Related papers: Towards Optimal Depth Reductions for Syntactically…

Optimizing the size and depth of CNOT circuits is an active area of research in quantum computing and is particularly relevant for circuits synthesized from the Clifford + T universal gate set. Although many techniques exist for finding…

We consider the following problem: given a program, find tight asymptotic bounds on the values of some variables at the end of the computation (or at any given program point) in terms of its input values. We focus on the case of…

The determinant can be computed by classical circuits of depth $O(\log^2 n)$, and therefore it can also be computed in classical space $O(\log^2 n)$. Recent progress by Ta-Shma [Ta13] implies a method to approximate the determinant of…

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone…

We present a quantum algorithm for multiplying two $n$-bit integers with overall circuit depth and $T$-depth both bounded by $O(\log^{2} n)$, while using $O(n^{2})$ gates and ancillary qubits. Our construction generates partial products via…

In this work we establish lower bounds on the size of Clifford circuits that measure a family of commuting Pauli operators. Our bounds depend on the interplay between a pair of graphs: the Tanner graph of the set of measured Pauli…

We consider the problem of computing the second elementary symmetric polynomial S^2_n(X) using depth-three arithmetic circuits of the form "sum of products of linear forms". We consider this problem over several fields and determine EXACTLY…

The approximate degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that approximates $f$ pointwise: $|f(x)-p(x)|\leq1/3$ for all $x\in\{0,1\}^n.$ For every $\delta>0,$ we construct CNF…

We first extend the results of Chatterjee,Kumar,Shi,Volk(Computational Complexity 2022) by showing that the degree $d$ elementary symmetric polynomials in $n$ variables have formula lower bounds of $\Omega(d(n-d))$ over fields of positive…

In this work, we prove the strongest known lower bounds for QAC$^0$, allowing polynomially many gates and ancillae. Our main results show that: (1) Depth-3 QAC$^0$ circuits cannot compute PARITY, and require $\Omega(\exp(\sqrt{n}))$ gates…

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the…

Valiant's famous VP vs. VNP conjecture states that the symbolic permanent polynomial does not have polynomial-size algebraic circuits. However, the best upper bound on the size of the circuits computing the permanent is exponential.…

Algebraic Branching Programs(ABPs) are standard models for computing polynomials. Syntactic multilinear ABPs (smABPs) are restrictions of ABPs where every variable is allowed to occur at most once in every path from the start to the…

In this paper, we study circuits and formulas for provenance polynomials of Datalog programs. We ask the following question: given an absorptive semiring and a fact of a Datalog program, what is the optimal depth and size of a…

We design polynomial size, constant depth (namely, $\mathsf{AC}^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, B\'ezout coefficients,…

For every constant $d$, we design a subexponential time deterministic algorithm that takes as input a multivariate polynomial $f$ given as a constant depth algebraic circuit over the field of rational numbers, and outputs all irreducible…

We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq…

We study the power of negation in the Boolean and algebraic settings and show the following results. * We construct a family of polynomials $P_n$ in $n$ variables, all of whose monomials have positive coefficients, such that $P_n$ can be…

We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and…

We show that there is a language in $\mathsf{S}_2\mathsf{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds…