Related papers: Smaller ACC0 Circuits for Symmetric Functions
Measures of circuit complexity are usually analyzed to ensure the computation of Boolean functions with economy and efficiency. One of these measures is energy complexity, which is related to the number of gates that output true in a…
We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to…
Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit…
One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC^0$, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing…
In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that…
This paper investigates the dynamics of the iterated sum-of-divisors function $\sigma_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $\sigma_k(m) \equiv 0…
Neural gates compute functions based on weighted sums of the input variables. The expressive power of neural gates (number of distinct functions it can compute) depends on the weight sizes and, in general, large weights (exponential in the…
Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$…
An efficient randomized polynomial identity test for noncommutative polynomials given by noncommutative arithmetic circuits remains an open problem. The main bottleneck to applying known techniques is that a noncommutative circuit of size…
In this work, we prove that for any $m>1$, there exists a family of good qudit quantum codes supporting transversal logical $\mathsf{C}^{m-1}\mathsf{Z}$ gates that can address specified logical qudits and be largely executed in parallel.…
Parallelization is a major challenge in quantum algorithms due to physical constraints like no-cloning. This is vividly illustrated by the conjecture of Moore and Nilsson from their seminal work on quantum circuit complexity [MN01,…
We give new, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix. A standard argument (e.g., the mixed product property of Kronecker products, or a generalization…
Fast combinational multipliers with large bit widths can occupy significant silicon area, which also drives up power consumption. Area can be reduced through resource sharing (i.e., folding) at the expense of lower throughput, which is…
Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is…
Any function F : {0,...,N-1} -> {-1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Mobius function mu in the sense that E_{0 <= x <= N-1} mu(x)F(x) = o(1). The proof combines…
We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $\mu \in [0,1]$ be the locality…
We address the black-box polynomial identity testing (PIT) problem for non-commutative polynomials computed by $+$-regular circuits, a class of homogeneous circuits introduced by [AJMR](STOC 2017, Theory of Computing 2019). These circuits…
For any $q > 1$, let $\MOD_q$ be a quantum gate that determines if the number of 1's in the input is divisible by $q$. We show that for any $q,t > 1$, $\MOD_q$ is equivalent to $\MOD_t$ (up to constant depth). Based on the case $q=2$, Moore…
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 study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of Controlled-NOT (C-NOT) gates required (single-qubit gates are free). We consider three…