The advent of quantum technologies has opened the possibility of using quantum systems for technological applications and has started a new era of quantum physics. This requires the development of experimental methods for controlling and manipulating very fragile states and dynamics of many-body systems with extreme precision. For the fulfillment of this goal, it is crucial to theoretically understand the precise mechanisms that control the evolution of a quantum system, as would be obtained by ab initio methods and a (numerical) solution of the Schrödinger equation. However, this is a hard computational problem, since the solution of a quantum problem on a classical computer requires resources that scale exponentially with the number of particles. Therefore, numerically efficient algorithms for computing the real-time evolution of many-body quantum systems are needed.The most efficient algorithms today are based on quantum Monte Carlo methods, where each physical particle is represented by an ensemble of fictitious particles (walkers) whose distribution in physical space reproduces the square of the wave function. In this way, a very accurate estimate for the stationary states has been obtained.

The efficiency of the algorithms is mainly due to a reduction of the many-body quantum problem to lower dimensions, by employing (artificial) many-body trial functions as a guide for the diffusion of the Monte Carlo walkers. However, this reduction comes at the price of losing essential quantum properties of the many-body system and, in particular, the quantum correlations (entanglement) among particles. As a consequence, the real-time evolution is beyond the reach of these methods. An alternative method is one in which the random walks occur in the wave-function (Hilbert) space, so that the physical particles move in a stochastic potential derived from the random wave functions. In this paper a combination of the above two methods is proposed: concurrent random walks of two ensembles—of particles and of wave-functions—are considered in a self-consistent way. The main idea is that each particle experiences a random local environment, the size of which is controlled by an effective window function: thus one has an ensemble of Monte Carlo walkers in physical space, each of which is affected by its own probability density, given by a concurrent walker in Hilbert space. The connection is given by the equations of Bohmian mechanics, which rule the evolution of a particle as determined by a pilot wave. The proposed method seems to be well-suited for the calculation of the real-time evolution of quantum averages and coherences for many-body quantum systems and already has been successfully tested on some simple systems for which the exact numerical solution of the many-body Schrödinger equation is available.

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