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^0[q], where n-ary Mod-q gates are also allowed. We show that it is possible to make a `cat' state on n qubits in constant depth if and...

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 propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum error-correcting codes, or...

We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^{-k} with k1. For the Wiener measure occurring in many applications we have k=2. We want to...

We exhibit a quantum circuit that performs the Quantum Fourier Transform on $n$ qubits in $ (n)$ depth. Thus, a parallel quantum computer can carry out the QFT in linear time. We conjecture that this can, in fact, be reduced to $ ({\rm log} n)$ depth, which would place the QFT in the class {\bf...

Given an initial distribution of sand in an Abelian sandpile, what final state does it relax to after all possible avalanches have taken place? In d = 3*, we show that this problem is P-complete, so that explicit simulation of the system is almost certainly necessary. We also show that the...