Classical Dynamics of Quantum Variations

Fabio Bagarello

Matija Medvidović, Dries Sels

Classical variational methods are used to simulate the dynamics of two-dimensional quantum models with continuous degrees of freedom using neural networks, pushing the limits of classical computation in simulating these systems.

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.

Maciej Zworski

Esther Bonet-Luz, Cesare Tronci

This paper presents the geometric setting of quantum variational principles and extends it to comprise the interaction between classical and quantum degrees of freedom. Euler-Poincaré reduction theory is applied to the Schrödinger, Heisenberg, and Wigner-Moyal dynamics of pure states. This construction leads to new variational principles for the description of mixed quantum states. The corresponding momentum map properties are presented as they arise from the underlying unitary symmetries. Finally, certain semidirect-product group structures are shown to produce new variational principles for Dirac’s interaction picture and the equations of hybrid classical-quantum dynamics.

D. C. Hutcgings