Related papers: Optimal quantum query bounds for almost all Boolea…
The claw problem is central in the fields of theoretical computer science as well as cryptography. The optimal quantum query complexity of the problem is known to be $\Omega\left(\sqrt{G}+(FG)^{1/3} \right)$ for input functions $f\colon…
We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables; given a linear Boolean function, the…
The quantum query models is one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including the Deutsch-Jozsa algorithm, the Simon algorithm, the Grover algorithm and…
We study the extremal Forrelation problem, where, provided with oracle access to Boolean functions $f$ and $g$ promised to satisfy either $\textrm{forr}(f,g)=1$ or $\textrm{forr}(f,g)=-1$, one must determine (with high probability) which of…
The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS 1998, J. ACM 2001), the adversary method by Ambainis (STOC 2000, J. Comput. Syst. Sci. 2002), and the compressed oracle method by Zhandry (CRYPTO 2019) have been shown…
The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(N\epsilon^{-2/3} + \epsilon^{-8/3})$…
Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and…
In this paper, we give a quadratic Goldreich-Levin algorithm that is close to optimal in the following ways. Given a bounded function $f$ on the Boolean hypercube $\mathbb{F}_2^n$ and any $\varepsilon>0$, the algorithm returns a quadratic…
The collision problem is to decide whether a function X:{1,..,n}->{1,..,n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Theta(n^{1/5}) on the number of queries needed by a quantum computer to…
Finite-sum optimization has wide applications in machine learning, covering important problems such as support vector machines, regression, etc. In this paper, we initiate the study of solving finite-sum optimization problems by quantum…
We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree $n$ over a finite field $\F_q$, the average-case complexity of our algorithm is an expected $O(n^{1 + o(1)} \log^{2 +…
The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a…
Simon in his FOCS'94 paper was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We study Simon's problem…
We show that a certain tensor norm, the completely bounded norm, can be expressed by a semidefinite program. This tensor norm recently attracted attention in the field of quantum computing, where it was used by Arunachalam, Bri\"{e}t and…
The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query…
We study the probability of making an error if, by querying an oracle a fixed number of times, we declare constant a randomly chosen n-bit Boolean function. We compare the classical and the quantum case, and we determine for how many…
The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson \cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and…
I improve the tight bound on quantum searching by Boyer et al. (quant-ph/9605034) to a matching bound, thus showing that for any probability of success Grovers quantum searching algorithm is optimal. E.g. for near certain success we have to…
We are presented with a graph, $G$, on $n$ vertices with $m$ edges whose edge set is unknown. Our goal is to learn the edges of $G$ with as few queries to an oracle as possible. When we submit a set $S$ of vertices to the oracle, it tells…
We study quantum algorithms for the hidden shift problem of complex scalar- and vector-valued functions on finite abelian groups. Given oracle access to a shifted function and the Fourier transform of the unshifted function, the goal is to…