Projection operator methods in linear stochastic differential equations

G. S. Agarwal

Alberto Barchielli, Anna Maria Paganoni

The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure [ 1 , 2 , 3 , 4 , 5 , 6 ], arisen inside the operational approach [ 1 , 7 ] to quantum mechanics, by using functional integrals [ 8 , 9 , 10 ], by using quantum stochastic differential equations [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 ] and by using classical stochastic differential equations (SDE’s) [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 4 , 19 , 20 , 21 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 6 ]. Various types of SDE’s are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both.

Alberto Barchielli, Anna Maria Paganoni

The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both. In this paper we introduce a class of linear SDE's for trace-class operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments and master equations and, so, to the general formulation of quantum mechanics. We do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum system are considered. Then we introduce the notion of a posteriori state and the non linear SDE satisfied by such states and we give conditions from which such equation is assured to preserve pure states and to send any mixed state into a pure one for large times. Finally we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case and we give some concrete examples of physical systems.

B. V. Rajarama Bhat, Kalyan B. Sinha

Yuri A. Rylov

Koichi Miyamoto

This work studies quantum algorithms to solve high-dimensional stochastic differential equations (SDEs) $\mathrm{d} \mathbf{X}_t = A(t) \mathbf{X}_t \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$. Aiming for a speed-up in the dimension $N$ of $\mathbf{X}_t$, we generate quantum states that encode $\mathbf{X}_t$ in the amplitudes, while most of the existing quantum methods for SDEs employ binary encoding. A key challenge is the amplitude encoding of the noise term, and we address this by utilizing the quantum circuit implementation of a pseudorandom number generator (PRNG). We propose two methods: the Dyson series-based method and the Euler-Maruyama (EM)-based method. In the former, we express the noise term via the Dyson series approximation of the time evolution operator, while in the latter, it is approximated using the EM time discretization. Both methods use the quantum linear systems solver to generate the amplitude-encoding state of $\mathbf{X}_t$, making only ${\rm polylog}(N)$ queries to the PRNG circuit and the block-encodings of $A$ and $B$. Additionally, going beyond state preparation, we present methods to estimate expectations of functions of $\mathbf{X}_t$ using the state.