Algebraic characterizations of dissipativity for linear quantum stochastic systems

Nivaldo Agostinho Lemos

V. P. Belavkin

A characterisation of the generators of quantum stochastic cocycles of completely positive (CP) maps is given in terms of the complete dissipativity (CD) of its form-generator. The pseudo-Hilbert dilation of the stochastic form-generator and the pre-Hilbert dilation of the corresponding dissipator is found. The general form of the linear continuous structural maps for the algebra of all bounded operators is derived and the quantum stochastic flow for the corresponding cocycle is outlined. It is proved that any w*-analytical bounded CD form-generator give rise to a quantum stochastic CP cocycle over a von Neumann algebra.

Vasily Evgenevich Tarasov

Suguru Endo, Jinzhao Sun, Ying Li +2 more

Variational quantum algorithms have been proposed to solve static and dynamic problems of closed many-body quantum systems. Here we investigate variational quantum simulation of three general types of tasks-generalized time evolution with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum system dynamics. The algorithm for generalized time evolution provides a unified framework for variational quantum simulation. In particular, we show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalized time evolution. Meanwhile, assuming a tensor product structure of the matrices, we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution. Finally, we introduce variational quantum simulation for open system dynamics. We variationally implement the stochastic Schrödinger equation, which consists of dissipative evolution and stochastic jump processes. We numerically test the algorithm with a 6-qubit 2D transverse field Ising model under dissipation.

M. Courbage

John Edward Gough

A theory of stochastic flows over the algebra of observables of a dynamical system is presented in which the main objective is to ensure that the overall canonical/symplectic structure on the algebra is preserved. We study both classical and quantum systems and the importance of physical interpretation in the Stratonovich interpretation is stressed. We find the natural formulation of quantum dissipative systems to be given in terms of quantum stochastic calculus. This treatment allows for a physically meaningful treatment of both constant and nonlinear dissipation. As an application, we quantize a mechanical system with the same nonlinear damping mechanism as the van der Pol oscillator.