Related papers: Circuit Depth Reductions

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against…

Recently, Gupta et.al. [GKKS2013] proved that over Q any $n^{O(1)}$-variate and $n$-degree polynomial in VP can also be computed by a depth three $\Sigma\Pi\Sigma$ circuit of size $2^{O(\sqrt{n}\log^{3/2}n)}$. Over fixed-size finite fields,…

We consider the problem of finding a near ground state of a $p$-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors' recent work [Gamarnik, Jagannath, Wein 2020], we establish that any…

We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were…

We prove a lower bound of $\Omega\left(n^{1.5}\right)$ for the number of product gates in non-commutative arithmetic circuits for an explicit $n$-variate degree-$n$ polynomial $f_{n}$ (over every field). We observe that this implies that…

Gurumuhkani et al. (CCC'24) introduced the local enumeration problem $Enum(k, t)$ as follows: for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment with Hamming weight less than $t(n)$,…

Agrawal and Vinay [AV08] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Korian…

We seek to develop better upper bound guarantees on the depth of quantum CZ gate, CNOT gate, and Clifford circuits than those reported previously. We focus on the number of qubits $n\,{\leq}\,$1,345,000 [1], which represents the most…

We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS…

Koiran showed that if a $n$-variate polynomial of degree $d$ (with $d=n^{O(1)}$) is computed by a circuit of size $s$, then it is also computed by a homogeneous circuit of depth four and of size $2^{O(\sqrt{d}\log(d)\log(s))}$. Using this…

We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function $f$ is the characteristic vector of the minimum sized set of negated variables needed in any…

In this paper, we show exponential lower bounds for the class of homogeneous depth-$5$ circuits over all small finite fields. More formally, we show that there is an explicit family $\{P_d : d \in \mathbb{N}\}$ of polynomials in…

In this paper, we investigate an approach to circuit lower bounds via bounded width circuits. The approach consists of two steps: (i) We convert circuits to (deterministic or nondeterministic) bounded width circuits. (ii) We prove lower…

We provide an $\Omega(log(n))$ lower bound for the depth of any quantum circuit generating the unique groundstate of Kitaev's spherical code. No circuit-depth lower bound was known before on this code in the general case where the gates can…

In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of {\it depth reduction} developed in the works of Agrawal-Vinay [AV08], Koiran…

In their paper on the "chasm at depth four", Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m). This…

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…

Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to prove VP not equal to VNP, it is sufficient to show that an explicit polynomial in…

In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree $n$ in $n^2$ variables such that any homogeneous depth 4 arithmetic circuit…

We prove two new upper bounds for depth-2 linear circuits computing the $N$th disjointness matrix $D^{\otimes N}$. First, we obtain a circuit of size $O\big(2^{1.24485N}\big)$ over $\{0,1\}$. Second, we obtain a circuit of degree…