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 , for any large real number . Then, for a sequence of processes each labeled by an integer size , we compare how the classical and quantum required memories scale with . In this setting, since both memories can diverge as , the efficiency metric tracks how fast they diverge. An extreme quantum memory advantage exists when the classical memory diverges in the limit , 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.
- Received 9 August 2017
- Revised 28 December 2017
DOI:https://doi.org/10.1103/PhysRevX.8.011025
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
From earthquakes to financial market crashes, rare events are often associated with catastrophe—from decimated social infrastructure and the substantial loss of life to global economic collapse. Though rare, the impact of these events cannot be ignored. Predicting and modeling such rare events is essential to mitigating their effects. However, their rarity makes this particularly challenging, often requiring vast data sets and massive computational resources. Here, we show how to circumvent these roadblocks by using a quantum algorithm that implements efficient rare-event sampling with substantially reduced memory requirements.
Biased sampling is a technique that is widely used in the analysis of rare events. It works by transforming a given distribution of events into a new distribution where events that were previously rare become typical. We extend this method to the realm of quantum computing and mathematically demonstrate a clear memory advantage over traditional computing algorithms. In one case, the memory required in a classical computer grows without bound as the size of the process increases, whereas in the quantum algorithm, the required memory reaches a finite limit.
Given the broad need for understanding rare events in physical, chemical, geological, biological, and social systems, our new sampling algorithm will impact many fields as quantum computers become available.
