Linear Equations

Andrei I. Maimistov, Ashat M. Basharov

A. J. Shaiju, Ian R. Petersen

This paper is concerned with the physical realizability of linear quantum stochastic differential equations (QSDEs). In a recent paper, James, Nurdin and Petersen gave necessary and sufficient conditions for the physical realizability of linear QSDEs with real coefficients. These conditions were algebraic conditions on the system state space matrices. In this paper, we first study general linear QSDEs with complex coefficients and explain how to relate them via unitary transformations to the class of linear QSDEs with real coefficients considered by James, Nurdin Petersen. We use this relation between linear QSDEs with real and complex coefficients to derive necessary and sufficient algebraic conditions for the physical realizability of the general complex linear QSDEs being considered. Then we prove the equivalence between these algebraic conditions and a condition that an associated linear system is (J, J')-unitary.

Simone Montangero

Jürgen Schmiegel, Martin Greiner, G. Soff

The Hamiltonian obtained from the linearization of the free collective Schr?dinger equation can be understood as a supercharge. The inclusion of a scalar potential shows that the linearization procedure is equivalent to the factorization method applied in supersymmetric quantum mechanics. Finally a supersymmetric Hamiltonian is constructed which includes the collective Schr?dinger equation as well as the linearized collective Schr?dinger equation.

Simone Montangero

Reuben Demirdjian, Thomas Hogancamp, Daniel Gunlycke

Herein, we present a polylogarithmic decomposition method to load the matrix from the linearized 1-dimensional Burgers' equation onto a quantum computer. First, we use the Carleman linearization method to map the nonlinear Burgers' equation into an infinite linear system of equations, which is subsequently truncated to order $α$. This new finite linear system is then embedded into a larger system of equations with the key property that its matrix can be decomposed into a linear combination of $\mathcal{O}(\log n_t + α^2\log n_x)$ terms for $n_t$ time steps and $n_x$ spatial grid points. While the terms in this linear combination are not unitary, each can be implemented using a simple block encoding procedure. A numerical simulation is performed by combining our approach with the variational quantuam linear solver demonstrating that accurate solutions are possible. Finally, a resource estimate shows that the upper bound of the Clifford and T gate counts scale like $\mathcal{O}(α(\log n_x)^2)$ and $\mathcal{O}((\log n_x)^2)$, respectively. This is therefore the first explicit polylogarithmic data loading method with respect to $n_x$ and $n_t$ for a Carleman linearized system.