Periodic Fourier representation of Boolean functions

In this work, we consider a new type of Fourier-like representation of Boolean function f\colon\{+1,-1\}^n\to\{+1,-1\}: f(x) = \cos(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i). This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing NMQCp. The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of f by \NMQCp. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the F_2-polynomial representation. In this work, we first show that Boolean functions related to ZZZZ-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least 2^{\deg_{\mathbb{F}_2}(f)}-1, which means that NMQCp efficiently computes a Boolean function $f$ if and only if F_2-degree of f is small. Furthermore, we show that any symmetric Boolean function, e.g., AND_n, Mod^3_n, Maj_n, etc, can be exactly computed by depth-2 NMQCp using a polynomial number of qubits, that implies exponential gaps between NMQCp and depth-2 NMQCp.

A fast exact quantum algorithm for solitude verification

Solitude verification is arguably one of the simplest fundamental problems in distributed computing, where the goal is to verify that there is a unique contender in a network. This paper devises a quantum algorithm that exactly solves the problem on an anonymous network, which is known as a network model with minimal assumptions [Angluin, STOC’80]. The algorithm runs in O(N) rounds if every party initially has the common knowledge of an upper bound N on the number of parties. This implies that all solvable problems can be solved in O(N) rounds on average without error (i.e., with zero-sided error) on the network. As a generalization, a quantum algorithm that works in O(N log_2 (max{k, 2})) rounds is obtained for the problem of exactly computing any symmetric Boolean function, over n distributed input bits, which is constant over all the n bits whose sum is larger than k for k belongs to {0, 1, . . . , N −1}. All these algorithms work with the bit complexities bounded by a polynomial in N.

Properties of Symmetric Boolean functions

In the present paper we consider symmetric Boolean functions with special property. We study properties of the maximal intervals of these functions. Later we show characteristics of corresponding interval graphs and simplified interval graphs. Specifically we prove, that these two graphs are isomorphic for symmetric Boolean function. Then we obtain the vertex degree of these graphs. We discuss also disjunctive normal forms.

The Weight and Nonlinearity of 2-rotation Symmetric Cubic Boolean Function

The conceptions of $\chi$-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the $k-th$ power of $\rho$.In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with $2n$ variables, which denoted by $F^{2n}(x^{2n})$. We give the recursive formula of weight of $F^{2n}(x^{2n})$, and prove that the weight of $F^{2n}(x^{2n})$ is the same as its nonlinearity.

Linear Recurrences and Asymptotic Behavior of Exponential Sums of Symmetric Boolean Functions

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.

Quantum communication complexity of block-composed functions

A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a {\em total} Boolean function in the two-party interactive model. Razborov's result ({\em Izvestiya: Mathematics}, 67(1):145--159, 2002) implies the conjectured negative answer for functions $F$ of the following form: $F(x, y)=f_n(x_1\cdot y_1, x_2\cdot y_2, ..., x_n\cdot y_n)$, where $f_n$ is a {\em symmetric} Boolean function on $n$ Boolean inputs, and $x_i$, $y_i$ are the $i$'th bit of $x$ and $y$, respectively. His proof critically depends on the symmetry of $f_n$. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions $F$ is the ``block-composition'' of a ``building block'' $g_k : \{0, 1\}^k \times \{0, 1\}^k \rightarrow \{0, 1\}$, and an $f_n : \{0, 1\}^n \rightarrow \{0, 1\}$, such that $F(x, y) = f_n( g_k(x_1, y_1), g_k(x_2, y_2), ..., g_k(x_n, y_n) )$, where $x_i$ and $y_i$ are the $i$'th $k$-bit block of $x, y\in\{0, 1\}^{nk}$, respectively. We show that as long as g_k itself is "hard'' enough, its block-composition with an arbitrary f_n has polynomially related quantum and classical communication complexities. For example, when g_k is the Inner Product function with k=\Omega(\log n), the deterministic communication complexity of its block-composition with any f_n is asymptotically at most the quantum complexity to the power of 7.