Extreme Quantum Memory Advantage for Rare-Event Sampling

ABSTRACT: We introduce a quantum algorithm for memory-efficient biased sampling of rare events generated by classical memoryful stochastic processes. Two efficiency metrics are used to compare quantum and classical resources for rare-event sampling. For a fixed stochastic process, the first is the classical-to-quantum ratio of required memory. We show for two example processes that there exists an infinite number of rare-event classes for which the memory ratio for sampling is larger than r, for any large real number r. Then, for a sequence of processes each labeled by an integer size N, we compare how the classical-to-quantum required memory ratio scales with N. In this setting, since both memories can diverge as N → ∞, the efficiency metric tracks how fast they diverge. An extreme quantum memory advantage exists when the classical memory diverges in the limit N → ∞, but the quantum memory has a finite bound. We then show that finite-state Markov processes and spin chains exhibit extreme memory advantage for sampling of almost all of their rare-event classes.