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Strange attractor simulated on a quantum computer

M. Terraneo, Bertrand Georgeot, Dima L. Shepelyansky

Published Mar 14, 2002• 7 citations
OpenAlex

Abstract

Starting from the work of Lorenz, it has been realized that the dynamics of many various dissipative systems converges to so-called strange attractors. These objects are characterized by fractal dimensions and chaotic unstable dynamics of individual trajectories. They appear in nature in very different contexts, including applications to turbulence and weather forecast, molecular dynamics, chaotic chemical reactions, multimode solid state lasers and complex dynamics in ecological systems and physiology. The efficient numerical simulation of such dissipative systems can therefore lead to many important practical applications. Here we study a simple deterministic model where dynamics converges to a strange attractor, and show that it can be efficiently simulated on a quantum computer. Even if the dynamics on the attractor is unstable, dissipative and irreversible, a realistic quantum computer can simulate it in a reversible way, and, already with 70 qubits, will provide access to new informations unaccessible for modern supercomputers.

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Starting from the work of Lorenz, it has been realized that the dynamics of many various dissipative systems converges to so-called strange attractors. These objects are characterized by fractal dimensions and chaotic unstable dynamics of individual trajectories. They appear in nature in very different contexts, including applications to turbulence and weather forecast, molecular dynamics, chaotic chemical reactions, multimode solid state lasers and complex dynamics in ecological systems and physiology. The efficient numerical simulation of such dissipative systems can therefore lead to many important practical applications. Here we study a simple deterministic model where dynamics converges to a strange attractor, and show that it can be efficiently simulated on a quantum computer. Even if the dynamics on the attractor is unstable, dissipative and irreversible, a realistic quantum computer can simulate it in a reversible way, and, already with 70 qubits, will provide access to new informations unaccessible for modern supercomputers.
In classical mechanics, driven systems with dissipation often exhibit complex, fractal dynamics known as strange attractors. This paper addresses the fundamental question of how such structures manifest in the quantum realm. We investigate the quantum Duffing oscillator, a paradigmatic chaotic system, using the Caldirola-Kanai framework, where dissipation is integrated directly into a time-dependent Hamiltonian. By employing the Husimi distribution to represent the quantum state in phase space, we present the first visualization of a quantum strange attractor within this model. Our simulations demonstrate how an initially simple Gaussian wave packet is stretched, folded, and sculpted by the interplay of chaotic dynamics and energy loss, causing it to localize onto a structure that beautifully mirrors the classical attractor. This quantum "photograph" is inherently smoothed, blurring the infinitely fine fractal details of its classical counterpart as a direct consequence of the uncertainty principle. We supplement this analysis by examining the out-of-time-ordered correlator, which shows that stronger dissipation clarifies the exponential growth associated with the classical Lyapunov exponent, thereby confirming the model's semiclassical behavior. This work offers a compelling geometric perspective on open chaotic quantum systems and sheds new light on the quantum-classical transition.
The effect of a large environment on a finite-size quantum mechanical system is two-fold: It brings dissipation, but also fluctuations of thermal and quantum origin. While dissipation tends to stabilize the dynamics, we question if and how environmental quantum fluctuations can generate chaos in an otherwise classically non-chaotic system. We work out a paradigmatic model of quantum optics: the dissipative Dicke model, where a large spin interacts with a dissipative harmonic mode. We dial in the classical/quantum correspondence by working in the semiclassical regime at large but finite spin. We demonstrate that, starting from a classically regular phase space in the superradiant regime, quantum noise can generate a strange attractor with fractal dimension and a positive Lyapunov exponent. We unveil the deep connection with shear-induced chaos that was recently developed in the mathematical community.
Nonlinear qubit master equations have recently been shown to exhibit rich dynamical phenomena such as period doubling, Hopf bifurcation, and strange attractors usually associated with classical nonlinear systems. Here we investigate nonlinear qubit models that support tunable Lorenz attractors. A Lorenz qubit could be realized experimentally by combining qubit torsion, generated by real or simulated mean field dynamics, with linear amplification and dissipation. This would extend engineered Lorenz systems to the quantum regime, allowing for their direct experimental study and possible application to quantum information processing.
Using the decoherence formalism of Gell-Mann and Hartle, a quantum system is found which is the equivalent of the classical chaotic Duffing oscillator. The similarities and the differences from the classical oscillator are examined; in particular, a new concept of quantum maps is introduced, and alterations in the classical strange attractor due to the presence of scale- dependent quantum effects are studied. Classical quantities such as the Lyapunov exponents and fractal dimension are examined, and quantum analogs are suggested. These results are generalized into a framework for quantum dissipative chaos, and there is a brief discussion of other work in this area.
Simulating nonlinear classical dynamics on a quantum computer is an inherently challenging task due to the linear operator formulation of quantum mechanics. In this work, we provide a systematic approach to alleviate this difficulty by developing an explicit quantum algorithm that implements the time evolution of a second-order time-discretized version of the Lorenz model. The Lorenz model is a celebrated system of nonlinear ordinary differential equations that has been extensively studied in the contexts of climate science, fluid dynamics, and chaos theory. Our algorithm possesses a recursive structure and requires only a linear number of copies of the initial state with respect to the number of integration time-steps. This provides a significant improvement over previous approaches, while preserving the characteristic quantum speed-up in terms of the dimensionality of the underlying differential equations system, which similar time-marching quantum algorithms have previously demonstrated. Notably, by classically implementing the proposed algorithm, we showcase that it accurately captures the structural characteristics of the Lorenz system, reproducing both regular attractors-limit cycles-and the chaotic attractor within the chosen parameter regime.
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, the quantum ergodic regime, and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates, and per gate dissipation rate induced by external decoherence. In the limit of strong dissipation the quantum algorithm generates a quantum attractor which may have a complex or simple structure. We also compare the effects of dissipative decoherence with the effects of static imperfections.