Non-linear Quantum Monte Carlo

The mean of a random variable can be understood as a $\textit{linear}$ functional on the space of probability distributions. Quantum computing is known to provide a quadratic speedup over classical Monte Carlo methods for mean estimation. In this paper, we investigate whether a similar quadratic speedup is achievable for estimating $\textit{non-linear}$ functionals of probability distributions. We propose a quantum-inside-quantum Monte Carlo algorithm that achieves such a speedup for a broad class of non-linear estimation problems, including nested conditional expectations and stochastic optimization. Our algorithm improves upon the direct application of the quantum multilevel Monte Carlo algorithm introduced by An et al.. The existing lower bound indicates that our algorithm is optimal up polylogarithmic factors. A key innovation of our approach is a new sequence of multilevel Monte Carlo approximations specifically designed for quantum computing, which is central to the algorithm's improved performance.

On Rearrangement of Items Stored in Stacks

There are $n \ge 2$ stacks, each filled with $d$ items (its full capacity), and one empty stack with capacity $d$. A robot arm, in one stack operation (move), may pop one item from the top of a non-empty stack and subsequently push it into a stack that is not at capacity. In a {\em labeled} problem, all $nd$ items are distinguishable and are initially randomly scattered in the $n$ stacks. The items must be rearranged using pop-and-push moves so that at the end, the $k^{\rm th}$ stack holds items $(k-1)d +1, \ldots, kd$, in that order, from the top to the bottom for all $1 \le k \le n$. In an {\em unlabeled} problem, the $nd$ items are of $n$ types of $d$ each. The goal is to rearrange items so that items of type $k$ are located in the $k^{\rm th}$ stack for all $1 \le k \le n$. In carrying out the rearrangement, a natural question is to find the least number of required pop-and-push moves. In terms of the required number of moves for solving the rearrangement problems, the labeled and unlabeled version have lower bounds $\Omega(nd + nd{\frac{\log d}{\log n}})$ and $\Omega(nd)$, respectively. Our main contribution is the design of an algorithm with a guaranteed upper bound of $O(nd)$ for both versions when $d \le cn$ for arbitrary fixed positive number $c$. In addition, a subroutine for a problem that we call the Rubik table problem is of independent interest, with applications to problems including multi-robot motion planning.