Nonlinear von neumann equations for quantum dissipative systems

Michael R. Berman, Ronnie Kosloff

P. Caldirola, E. Montaldi

Martin Hensel, H. J. Korsch

A recently proposed generalized von Neumann equation describing dissipative time evolution of quantum systems is applied to a damped and driven harmonic oscillator. Utilizing Lie algebraic methods the nonlinear operator equation is solved and from this solution the behaviour of the mean value (x) is extracted, including both amplitude and phaseshift relative to the driving force. A differential equation for (x) is derived and the damping constant is identified.

Daniel F. Walls, Gerard J Milburn

I. Novikau, Ilon Joseph

We propose an explicit algorithm based on the Linear Combination of Hamiltonian Simulations technique to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman--von Neumann formulation of nonlinear dynamics. By including dissipation into the model, through an upwind discretization of the advection operator, we avoid spurious parasitic oscillations which usually accompany standard finite difference discretizations of the advection equation. In contrast to prior works on quantum simulation of nonlinear problems, we explain in detail how different components of the algorithm can be implemented by using the Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) methods. In addition, we discuss the general method for implementing the block-encoding (BE) required for QSP and QSVT circuits and provide explicit implementations of the BE oracles tailored to our specific test cases. We simulate the resulting circuit on a digital emulator of quantum fault-tolerant computers and investigate its complexity and success probability. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations including the KvN and Carleman embeddings of nonlinear systems, the semiclassical Koopman-van Hove (KvH) equation, as well as the advection and Liouville equations.

Daniel F. Walls, Gerard J Milburn