Related papers: Polynomial representation of general partial Boole…
The algebraic degree of Boolean functions (or vectorial Boolean functions) is an important cryptographic parameter that should be computed by fast algorithms. They work in two main ways: (1) by computing the algebraic normal form and then…
In Exact Quantum Query model, almost all of the Boolean functions for which non-trivial query algorithms exist are symmetric in nature. The most well known techniques in this domain exploit parity decision trees, in which the parity of two…
The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly…
This paper introduces and analyzes symmetric and anti-symmetric quantum binary functions. Generally, such functions uniquely convert a given computational basis state into a different basis state, but with either a plus or a minus sign.…
Deutsch's algorithm for two qubits (one control qubit plus one auxiliary qubit) is extended to two $d$-dimensional quantum systems or qudits for the case in which $d$ is equal to $2^n$, $n=1,2,...$ . This allows one to classify a certain…
This paper demonstrates the use of entanglement resources in quantum speedup by presenting an algorithm which is the generalization of an algorithm proposed by Goswami and Panigrahi [arXiv:1706.09489 (2017)]. We generalize the algorithm and…
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions. In this paper, we…
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…
We compare the failure probabilities of ensemble implementations of quantum algorithms which use pseudo-pure initial states, quantified by their polarization, to those of competing classical probabilistic algorithms. Specifically we…
We study the query complexity on slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number…
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum…
Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a…
The weight decision problem, which requires to determine the Hamming weight of a given binary string, is a natural and important problem, with applications in cryptanalysis, coding theory, fault-tolerant circuit design and so on. In…
We consider a family of unstructured problems, for which we propose a method for constructing analog, continuous-time quantum algorithms that are more efficient than their classical counterparts. In this family of problems, which we refer…
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough.…
We study randomized and quantum query (a.k.a. decision tree) complexity for all total Boolean functions, with emphasis to derandomization and dequantization (removing quantumness from algorithms). Firstly, we show that $D(f) = O(Q_1(f)^3)$…
We study the computation complexity of Boolean functions in the quantum black box model. In this model our task is to compute a function $f:\{0,1\}\to\{0,1\}$ on an input $x\in\{0,1\}^n$ that can be accessed by querying the black box.…
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…
A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is: how big is the advantage of exact quantum algorithms over their classical counterparts: deterministic…