Related papers: A Fast Quantum Algorithm for the Affine Boolean Fu…
We introduce an efficient and accurate readout measurement scheme for single and multi-qubit states. Our method uses Bayesian inference to build an assignment probability distribution for each qubit state based on a reference…
This paper considers the problem of approximating a Boolean function $f$ using another Boolean function from a specified class. Two classes of approximating functions are considered: $k$-juntas, and linear Boolean functions. The $n$ input…
The r-th order nonlinearity of a Boolean function is the minimum number of elements that have to be changed in its truth table to arrive at a Boolean function of degree at most r. It is shown that the (suitably normalised) r-th order…
A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the…
We introduce a randomized algorithm for computing the minimal-norm solution to an underdetermined system of linear equations. Given an arbitrary full-rank m x n matrix A with m<n, any m x 1 vector b, and any positive real number epsilon…
Boolean functions are mathematical objects with numerous applications in domains like coding theory, cryptography, and telecommunications. Finding Boolean functions with specific properties is a complex combinatorial optimization problem…
One of the most basic computational problems is the task of finding a desired item in an ordered list of N items. While the best classical algorithm for this problem uses log_2 N queries to the list, a quantum computer can solve the problem…
We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm.…
We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an $O(\log n)$-approximation algorithm, based on the submodular goal-value approach of…
We prove that, to compute a Boolean function $f$ on $N$ variables with error probability $\epsilon$, any quantum black-box algorithm has to query at least $\frac{1 - 2\sqrt{\epsilon}}{2} \rho_f N = \frac{1 - 2\sqrt{\epsilon}}{2} \bar{S}_f$…
We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose…
Boolean matrix factorisation aims to decompose a binary data matrix into an approximate Boolean product of two low rank, binary matrices: one containing meaningful patterns, the other quantifying how the observations can be expressed as a…
We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly…
A fast and accurate algorithm for solving a Bernstein-Vandermonde linear system is presented. The algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of Neville…
We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$ uniformly random examples of an unknown function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$, our…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
A new quantum algorithm for a search problem and its computational complexity are discussed. It is shown in the search problem containing 2^n objects that our algorithm runs in polynomial time.
It is believed that there is no efficient classical algorithm to determine the linear structure of Boolean function. We investigate an extension of Simon's period-finding quantum algorithm, and propose an efficient quantum algorithm to…
Traditional cryptography is suffering a huge threat from the development of quantum computing. While many currently used public-key cryptosystems would be broken by Shor's algorithm, the effect of quantum computing on symmetric ones is…
The problem of quantum state filtering consists of determining whether an unknown quantum state, which is chosen from a known set of states, is either a particular, specified state, or not. We consider this problem for the case that the…