Analog classical simulation of closed quantum systems

Abstract

We develop an analog classical simulation algorithm of noiseless quantum dynamics. By formulating the Schrödinger equation into a linear system of real-valued ordinary differential equations (ODEs), the probability amplitudes of a complex state vector can be encoded in the continuous physical variables of an analog computer. Our algorithm reveals the full dynamics of complex probability amplitudes. Such real-time simulation is impossible in quantum simulation approaches without collapsing the state vector, and it is relatively computationally expensive for digital classical computers. For a real symmetric time-independent Hamiltonian, the ODEs may be solved by a simple analog mechanical device such as a one-dimensional spring-mass system. Since the underlying dynamics of quantum computers is governed by the Schrödinger equation, our findings imply that analog computers can also perform quantum algorithms. We illustrate how to simulate the Schrödinger equation in such a paradigm, with an application to quantum approximate optimization algorithm. This may pave the way to emulate quantum algorithms with physical computing devices, including analog, continuous-time circuits.