# Interfering trajectories in experimental quantum-enhanced stochastic simulation

## TL;DR

This paper demonstrates a quantum simulation of a stochastic process over multiple time steps using less memory than classical methods, leveraging quantum superposition and interference to compare statistical futures.

## Contribution

It introduces a multi-time-step quantum simulation method that maintains coherence and demonstrates quantum interference of statistical futures, surpassing classical memory limits.

## Key findings

- Achieved quantum simulation with reduced memory usage.
- Demonstrated high-visibility interference of quantum states.
- Validated quantum advantage in stochastic process simulation.

## Abstract

Simulations of stochastic processes play an important role in the quantitative sciences, enabling the characterisation of complex systems. Recent work has established a quantum advantage in stochastic simulation, leading to quantum devices that execute a simulation using less memory than possible by classical means. To realise this advantage it is essential that the memory register remains coherent, and coherently interacts with the processor, allowing the simulator to operate over many time steps. Here we report a multi-time-step experimental simulation of a stochastic process using less memory than the classical limit. A key feature of the photonic quantum information processor is that it creates a quantum superposition of all possible future trajectories that the system can evolve into. This superposition allows us to introduce, and demonstrate, the idea of comparing statistical futures of two classical processes via quantum interference. We demonstrate interference of two 16-dimensional quantum states, representing statistical futures of our process, with a visibility of 0.96 $\pm$ 0.02.