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arXiv:1907.00841 · Physical Review Letters

Local-in-Time Error in Variational Quantum Dynamics

Rocco Martinazzo, Irène Burghardt

Published Apr 13, 2020• 34 citations
arXivHTMLPDF

Abstract

The McLachlan "minimum-distance" principle for optimizing approximate solutions of the time-dependent Schrödinger equation is revisited, with a focus on the local-in-time error accompanying the variational solutions. Simple, exact expressions are provided for this error, which are then evaluated in illustrative cases, notably the widely used mean-field approach and the adiabatic quantum molecular dynamics. Based on these findings, we demonstrate the rigorous formulation of an adaptive scheme that resizes on the fly the underlying variational manifold and, hence, optimizes the overall computational cost of a quantum dynamical simulation. Such adaptive schemes are a crucial requirement for devising and applying direct quantum dynamical methods to molecular and condensed-phase problems.

DOI
10.1103/physrevlett.124.150601

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We propose a general-purpose, self-adaptive approach to construct a variational wave-function ansatz for highly accurate quantum dynamics simulations based on McLachlan’s variational principle. The key idea is to dynamically expand the variational ansatz along the time-evolution path such that the “McLachlan distance”, which is a measure of the simulation accuracy, remains below a set threshold. We apply this adaptive variational quantum dynamics simulation (AVQDS) approach to the integrable Lieb-Schultz-Mattis spin chain and the nonintegrable mixed-field Ising model, where it captures both finite-rate and sudden post-quench dynamics with high fidelity. The AVQDS quantum circuits that prepare the time-evolved state are much shallower than those obtained from first-order Trotterization and contain up to 2 orders of magnitude fewer cnot gate operations. We envision that a wide range of dynamical simulations of quantum many-body systems on near-term quantum-computing devices will be made possible through the AVQDS framework.
We propose a general-purpose, self-adaptive approach to construct a variational wave-function ansatz for highly accurate quantum dynamics simulations based on McLachlan's variational principle.The key idea is to dynamically expand the variational ansatz along the time-evolution path such that the "McLachlan distance", which is a measure of the simulation accuracy, remains below a set threshold.We apply this adaptive variational quantum dynamics simulation (AVQDS) approach to the integrable Lieb-Schultz-Mattis spin chain and the nonintegrable mixed-field Ising model, where it captures both finite-rate and sudden post-quench dynamics with high fidelity.The AVQDS quantum circuits that prepare the timeevolved state are much shallower than those obtained from first-order Trotterization and contain up to 2 orders of magnitude fewer CNOT gate operations.We envision that a wide range of dynamical simulations of quantum many-body systems on near-term quantum-computing devices will be made possible through the AVQDS framework.
Approximate solutions to the time-dependent Schr\"odinger equation can be sought in terms of time-dependent parameters. In this work, a reasonable definition of the "best" values for these parameters is given by requiring that the time integral of the deviation of the approximate solution from the exact solution be minimized. A procedure for specifically evaluating the parameters to optimize the approximate solution is given in terms of equations analogous to Hamilton's equations of classical mechanics, which produce a minimum-deviation integral in analogy to Hamilton's principle. A comparison of this technique with previous approaches is made, both in terms of general principles, and in terms of a simple hypothetical example of a system subjected to a time-dependent perturbation.