相关论文: Query complexity of Boolean functions on the middl…

We study the deterministic query complexity of Boolean functions on slices of the hypercube. The $k^{th}$ slice $\binom{[n]}{k}$ of the hypercube $\{0,1\}^n$ is the set of all $n$-bit strings with Hamming weight $k$. We show that there…

In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…

We provide two sufficient and necessary conditions to characterize any $n$-bit partial Boolean function with exact quantum 1-query complexity. Using the first characterization, we present all $n$-bit partial Boolean functions that depend on…

We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these…

We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called $\Sigma\Pi\Sigma$…

It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost…

We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends…

The (low soundness) linearity testing problem for the middle slice of the Boolean cube is as follows. Let $\varepsilon>0$ and $f$ be a function on the middle slice on the Boolean cube, such that when choosing a uniformly random quadruple…

We establish a lower bound of $\Omega{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $\Omega(\sqrt{D(f)})$ is a lower bound for all Boolean functions.…

We show that, for almost all N-variable Boolean functions f, at least N/4-O(\sqrt{N} log N) queries are required to compute f in quantum black-box model with bounded error.

Early in 1992, Deutsch-Jozsa algorithm computed a symmetric partial Boolean function with a single quantum query, and thus achieved the best separation between classical deterministic and exact quantum query complexity. Until recent years,…

We initiate the study of a new model of query complexity of Boolean functions where, in addition to 0 and 1, the oracle can answer queries with ``unknown''. The query algorithm is expected to output the function value if it can be…

We consider the problem of exact identification for read-once functions over arbitrary Boolean bases. We introduce a new type of queries (subcube identity ones), discuss its connection to previously known ones, and study the complexity of…

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree functions. Here we provide an alternative proof for general low-degree functions, with no constraints on the influences.…

We show that for any constant $c>0$, any (two-sided error) adaptive algorithm for testing monotonicity of Boolean functions must have query complexity $\Omega(n^{1/2-c})$. This improves the $\tilde\Omega(n^{1/3})$ lower bound of [CWX17] and…

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…

We apply the pigeonhole principle to show that there must exist Boolean functions on 7 inputs with a multiplicative complexity of at least 7, i.e., that cannot be computed with only 6 multiplications in the Galois field with two elements.

This paper describes a purely functional library for computing level-$p$-complexity of Boolean functions, and applies it to two-level iterated majority. Boolean functions are simply functions from $n$ bits to one bit, and they can describe…

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…

We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dam's oracle interrogation is…