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 $ΣΠΣ$ circuit of size $2^{O(\sqrt{n}\log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $ΣΠΣ$ circuit that computes $Det_n$ (or $Perm_n$) must be of size $2^{Ω(n)}$ [GK1998]. In this paper, we prove that over fixed-size finite fields, any $ΣΠΣ$ circuit for computing the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\times n$, must be of size $2^{Ω(n\log n)}$. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the $n^{O(1)}$-variate and $n$-degree polynomials in VP by depth 3 circuits of size $2^{o(n\log n)}$. The result [GK1998] can only rule out such a possibility for depth 3 circuits of size $2^{o(n)}$.
We also give an example of an explicit polynomial ($NW_{n,ε}(X)$) in VNP (not known to be in VP), for which any $ΣΠΣ$ circuit computing it (over fixed-size fields) must be of size $2^{Ω(n\log n)}$. The polynomial we consider is constructed from the combinatorial design. An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger circuit size lower bounds than Permanent in a reasonably strong model of computation.
Next, we prove that any depth 4 $ΣΠ^{[O(\sqrt{n})]}ΣΠ^{[\sqrt{n}]}$ circuit computing $NW_{n,ε}(X)$ (over any field) must be of size $2^{Ω(\sqrt{n}\log n)}$. To the best of our knowledge, the polynomial $NW_{n,ε}(X)$ is the first example of an explicit polynomial in VNP such that it requires $2^{Ω(\sqrt{n}\log n)}$ size depth four circuits, but no known matching upper bound.
The best known size lower bounds against unrestricted circuits have remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than $5n$. In this work, we propose a non-gate-elimination approach for obtaining circuit lower bounds, via certain depth-three lower bounds. We prove that every (unbounded-depth) circuit of size $s$ can be expressed as an OR of $2^{s/3.9}$ $16$-CNFs. For DeMorgan formulas, the best known size lower bounds have been stuck at around $n^{3-o(1…
Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result als…
We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($ΣΠΣΠ$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = ω(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas~\cite{T15} for general circuits, and is known to be asymptotically optimal in the exponent when $d < n^ε$ for a small enough constant $ε$. Our upper bound matches the lower bound of $\exp\left({Ω\left(\sqrt{n\…
We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $Δ+1$ to one of product-depth $Δ$ in the multilinear setting. We show that for every positive $Δ= Δ(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(Δ)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $Δ+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $Δ$ and size less than $\exp(n^{Ω(1/Δ)})$. This result is tight up to the constant implicit in the double exponent for all $Δ= o(\log n/\log \log n).$ This strengthens a…
We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by small-depth circuits. For constant-depth circuits of polynomial size (i.e. $\mathsf{AC^0}$ tampering functions), our codes have codeword length $n = k^{1+o(1)}$ for a $k$-bit message. This is an exponential improvement of the previous best construction due to Chattopadhyay and Li (STOC 2017), which had codeword length $2^{O(\sqrt{k})}$. Our construction remains efficient for circuit depths as large as $Θ(\log(n)/\log\log(n))$ (indeed, our codeword length remains $n\leq k^{1+ε})$, a…
We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant $\sum_{i,j,k,\ell\in[n]} z_{ijk\ell}x_ix_j x_k x_\ell - β=0$, for Boolean variables, does not have polynomial-size IPS refutations where the refutations are multilinear and written as constant-depth circuits. Andrews and Forbes (STOC'22) established recently a constant-depth IPS lower bound, but their hard instance does not have itself small constant-depth circuits, while our inst…
We show that there is a randomized algorithm that, when given a small constant-depth Boolean circuit $C$ made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of constant-degree polynomials), counts the number of satisfying assignments to $C$ in significantly better than brute-force time. Formally, for any constants $d,k$, there is an $ε> 0$ such that the algorithm counts the number of satisfying assignments to a given depth-$d$ circuit $C$ made up of $k$-PTF gates such that $C$ has size at most $n^{1+ε}$. The algorithm…
We study the complexity of small-depth Frege proofs and give the first tradeoffs between the size of each line and the number of lines. Existing lower bounds apply to the overall proof size -- the sum of sizes of all lines -- and do not distinguish between these notions of complexity. For depth-$d$ Frege proofs of the Tseitin principle on the $n \times n$ grid where each line is a size-$s$ formula, we prove that $\exp(n/2^{Ω(d\sqrt{\log s})})$ many lines are necessary. This yields new lower bounds on line complexity that are not implied by Håstad's recent $\exp(n^{Ω(1/d)})$ lower bound on the …
We show that over the field of complex numbers, \emph{every} homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising since there exist homogeneous polynomials $P$ on $n$ variables of degree $2$, such that any depth-$3$ arithmetic circuit computing $P$ must have top fan-in at least $Ω(n)$. As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal and Saptharishi …
Recently Brakerski, Christiano, Mahadev, Vazirani and Vidick (FOCS 2018) have shown how to construct a test of quantumness based on the learning with errors (LWE) assumption: a test that can be solved efficiently by a quantum computer but cannot be solved by a classical polynomial-time computer under the LWE assumption. This test has lead to several cryptographic applications. In particular, it has been applied to producing certifiable randomness from a single untrusted quantum device, self-testing a single quantum device and device-independent quantum key distribution. In this paper, we show …
Decoherence -- in the current physical implementations of quantum computers -- makes depth reduction a vital task in quantum-circuit design. Moore and Nilsson (SIAM Journal of Computing, 2001) demonstrated that additional qubits -- known as ancillae -- can be used to provide an extended space to parallelize quantum circuits. Specifically, they proved that, with $O(n^2)$ ancillae, any $n$-qubit CNOT circuit can be transformed into an equivalent one of $O(\log n)$ depth. However, the near-term quantum technologies can only support a limited amount of qubits, making space-depth trade-off a fundam…
In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing results are limited to depths $2$ and $3$, and achieving results for higher depths has been an important open question. In this paper, we focus on feedforward ReLU networks, and prove fundamental barriers to proving such results beyond depth $4$, by reduction to open problems and natural-proof barriers in circuit complexity. To show this, we study a seemingly u…
