@article {3138, title = {Quantum divide and conquer}, year = {2022}, month = {10/12/2022}, abstract = {

The divide-and-conquer framework, used extensively in classical algorithm design, recursively breaks a problem of size n into smaller subproblems (say, a copies of size n/b each), along with some auxiliary work of cost Caux(n), to give a recurrence relation
C(n)\≤aC(n/b)+Caux(n)
for the classical complexity C(n). We describe a quantum divide-and-conquer framework that, in certain cases, yields an analogous recurrence relation
CQ(n)\≤a\−\−\√CQ(n/b)+O(CauxQ(n))
that characterizes the quantum query complexity. We apply this framework to obtain near-optimal quantum query complexities for various string problems, such as (i) recognizing regular languages; (ii) decision versions of String Rotation and String Suffix; and natural parameterized versions of (iii) Longest Increasing Subsequence and (iv) Longest Common Subsequence.

}, keywords = {Data Structures and Algorithms (cs.DS), FOS: Computer and information sciences, FOS: Physical sciences, Quantum Physics (quant-ph)}, doi = {10.48550/ARXIV.2210.06419}, url = {https://arxiv.org/abs/2210.06419}, author = {Andrew M. Childs and Kothari, Robin and Kovacs-Deak, Matt and Sundaram, Aarthi and Wang, Daochen} } @article {3022, title = {Quantum Algorithms for Reinforcement Learning with a Generative Model}, journal = {Proceedings of the 38th International Conference on Machine Learning, PMLR}, volume = {139}, year = {2021}, month = {12/15/2021}, chapter = {10916-10926}, abstract = {

Reinforcement learning studies how an agent should interact with an environment to maximize its cumulative reward. A standard way to study this question abstractly is to ask how many samples an agent needs from the environment to learn an optimal policy for a γ-discounted Markov decision process (MDP). For such an MDP, we design quantum algorithms that approximate an optimal policy (π\∗), the optimal value function (v\∗), and the optimal Q-function (q\∗), assuming the algorithms can access samples from the environment in quantum superposition. This assumption is justified whenever there exists a simulator for the environment; for example, if the environment is a video game or some other program. Our quantum algorithms, inspired by value iteration, achieve quadratic speedups over the best-possible classical sample complexities in the approximation accuracy (ϵ) and two main parameters of the MDP: the effective time horizon (11\−γ) and the size of the action space (A). Moreover, we show that our quantum algorithm for computing q\∗ is optimal by proving a matching quantum lower bound.

}, keywords = {FOS: Computer and information sciences, FOS: Physical sciences, Machine Learning (cs.LG), Quantum Physics (quant-ph)}, doi = {10.48550/ARXIV.2112.08451}, url = {https://arxiv.org/abs/2112.08451}, author = {Daochen Wang and Sundaram, Aarthi and Kothari, Robin and Kapoor, Ashish and Roetteler, Martin} }