Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones

We show that a wide variety of non-linear cellular automata (CAs) can be decomposed into a quasidirect product of linear ones. These CAs can be predicted by parallel circuits of depth O(log^2 t) using gates with binary inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of inputs are allowed. Thus these CAs can be predicted by (idealized) parallel computers much faster than by explicit simulation, even though they are non-linear. This class includes any CA whose rule, when written as an algebra, is a solvable group. We also show that CAs based on nilpotent groups can be predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p'' gates respectively. We use these techniques to give an efficient algorithm for a CA rule which, like elementary CA rule 18, has diffusing defects that annihilate in pairs. This can be used to predict the motion of defects in rule 18 in O(log^2 t) parallel time

Counting, Fanout, and the Complexity of Quantum ACC

We propose definitions of \QAC^0, the quantum analog of the classical class \AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and \QACC[q], the analog of the class \ACC[q] where \Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum \MOD_q gates in constant depth for any $q$, so \QACC[2] = \QACC. More generally, we show that for any $q,p > 1$, \MOD_q is equivalent to \MOD_p (up to constant depth). This implies that \QAC^0 with unbounded fanout gates, denoted \QACwf^0, is the same as \QACC[q] and \QACC for all $q$. Since \ACC[p] \ne \ACC[q] whenever $p$ and $q$ are distinct primes, \QACC[q] is strictly more powerful than its classical counterpart, as is \QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for \QACC^0 in terms of related language classes. We define classes of languages \EQACC, \NQACC and \BQACC_{\rats}. We define a notion of $\log$-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