Related papers: Lower Bounds for XOR of Forrelations
We obtain the strongest separation between quantum and classical query complexity known to date -- specifically, we define a black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved…
Quantum machine learning applies principles such as superposition and entanglement to data processing and optimization. Variational quantum models operate on qubits in high-dimensional Hilbert spaces and provide an alternative approach to…
Computational models typically assume that operations are applied in a fixed sequential order. In recent years several works have looked at relaxing this assumption, considering computations without any fixed causal structure and showing…
There are few explicit examples of two player nonlocal games with a large gap between classical and quantum value. One of the reasons is that estimating the classical value is usually a hard computational task. This paper is devoted to…
In this paper, we prove a strong XOR lemma for bounded-round two-player randomized communication. For a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow\{0,1\}$, the $n$-fold XOR function $f^{\oplus n}:\mathcal{X}^n\times…
The ability of the Quantum Approximate Optimization Algorithm (QAOA) to deliver a quantum advantage on combinatorial optimization problems is still unclear. Recently, a scaling advantage over a classical solver was postulated to exist for…
We show a partial Boolean function $f$ together with an input $x\in f^{-1}\left(*\right)$ such that both $C_{\bar{0}}\left(f,x\right)$ and $C_{\bar{1}}\left(f,x\right)$ are at least $C\left(f\right)^{2-o\left(1\right)}$. Due to recent…
Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86}, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently,…
Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2+sqrt(N) calls to the oracle are sufficient to guess the whole content of the oracle (being an N bit string) with probability greater than…
In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum…
In this paper, we theoretically and experimentally analyze sequential processors with limited communication between parts. We compare the expressivity of sequential quantum and classical processors under the same constraints. They consist…
We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower bounded by a…
Given a prior probability distribution over a set of possible oracle functions, we define a number of queries to be useless for determining some property of the function if the probability that the function has the property is unchanged…
We study the communication complexity of symmetric XOR functions, namely functions $f: \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ that can be formulated as $f(x,y)=D(|x\oplus y|)$ for some predicate $D: \{0,1,...,n\} \rightarrow…
Let X = (x_0,...,x_{n-1})$ be a sequence of n numbers. For \epsilon > 0, we say that x_i is an \epsilon-approximate median if the number of elements strictly less than x_i, and the number of elements strictly greater than x_i are each less…
Suppose we have a two-party communication protocol for $f$ which allows the parties to make queries to an oracle computing $g$; for example, they may query an Equality oracle. To translate this protocol into a randomized protocol, we must…
Cryptographic problems can often be reduced to solving Boolean polynomial systems, whose equivalent logical formulas can be treated using SAT solvers. Given the algebraic nature of the problem, the use of the logical XOR operator is common…
We prove a query complexity lower bound for $\mathsf{QMA}$ protocols that solve approximate counting: estimating the size of a set given a membership oracle. This gives rise to an oracle $A$ such that $\mathsf{SBP}^A \not\subset…
We study classical query algorithms with post-selection, and find that they are closely connected to rational functions with nonnegative coefficients. We show that the post-selected classical query complexity of a Boolean function is equal…
This paper presents a quantum algorithm for solving the fractional Poisson equation \((-\Delta)^s u = f\) with \(s \in (0,1)\) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system…