Counting,
fanout and the complexity of quantum ACC
(pp35-65)
F. Green, S. Homer, C. Moore, and C. Pollett
doi:
https://doi.org/10.26421/QIC2.1-3
Abstracts:
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 and polynomial size). 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 in terms of related language classes.
We define classes of languages closely related to QACC[2] and show that
restricted versions of them can be simulated by polynomial-size
circuits. With further restrictions, language classes related to
QACC[2] operators can be simulated by classical threshold circuits of
polynomial size and constant depth.
Key words: quantum
computation, quantum & circuit complexity, threshold circuit |