Extreme Quantum Memory Advantage for Rare-Event Sampling

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 and quantum required memories scale with N. In this setting, since both memories can diverge as N -> infinity , the efficiency metric tracks how fast they diverge. An extreme quantum memory advantage exists when the classical memory diverges in the limit N -> infinity, but the quantum memory has a finite bound. We then show that finite-state Markov processes and spin chains exhibit memory advantage for sampling of almost all of their rare-event classes.