Related papers: Quantum Speedups for Polynomial-Time Dynamic Progr…
We create a variety of new quantum algorithms that use Grover's algorithm and similar techniques to give polynomial speedups over their classical counterparts. We begin by introducing a set of tools that carefully minimize the impact of…
In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic…
Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best…
We present a quantum algorithm to solve dynamic programming problems with convex value functions. For linear discrete-time systems with a $d$-dimensional state space of size $N$, the proposed algorithm outputs a quantum-mechanical…
We discuss the problem of finite-horizon dynamic programming (DP) on a quantum computer. We introduce a query model for studying quantum and classical algorithms for solving DP problems, and provide example oracle constructions for the…
Designing quantum algorithms with a speedup over their classical analogs is a central challenge in quantum information science. Motivated by recent experimental observations of a superlinear quantum speedup in solving the Maximum…
Linear regression is one of the most fundamental linear algebra problems. Given a dense matrix $A \in \mathbb{R}^{n \times d}$ and a vector $b$, the goal is to find $x'$ such that $ \| Ax' - b \|_2^2 \leq (1+\epsilon) \min_{x} \| A x - b…
Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube…
Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of…
We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time $n^{\frac{1}{2}} m^{\frac{1}{2}} s^2 \text{poly}(\log(n), \log(m), R, r, 1/\delta)$, with $n$ and $s$ the dimension and row-sparsity of the…
We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a…
We initiate a systematic study of the time complexity of quantum divide and conquer algorithms for classical problems. We establish generic conditions under which search and minimization problems with classical divide and conquer algorithms…
We give quantum speedups of several general-purpose numerical optimisation methods for minimising a function $f:\mathbb{R}^n \to \mathbb{R}$. First, we show that many techniques for global optimisation under a Lipschitz constraint can be…
Recent works have shown that quantum computers can polynomially speed up certain SAT-solving algorithms even when the number of available qubits is significantly smaller than the number of variables. Here we generalise this approach. We…
We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^m$, we propose classical…
While it seems possible that quantum computers may allow for algorithms offering a computational speed-up over classical algorithms for some problems, the issue is poorly understood. We explore this computational speed-up by investigating…
Mixed Integer Programs (MIPs) model many optimization problems of interest in Computer Science, Operations Research, and Financial Engineering. Solving MIPs is NP-Hard in general, but several solvers have found success in obtaining…
Some approaches to solving challenging dynamic programming problems, such as Q-learning, begin by transforming the Bellman equation into an alternative functional equation, in order to open up a new line of attack. Our paper studies this…
In this work, we design quantum algorithms that are more efficient than classical algorithms to solve time-dependent and finite-horizon Markov Decision Processes (MDPs) in two distinct settings: (1) In the exact dynamics setting, where the…
The simulation of large-scale classical systems in exponentially small space on quantum computers has gained attention. The prior work demonstrated that a quantum algorithm offers an exponential speedup over any classical algorithm in…