Practical Unitary Simulator for Non-Markovian Complex Processes

Abstract

Stochastic processes are as ubiquitous throughout the quantitative sciences as they are notorious for being difficult to simulate and predict. In this Letter, we propose a unitary quantum simulator for discrete-time stochastic processes which requires less internal memory than any classical analogue throughout the simulation. The simulator's internal memory requirements equal those of the best previous quantum models. However, in contrast to previous models, it only requires a (small) finite-dimensional Hilbert space. Moreover, since the simulator operates unitarily throughout, it avoids any unnecessary information loss. We provide a stepwise construction for simulators for a large class of stochastic processes hence directly opening the possibility for experimental implementations with current platforms for quantum computation. The results are illustrated for an example process.

Dimensional Quantum Memory Advantage in the Simulation of Stochastic Processes

“…To simulate a stochastic system's future behaviour, information about its past must be stored [3,4]and thus memory is a key resource. Quantum information processing promises a memory advantage for stochastic simulation [5][6][7][8][9][10][11][12][13][14][15] that has been validated in recent proof-of-concept experiments [16,17]. Yet, in all past works, the memory saving would only become accessible in the limit of a large number of parallel simulations [6,18], because the memory registers of individual quantum simulators had the same dimensionality as their classical counterparts.…”

“…By "single-shot" we mean that any individual simulator obtains an advantage, rather than requiring an asymptotically large array of simulators. We investigate a specific stochastic process, while noting that it is theoretically known that the advantage holds for a range of other simulation tasks [14]. The process we simulate here can be understood as the output of a biased perturbed coin after post-processing [15](see Fig.…”

Memory-efficient tracking of complex temporal and symbolic dynamics with quantum simulators

“…While the information cost of of a quantum model constructed by our protocol is lower than any classical model, there is no claim of optimality over all quantum models. By accounting for longer-range temporal correlations [27,28,35], it has been found that discrete-time q-machines can reduce their information cost, and analogous constructs may be possible in the continuous-time case, providing further memory savings. Other reductions may be possible by exploiting the possibility of using complex amplitudes in the QMS.…”

Robust inference of memory structure for efficient quantum modeling of stochastic processes

“…A set of quantum memory states and corresponding interaction can be found through a recursive expression for the overlaps of the states and employing a reverse Gram-Schmidt procedure [20]. The estimated quantum statistical memory is then given by the von Neumann entropy of the corresponding stationary state of the memory:…”