Time-Dependent Problems

Yui Tsuyumine, Kenichi Masuda, Takeshi Hachikawa +5 more

The Vehicle Routing Problem (VRP) is an extension of the Travelling Salesperson Problem (TSP). A VRP determines the delivery sequence where multiple vehicles are available, delivery should be made to multiple points, and each point should be visited once. However, road conditions change depending on the time of day, and the time required to travel between two points changes accordingly. Unless these changes are considered, a calculated minimum cost is not realized in actual logistics. Aiming to apply the Time-Dependent Vehicle Routing Problem with Time Windows (TDVRPTW), which is a VRP extension that counts changes in the time required to travel between two points depending on the time of day, to Ising machines including quantum annealers, this paper proposes an effective formulation to realize TDVRPTW, composed of several objective functions and constraints. Further, this paper compares and evaluates the results obtained by a physical Ising machine and confirms the validity of the formulation, thus verifying the applicability of the TDVRPTW to quantum computing.

Grigori Karlovich Giorgadze

In this paper, we investigate the problem of simulation of quantum-mechanical dynamical system with prescribed properties using of exactly solvable Hamiltonians. A method for solving time-dependent matrix Schrodinger equations in an explicit analytical form will be developed. It is based on exactly solvable time-independent problems, a certain choice of the initial conditions, and special time-dependent gauge transformations converting time-independent problems to time-dependent ones. An application of the exactly solvable time-dependent problems thus obtained to the construction of a universal set of gates for quantum computers is also presented.

Jerry Goodisman

Luca Salasnich, Francesco Lorenzi

Tobias Kramer, M. Moshińsky

Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: one class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like e.g. the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which can be used to map the solution of the time-dependent problem to that of a related time-independent one. In this paper we discuss and develop analytic methods for solving time-dependent tunnelling problems, which cannot be addressed by using quadratic Hamiltonians. Specifically, we give an analytic solution to the problem of tunnelling from an attractive time-dependent potential which is embedded in a long-range repulsive potential. Recent progress in atomic physics makes it possible to observe experimentally time-dependent phenomena and record the probability distribution over a long range of time. Of special interest is the observation of macroscopical quantum-tunnelling phenomena in Bose–Einstein condensates with time-dependent trapping potentials. We apply our model to such a case in the last section.