Related papers: Prefix Sums via Kronecker Products
An algorithm for reversible logic synthesis is proposed. The task is, for a given $n$-bit substitution map $P_n: \{0,1\}^n \rightarrow \{0,1\}^n$, to find a sequence of reversible logic gates that implements the map. The gate library…
We present a collection of new results on problems related to 3SUM, including: 1. The first truly subquadratic algorithm for $\ \ \ \ \ $ 1a. computing the (min,+) convolution for monotone increasing sequences with integer values bounded by…
Given a boolean n by n matrix A we consider arithmetic circuits for computing the transformation x->Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative…
We consider the numerical approximation of $f({\cal A})b$ where $b\in{\mathbb R}^{N}$ and $\cal A$ is the sum of Kronecker products, that is ${\cal A}=M_2 \otimes I + I \otimes M_1\in{\mathbb R}^{N\times N}$. Here $f$ is a regular function…
We present quantum circuits for comparison and increment operations that achieve an asymptotically optimal gate count of $\Theta(n)$ and depth of $\Theta(\log n)$ over the Clifford+Toffoli gate set, while using a provably minimal number of…
We consider the problem of the variational quantum circuit synthesis into a gate set consisting of the CNOT gate and arbitrary single-qubit (1q) gates with the primary target being the minimization of the CNOT count. First we note that…
The matrices that can be exactly represented by a circuit over the Toffoli-Hadamard gate set are the orthogonal matrices of the form $M/ \sqrt{2}{}^k$, where $M$ is an integer matrix and $k$ is a nonnegative integer. The exact synthesis…
The paper proposes an implicit (i.e., machine-independent) complexity approach to studying computation by polynomial-size, constant-depth circuits with gates counting modulo a constant through the lens of discrete ordinary differential…
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent $\omega$ of matrix multiplication. The Kronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the $3\times 3$…
We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range, which…
We present an n-bit Toffoli gate quantum circuit based on the realization proposed by Barenco, where some of the Toffoli gates in their construction are replaced with Peres gates. This results in a significant cost reduction. Our main…
Quantum computation offers the potential to solve fundamental yet otherwise intractable problems across a range of active fields of research. Recently, universal quantum-logic gate sets - the building blocks for a quantum computer - have…
We consider the problem of constructing fast and small binary adder circuits. Among widely-used adders, the Kogge-Stone adder is often considered the fastest, because it computes the carry bits for two $n$-bit numbers (where $n$ is a power…
We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix…
We introduce an infinite family of Kronecker series twisted by characters. As an application, we give a closed formula for the sum of all Hecke eigenforms on ${\Gamma}_0(N) $ multiplied by their twisted period polynomials in terms of the…
Consider a pair of elements $f$ and $g$ in a commutative ring $Q$. Given a matrix factorization of $f$ and another of $g$, the tensor product of matrix factorizations, which was first introduced by Kn\"orrer and later generalized by…
We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassen's algorithm for matrix multiplication. We build on a previous framework that establishes communication…
The Kronecker product-based algorithm for context-free path querying (CFPQ) was proposed by Orachev et al. (2020). We reduce this algorithm to operations over Boolean matrices and extend it with the mechanism to extract all paths of…
We provide evidence that commonly held intuitions when designing quantum circuits can be misleading. In particular we show that: a) reducing the T-count can increase the total depth; b) it may be beneficial to trade CNOTs for measurements…
We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values $(x,y)$ to $(x,x \oplus y)$) is universal in the sense that all unitary operations on arbitrarily many…