Classical and quantum dynamics with discrete time : theory and algorithms

Other Contributor-Ludwig Streit

Minchen Qiao, Zi-Ming Li, Yu-xi Liu

The most widely used approach for simulating the dynamics of time-dependent Hamiltonians via quantum computation depends on the quantum-classical hybrid variational quantum time evolution algorithm, in which ordinary differential equations of the variational coefficients for determining time evolution are solved via classical simulations with a time discretization method. We here present a full-quantum approach, in which ordinary differential equations of the variational coefficients are transformed into static linear equations via the Chebyshev spectral discretization method and then solved via the quantum singular value transformation algorithm. Our full quantum algorithm avoids classical feedback, achieves exponential convergence for smooth Hamiltonians, and yields a quantum circuit depth that is independent of the number of time steps. We demonstrate two implementation strategies, with a global formulation designed for fault-tolerant architectures and a sequential formulation tailored to near-term devices, and validate the approach through numerical simulations of proton-hydrogen charge-transfer dynamics, a prototypical time-dependent quantum chemistry problem. This work establishes a systematic pathway from quantum-classical hybrid variational quantum algorithms to full-quantum solvers for general time-dependent Hamiltonians, particularly those whose dynamics admit compact variational descriptions, opening a route toward full quantum computational advantages in time-dependent simulations.

Eric R. Bittner, Peter J. Rossky

The role of quantum coherence loss in mixed quantum-classical dynamical systems is explored in the context of the theory of quantum decoherence introduced recently by Bittner and Rossky [J. Chem. Phys. 103, 8130 (1995)]. This theory, which is based upon the consistent histories interpretation of quantum mechanics, introduces decoherence in the quantum subsystem by carefully considering the relevant time and length scales over which one must consider the effects of phase interference between alternative histories of the classical subsystem. Such alternative histories are an integral part of any quantum-classical computational scheme which employs transitions between discrete quantum states; consequently, the coherences between alternative histories have a profound effect on the transition probability between quantum states. In this paper, we review the Bittner–Rossky theory and detail a computational algorithm suitable for large-scale quantum molecular dynamics simulations which implements this theory. Application of the algorithm towards the relaxation of a photoexcited aqueous electron compare well to previous estimates of the excited state survival time as well as to the experimental measurements.

Ivan Bardet

Efstratios Koukoutsis, George Vahala, Min Soe +3 more

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.