The famous least squares Monte Carlo (LSM) algorithm combines linear least
square regression with Monte Carlo simulation to approximately solve problems
in stochastic optimal stopping theory. In this work, we propose a quantum LSM
based on quantum access to a stochastic process, on quantum circuits for
computing the optimal stopping times, and on quantum techniques for Monte
Carlo. For this algorithm, we elucidate the intricate interplay of function
approximation and quantum algorithms for Monte Carlo. Our algorithm achieves a
nearly quadratic speedup in the runtime compared to the LSM algorithm under
some mild assumptions. Specifically, our quantum algorithm can be applied to
American option pricing and we analyze a case study for the common situation of
Brownian motion and geometric Brownian motion processes.Comment: 45 pages; v2: title slightly changed, typos fixed, references adde
