Related papers: Efficient quantum protocols for XOR functions

The level-$k$ $\ell_1$-Fourier weight of a Boolean function refers to the sum of absolute values of its level-$k$ Fourier coefficients. Fourier growth refers to the growth of these weights as $k$ grows. It has been extensively studied for…

We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and non-deterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound…

We call $F:\{0, 1\}^n\times \{0, 1\}^n\to\{0, 1\}$ a symmetric XOR function if for a function $S:\{0, 1, ..., n\}\to\{0, 1\}$, $F(x, y)=S(|x\oplus y|)$, for any $x, y\in\{0, 1\}^n$, where $|x\oplus y|$ is the Hamming weight of the bit-wise…

It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Omega(log n), and that this bound is achieved for some functions. In this paper we study the case of…

In this paper, we focus on the quantum communication complexity of functions of the form $f \circ G = f(G(X_1, Y_1), \ldots, G(X_n, Y_n))$ where $f: \{0, 1\}^n \to \{0, 1\}$ is a symmetric function, $G: \{0, 1\}^j \times \{0, 1\}^k \to \{0,…

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of…

The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication…

We show a new duality between the polynomial margin complexity of $f$ and the discrepancy of the function $f \circ \textsf{XOR}$, called an $\textsf{XOR}$ function. Using this duality, we develop polynomial based techniques for…

We give a tight lower bound of Omega(\sqrt{n}) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(\log n) qubits for…

This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication…

$\newcommand{\sp}{\mathsf{sparsity}}\newcommand{\s}{\mathsf{s}}\newcommand{\al}{\mathsf{alt}}$ The well-known Sensitivity Conjecture states that for any Boolean function $f$, block sensitivity of $f$ is at most polynomial in sensitivity of…

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the…

This paper studies the gap between quantum one-way communication complexity $Q(f)$ and its classical counterpart $C(f)$, under the {\em unbounded-error} setting, i.e., it is enough that the success probability is strictly greater than 1/2.…

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…

We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a…

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 study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Let f be a Boolean function with…

In this work we revisit the Boolean Hidden Matching communication problem, which was the first communication problem in the one-way model to demonstrate an exponential classical-quantum communication separation. In this problem, Alice's…

In this note, we study the relation between the parity decision tree complexity of a boolean function $f$, denoted by $\mathrm{D}_{\oplus}(f)$, and the $k$-party number-in-hand multiparty communication complexity of the XOR functions…

This paper gives a nearly tight characterization of the quantum communication complexity of the permutation-invariant Boolean functions. With such a characterization, we show that the quantum and randomized communication complexity of the…