Rigorous representation and exact simulation of real Gaussian stationary processes

Manuel J. Miranda, Paolo Bocchini

A. I. Oksak

F Rousse, M Fasi, A Dmytryshyn +3 more

We use the Gaussian Phase-Space Representation to solve the real-time dynamic of interacting fermions in 1D, 2D, and 3D systems. The method is exact up to a spiking point, which represents a limit on the practical simulation time. The spiking can be delayed, and the practical simulation time extended, by adjusting the gauges of the representation, resulting in different equivalent stochastic differential equations. Here, we work on the so-called diffusion gauge and propose an algorithm to find efficiently new implementations of the noise terms. Compared with our initial results [F. Rousse \textit{et al.} 2024, J. Phys. A: Math. Theor. \textbf{57}, 015303], the new method achieves a significantly longer practical simulation time and can be applied to significantly larger systems.

Robert I. Booth, Titouan Carette, Cole Comfort

We give generators and relations for the hypergraph props of Gaussian relations and positive affine Lagrangian relations. The former extends Gaussian probabilistic processes by completely-uninformative priors, and the latter extends Gaussian quantum mechanics with infinitely-squeezed states. These presentations are given by adding a generator to the presentation of real affine relations and of real affine Lagrangian relations which freely codiscards effects, as well as certain rotations. The presentation of positive affine Lagrangian relations provides a rigorous justification for many common yet informal calculations in the quantum physics literature involving infinite-squeezing. Our presentation naturally extends Menicucci et al.'s graph-theoretic representation of Gaussian quantum states with a representation for Gaussian transformations. Using this graphical calculus, we also give a graphical proof of Braunstein and Kimble's continuous-variable quantum teleportation protocol. We also interpret the LOv-calculus, a diagrammatic calculus for reasoning about passive linear-optical quantum circuits in our graphical calculus. Moreover, we show how our presentation allows for additional optical operations such as active squeezing.

Göran Lindblad

We define a stationary Gaussian quantum stochastic process (GQSP) on the C*-algebra of the canonical commutation relations over a real symplectic Hilbert space. Physically GQSPs can describe diffusion with non-Markovian memory effects in quantum harmonic oscillators of arbitrary dimension. We find that in analogy with the commutative case a GQSP is completely specified by a operator-valued autocovariance function satisfying certain positive definiteness and reality conditions. The autocovariance function also determines the response of the system to a class of time-dependent generalized forces, and it has a spectral representation in terms of a positive operator-valued measure.