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arXiv:0803.3267 · Physical Review A

Efficient simulation of quantum evolution using dynamical coarse graining

Michael Khasin, Ronnie Kosloff

Published Jul 11, 2008• 7 citations
arXivHTMLPDF

Abstract

A scheme to simulate the evolution of a restricted set of observables of a quantum system is proposed. The set comprises the spectrum-generating algebra of the Hamiltonian. Focusing on the simulation of the restricted set allows to drastically reduce the cost of the simulation. This reduction is the result of replacing the original unitary dynamics by a special open-system evolution. This open-system evolution can be interpreted as a process of weak measurement of the distinguished observables performed on the evolving system of interest. Under the condition that the observables are ``classical'' and the Hamiltonian is moderately nonlinear, the open-system dynamics displays a large time-scale separation between the relaxation of the observables and the decoherence of a generic state. This time-scale separation allows the unitary dynamics of the observables to be efficiently simulated by the open-system dynamics on the intermediate time scale. The simulation employs unraveling of the corresponding master equations into pure-state evolutions, governed by the stochastic nonlinear Schr\"odinger equation. The stochastic pure-state evolution can be simulated efficiently using a representation of the state in the time-dependent basis of the generalized coherent states, associated with the spectrum-generating algebra.

DOI
10.1103/physreva.78.012321

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A scheme to simulate the evolution of a restricted set of observables of a quantum system is proposed. The set comprises the spectrum-generating algebra of the Hamiltonian. Focusing on the simulation of the restricted set allows to drastically reduce the cost of the simulation. This reduction is the result of replacing the original unitary dynamics by a special open-system evolution. This open-system evolution can be interpreted as a process of weak measurement of the distinguished observables performed on the evolving system of interest. Under the condition that the observables are ``classical'' and the Hamiltonian is moderately nonlinear, the open-system dynamics displays a large time-scale separation between the relaxation of the observables and the decoherence of a generic state. This time-scale separation allows the unitary dynamics of the observables to be efficiently simulated by the open-system dynamics on the intermediate time scale. The simulation employs unraveling of the corresponding master equations into pure-state evolutions, governed by the stochastic nonlinear Schr"odinger equation. The stochastic pure-state evolution can be simulated efficiently using a representation of the state in the time-dependent basis of the generalized coherent states, associated with the spectrum-generating algebra.
An algorithm for simulation of quantum many-body dynamics having su(2)(2)(2) spectrum-generating algebra is developed. The algorithm is based on the idea of dynamical coarse-graining. The original unitary dynamics of the target observables---the elements of the spectrum-generating algebra---is simulated by a surrogate open-system dynamics, which can be interpreted as weak measurement of the target observables, performed on the evolving system. The open-system state can be represented by a mixture of pure states, localized in the phase space. The localization reduces the scaling of the computational resources with the Hilbert-space dimension nnn by factor ${n}^{3/2}(\mathrm{ln} n){}^{\ensuremath{-}1}$ compared to conventional sparse-matrix methods. The guidelines for the choice of parameters for the simulation are presented and the scaling of the computational resources with the Hilbert-space dimension of the system is estimated. The algorithm is applied to the simulation of the dynamics of systems of $2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ and $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ cold atoms in a double-well trap, described by the two-site Bose-Hubbard model.
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.
Constrained Hamiltonian dynamics is exploited to provide the mathematical framework of a coarse-grained description of the quantum system of nonlinear interacting oscillators. The coarse graining is treated as an equivalence relation on the set of quantum states resulting in the emergence of classical phase space. The equivalence relation imposes constraints on the Hamiltonian dynamics of the quantum system. It is seen that the evolution of the coarse-grained system preserves constant and minimal quantum fluctuations of the fundamental observables. This leads to the emergence of the corresponding classical system on a sufficiently large scale.
We introduce a scheme for the quantum simulation of many-body decoherence based on the unitary evolution of a stochastic Hamiltonian. Modulating the strength of the interactions with stochastic processes, we show that the noise-averaged density matrix simulates an effectively open dynamics governed by k-body Lindblad operators. Markovian dynamics can be accessed with white-noise fluctuations; non-Markovian dynamics requires colored noise. The time scale governing the fidelity decay under many-body decoherence is shown to scale as N^{-2k} with the system size N. Our proposal can be readily implemented in a variety of quantum platforms including optical lattices, superconducting circuits, and trapped ions.
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.
We develop an interacting extension of the Double Covariance Model (DCM), a stochastic subquantum framework in which macroscopic quantum dynamics emerge through coarse-graining of correlated microscopic fluctuations. Starting from local stochastic differential equations on subsystem Hilbert spaces, we derive a closed evolution equation for a coarse-grained double covariance operator using multi-scale Itô calculus and sliding-window averaging. The construction explicitly incorporates two separated temporal scales: a fast microscopic fluctuation scale governing subquantum stochastic processes and a slower macroscopic observation scale associated with coarse-grained dynamics. Within the hydrodynamic limit, where the ratio between microscopic correlation time and averaging-window scale vanishes, rapidly fluctuating corrections disappear and the effective dynamics converges to a deterministic macroscopic transport equation. We show that the emergent macroscopic dynamics has the exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form: coherent Hamiltonian evolution arises from deterministic subquantum flow, while dissipative channels emerge from quadratic noise correlations. The framework further demonstrates how non-separable interaction Hamiltonians can arise from strictly local, state-dependent stochastic feedback fields. In the fluctuation-free limit, the model reduces naturally to the standard von Neumann equation, providing a unified stochastic foundation for both open and closed quantum dynamics.