We introduce the problem of determining if the mode of the output distribution of a quantum circuit (given as a black-box) is larger than a given threshold, named HighDist, and a similar problem based on the absolute values of the amplitudes, named HighAmp. We design quantum algorithms for promised versions of these problems whose space complexities are logarithmic in the size of the domain of the distribution, but the query complexities are independent.
Using these, we further design algorithms to estimate the largest probability and the largest amplitude among the output distribution of a quantum black-box. All of these allow us to improve the query complexity of a few recently studied problems, namely, $k$-distinctness and its gapped version, estimating the largest frequency in an array, estimating the min-entropy of a distribution, and the non-linearity of a Boolean function, in the $\tilde{O}(1)$-qubits scenario. The time-complexities of almost all of our algorithms have a small overhead over their query complexities making them efficiently implementable on currently available quantum backends.
The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of $O(1)$ vs. $\sqrt{N}$ between quantum and randomized query complexities remain the state-of-the-art (where $N$ is the input length), leaving open the question of whether $O(1)$ vs. $N^{1/2+Ω(1)}$ separations are possible? We answer this question in …
Simon's problem is an essential example demonstrating the faster speed of quantum computers than classical computers for solving some problems. The optimal separation between exact quantum and classical query complexities for Simon's problem has been proved by Cai $\&$ Qiu. Generalized Simon's problem can be described as follows. Given a function $f:{\{0, 1\}}^n \to {\{0, 1\}}^m$, with the property that there is some unknown hidden subgroup $S$ such that $f(x)=f(y)$ iff $x \oplus y\in S$, for any $x, y\in {\{0, 1\}}^n$, where $|S|=2^k$ for some $0\leq k\leq n$. The goal is to find $S$. For the…
Variational quantum algorithms, which have risen to prominence in the noisy intermediate-scale quantum setting, require the implementation of a stochastic optimizer on classical hardware. To date, most research has employed algorithms based on the stochastic gradient iteration as the stochastic classical optimizer. In this work we propose instead using stochastic optimization algorithms that yield stochastic processes emulating the dynamics of classical deterministic algorithms. This approach results in methods with theoretically superior worst-case iteration complexities, at the expense of gr…
Quantum Algorithms for Bandits with Knapsacks with Improved Regret and Time Complexities
Center on Frontiers of Computing Studies, Peking University School of Computer Science, Peking University（前沿计算研究所以及北京大学计算机科学学院，北京大学）；Guanghua School of Management, Peking University（光华管理学院，北京大学）；Department of Management Science and Engineering, Stanford University（管理科学与工程系，斯坦福大学）
We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: - Longest Common Substring can be solved by a quantum algorithm in $\tilde O(n^{2/3})$ time, improving upon the rece…
We apply our recent work on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm -- the Louvain algorithm for community detection -- and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs. We find that this analysis yields insights not available to us via the asymptotic analysis, further emphasising the utility in such an empirical approach. In particular, we observe that a complicat…
We investigate time complexities of finite difference methods for solving the multiscale transport equation with quantum algorithms. We find that the time complexities of both the classical treatment and quantum treatment for a standard explicit scheme scale as $\mathcal{O}(1/\varepsilon)$, where $\varepsilon$ is the small scaling parameter, while the complexities for the even-odd parity based Asymptotic-Preserving (AP) scheme do not depend on $\varepsilon$. This indicates that it is still of great importance to use AP (and probably other efficient multiscale) schemes for multiscale problems i…
An important aspect of quantum simulation is the preparation of physically interesting states on a quantum computer, and this task can often be costly or challenging to implement. A digital, ``site-by-site'' scheme of state preparation was introduced in arXiv:1911.03505 as a way to prepare the vacuum state of certain fermionic field theory Hamiltonians with a mass gap. More generally, this algorithm may be used to prepare ground states of Hamiltonians by adding one site at a time as long as successive intermediate ground states share a non-zero overlap and the Hamiltonian has a non-vanishing s…
Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE - the heat equation in a rectangular region - and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension $d \ge 2$, there is an at most quadratic quantum speedup using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative ap…
