Quantum Dynamics in Phase Space using Projected von Neumann Bases
Shai Machnes, Elie Ass\'emat, Henrik R. Larsson, David Tannor
TL;DR
This paper explores the mathematical foundations of the biorthogonal von Neumann method for quantum simulations, focusing on non-orthogonal projections and their implications for representing the Schrödinger equation.
Contribution
It provides a detailed analysis of non-orthogonal projections in the PvB method and compares different Schrödinger equation representations in reduced bases.
Findings
Analysis of non-orthogonal projection differences from orthogonal projection
Comparison of Schrödinger equation representations in reduced bases
Discussion of challenges and future outlook for PvB method
Abstract
We describe the mathematical underpinnings of the biorthogonal von Neumann method for quantum mechanical simulations (PvB). In particular, we present a detailed discussion of the important issue of non-orthogonal projection onto subspaces of biorthogonal bases, and how this differs from orthogonal projection. We present various representations of the Schr\"odinger equation in the reduced basis and discuss their relative merits. We conclude with illustrative examples and a discussion of the outlook and challenges ahead for the PvB representation.
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