Some Notes on Parallel Quantum Computation

We exhibit some simple gadgets useful in designing shallow parallel circuits for quantum algorithms. We prove that any quantum circuit composed entirely of controlled-not gates or of diagonal gates can be parallelized to logarithmic depth, while circuits composed of both cannot. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we exhibit a simple quantum circuit related to it that we believe cannot be parallelized to less than linear depth, and therefore might be used to prove that QNC < QP

On the Complexity of Quantum ACC

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 \cite{moore99} has shown that quantum analogs of AC$^{(0)}$, ACC$[q]$, and ACC, denoted QAC$^{(0)}_{wf}$, QACC$[2]$, QACC respectively, define the same class of operators, leaving $q > 2$ as an open question. Our result resolves this question, proving that QAC$^{(0)}_{wf} =$ QACC$[q] =$ QACC for all $q$. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC and BQACC_{\rats}. We define a notion $\log$-planar QACC operators and show the appropriately restricted versions of EQACC and NQACC are contained in P/poly. We also define a notion of $\log$-gate restricted QACC operators and show the appropriately restricted versions of EQACC and NQACC are contained in TC$^{(0)}$. To do this last proof, we show that TC$^{(0)}$ can perform iterated addition and multiplication in certain field extensions. We also introduce the notion of a polynomial-size tensor graph and show that families of such graphs can encode the amplitudes resulting from apply an arbitrary QACC operator to an initial state.Comment: 22 pages, 4 figures This version will appear in the July 2000 Computational Complexity conference. Section 4 has been significantly revised and many typos correcte

Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials

We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over $\mathbb{R}$) of functions from some "simple" class ${\cal C}$. In particular, given ${\cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${\cal C}$ is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when ${\cal C}$ is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where ${\cal C}$ is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts. We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. For example, we show there are functions in nondeterministic quasi-polynomial time that require super-polynomial size: $\bullet$ Depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds. $\bullet$ Depth-two neural networks with ReLU activation function. $\bullet$ $\mathbb{R}$-linear combinations of $O(1)$-degree $\mathbb{F}_p$-polynomials, for every prime $p$ (related to problems regarding Higher-Order "Uncertainty Principles"). We also obtain a function in $E^{NP}$ requiring $2^{\Omega(n)}$ linear combinations. $\bullet$ $\mathbb{R}$-linear combinations of $ACC \circ THR$ circuits of polynomial size (further generalizing the recent lower bounds of Murray and the author). (The above is a shortened abstract. For the full abstract, see the paper.

Depth Reduction for Circuits with a Single Layer of Modular Counting Gates

We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e ${AC}^0 circ {MOD}_m$ circuits. We show that the following holds for several types of gates $G$: by adding a gate of type $G$ at the output, it is possible to obtain an equivalent randomized depth 2 circuit of quasipolynomial size consisting of a gate of type $G$ at the output and a layer of modular counting gates, i.e $G circ {MOD}_m$ circuits. The types of gates $G$ we consider are modular counting gates and threshold-style gates. For all of these, strong lower bounds are known for (deterministic) $G circ {MOD}_m$ circuits