Related papers: Separating OR, SUM, and XOR Circuits
We describe an implementation of Shor's quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The…
An algorithm for the reduction of one-loop n-point tensor integrals to basic integrals is proposed. We transform tensor integrals to scalar integrals with shifted dimension and reduce these by recurrence relations to integrals in generic…
We reformulate the O(N) sigma model as a loop model whose configurations are the all-order strong coupling graphs of the original model. The loop configurations are represented by a pointer list in the computer and a Monte Carlo update…
We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra…
The soft and collinear singularities of general scalar and tensor one-loop N-point integrals are worked out explicitly. As a result a simple explicit formula is given that expresses the singular part in terms of 3-point integrals. Apart…
A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f is a read-k circuit if the polynomial produced (purely syntactically) by the arithmetic (+,x) version of the circuit has the property that for every prime implicant…
In the classical transformer attention scheme, we are given three $n \times d$ size matrices $Q, K, V$ (the query, key, and value tokens), and the goal is to compute a new $n \times d$ size matrix $D^{-1} \exp(QK^\top) V$ where $D =…
We present a fit-for-purpose introduction to tensors and their operations. It is envisaged to help the reader become acquainted with its underpinning concepts for the study of path signatures. The text includes exercises, solutions and many…
In this paper we consider the time complexity of computing the sum and product of two $n$-bit numbers within the tile self-assembly model. The (abstract) tile assembly model is a mathematical model of self-assembly in which system…
In this work, we revisit prefix sums through the lens of linear algebra. We describe an identity that decomposes triangular all-ones matrices as a sum of two Kronecker products, and apply it to design recursive prefix sum algorithms and…
We present sharp bounds for $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ in terms of the variance of the vector $(x_1^{1/2},...,x_n^{1/2})$.
We consider the following problem: given three sets of real numbers, output a word-RAM data structure from which we can efficiently recover the sign of the sum of any triple of numbers, one in each set. This is similar to a previous work by…
We propose an efficient algorithmic framework for time domain circuit simulation using exponential integrator. This work addresses several critical issues exposed by previous matrix exponential based circuit simulation research, and makes…
Probabilistic circuits are a unifying representation of functions as computation graphs of weighted sums and products. Their primary application is in probabilistic modeling, where circuits with non-negative weights (monotone circuits) can…
Our work explores the hardness of $3$SUM instances without certain additive structures, and its applications. As our main technical result, we show that solving $3$SUM on a size-$n$ integer set that avoids solutions to $a+b=c+d$ for $\{a,…
In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, non-interacting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we…
We show how to evaluate tensor one-loop integrals in momentum space avoiding the usual plague of Gram determinants. We do this by constructing combinations of $n$- and $(n-1)$-point scalar integrals that are finite in the limit of vanishing…
We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in…
We give a definition of an integer-valued function $\sum_i \alpha_i x ^*_i$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module…
We perform formal verification of quantum circuits by integrating several techniques specialized to particular classes of circuits. Our verification methodology is based on the new notion of a reversible miter that allows one to leverage…