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 QACwf(0)β, QACC[2], QACC respectively,
define the same class of operators, leaving q>2 as an open question. Our
result resolves this question, proving that QACwf(0)β= 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