  Stochastic processes underlie a vast range of natural and social phenomena.
Some processes such as atomic decay feature intrinsic randomness, whereas other
complex processes, e.g. traffic congestion, are effectively probabilistic
because we cannot track all relevant variables. To simulate a stochastic
system's future behaviour, information about its past must be stored and thus
memory is a key resource. Quantum information processing promises a memory
advantage for stochastic simulation that has been validated in recent
proof-of-concept experiments. Yet, in all past works, the memory saving would
only become accessible in the limit of a large number of parallel simulations,
because the memory registers of individual quantum simulators had the same
dimensionality as their classical counterparts. Here, we report the first
experimental demonstration that a quantum stochastic simulator can encode the
relevant information in fewer dimensions than any classical simulator, thereby
achieving a quantum memory advantage even for an individual simulator. Our
photonic experiment thus establishes the potential of a new, practical resource
saving in the simulation of complex systems.
