abstract

Abstract Quantum collision models describe open quantum systems through repeated interactions with a coarse-grained environment. However, a complete certification of these models is lacking, as no complete error bounds on the simulation of system observables have been established. Here, we show that Markovian and non-Markovian collision models can be recovered analytically from chain mapping techniques starting from a general microscopic Hamiltonian. This derivation reveals a previously unidentified source of error—induced by an unfaithful sampling of the environment—in dynamics obtained with collision models that can become dominant for small but finite time-steps. With the complete characterization of this error, all collision models errors are now identified and quantified, which enables the promotion of collision models to the class of numerically exact methods. To confirm the predictions of our equivalence results, we implemented a non-Markovian collision model of the Spin Boson Model, and identified, as predicted, a regime in which the collision model is fundamentally inaccurate.

DOI
10.1038/s42005-025-02201-2

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Abstract Quantum collision models describe open quantum systems through repeated interactions with a coarse-grained environment. However, a complete certification of these models is lacking, as no complete error bounds on the simulation of system observables have been established. Here, we show that Markovian and non-Markovian collision models can be recovered analytically from chain mapping techniques starting from a general microscopic Hamiltonian. This derivation reveals a previously unidentified source of error—induced by an unfaithful sampling of the environment—in dynamics obtained with collision models that can become dominant for small but finite time-steps. With the complete characterization of this error, all collision models errors are now identified and quantified, which enables the promotion of collision models to the class of numerically exact methods. To confirm the predictions of our equivalence results, we implemented a non-Markovian collision model of the Spin Boson Model, and identified, as predicted, a regime in which the collision model is fundamentally inaccurate.
We develop randomized quantum algorithms to simulate quantum collision models, also known as repeated interaction schemes, which provide a rich framework to model various open-system dynamics. The underlying technique involves composing time evolutions of the total (system, bath, and interaction) Hamiltonian and intermittent tracing out the environment degrees of freedom. This results in a unified framework where any near-term Hamiltonian simulation algorithm can be incorporated to implement an arbitrary number of such collisions on early fault-tolerant quantum computers: we do not assume access to specialized oracles such as block encodings and minimize the number of ancilla qubits needed. In particular, using the correspondence between Lindbladian evolution and completely positive trace-preserving maps arising out of memoryless collisions, we provide an end-to-end quantum algorithm for simulating Lindbladian dynamics. For a system of nnn-qubits, we exhaustively compare the circuit depth needed to estimate the expectation value of an observable with respect to the reduced state of the system after time ttt while employing different near-term Hamiltonian simulation techniques, requiring at most n+2n+2n+2 qubits in all. We compare the cnot gate counts of the various approaches for estimating the transverse field magnetization of a 10-qubit XX-Heisenberg spin chain under amplitude damping. Finally, we also develop a framework to efficiently simulate an arbitrary number of memory-retaining collisions, i.e., where environments interact, leading to non-Markovian dynamics. Overall, our methods can leverage quantum collision models for both Markovian and non-Markovian dynamics on early fault-tolerant quantum computers, shedding light on the advantages and limitations of simulating open systems dynamics using this framework.
Non-Markovian effects are important in modeling the behavior of open quantum systems arising in solid-state physics, quantum optics as well as in study of biological and chemical systems. A common approach to the analysis of such systems is to approximate the non-Markovian environment by discrete bosonic modes thus mapping it to a Lindbladian or Hamiltonian simulation problem. While systematic constructions of such modes have been proposed in previous works [D. Tamascelli et al, PRL (2012), A. W. Chin et al, J. of Math. Phys (2010)], the resulting approximation lacks rigorous convergence guarantees. In this paper, we initiate a rigorous study of the convergence properties of these methods. We show that under some physically motivated assumptions on the system-environment interaction, the finite-time dynamics of the non-Markovian open quantum system computed with a sufficiently large number of modes is guaranteed to converge to the true result. Furthermore, we show that, for most physically interesting models of non-Markovian environments, the approximation error falls off polynomially with the number of modes. Our results lend rigor to numerical methods used for approximating non-Markovian quantum dynamics and allow for a quantitative assessment of classical as well as quantum algorithms in simulating non-Markovian quantum systems.
A Dissertation on open quantum systems, containing a general review of the current standing of the field, in particular regarding systems interacting with Gaussian environments, as well as original analytical and numerical results. The problem of the impact of a variation in the environmental properties on the reduced dynamics of an open system is investigated analytically, and a new computational technique for the simulation of non-perturbative, non-Markovian open quantum system dynamics based on devising error-certified effective environments is presented.
We present a theoretical framework to tackle quantum non-Markovian dynamics based on a microscopic collision model (CM), where the bath consists of a large collection of initially uncorrelated ancillas. Unlike standard memoryless CMs, we endow the bath with memory by introducing interancillary collisions between next system-ancilla interactions. Our model interpolates between a fully Markovian dynamics and the continuous interaction of the system with a single ancilla, i.e., a strongly non-Markovian process. We show that in the continuous limit one can derive a general master equation, which, while keeping such features, is guaranteed to describe an unconditionally completely positive and trace-preserving dynamics. We apply our theory to an atom in a dissipative cavity for a Lorentzian spectral density of bath modes, a dynamics which can be exactly solved. The predicted evolution shows a significant improvement in approaching the exact solution with respect to two well-known memory-kernel master equations.
Recently, a large class of quantum non-Markovian piecewise dynamics for an open quantum system obeying closed evolution equations has been introduced [1]. These dynamics have been defined in terms of a waiting-time distribution between quantum jumps, along with quantum maps describing the effect of jumps and the system evolution between them. Here, we present a quantum collision model with memory, whose reduced dynamics in the continuous-time limit reproduces the above class of non-Markovian piecewise dynamics, thus providing an explicit microscopic realization.
The quantum collision models are a useful method to describe the dynamics of an open quantum system by means of repeated interactions between the system and some particles of the environment, which are usually termed “ancillas”. In this paper, we review the main collision models for the dynamics of multipartite open quantum systems, which are composed of several subsystems. In particular, we are interested in models that are based on elementary collisions between the subsystems and the ancillas, and that simulate global and/or local Markovian master equations in the limit of infinitesimal timestep. After discussing the mathematical details of the derivation of a generic collision-based master equation, we provide the general ideas at the basis of the collision models for multipartite systems, we discuss their strengths and limitations, and we show how they may be simulated on a quantum computer. Moreover, we analyze some properties of a collision model based on entangled ancillas, derive the master equation it generates for small timesteps, and prove that the coefficients of this master equation are subject to a constraint that limits their generality. Finally, we present an example of such collision model with two bosonic ancillas entangled in a two-mode squeezed thermal state.