Counting, Fanout, and the Complexity of Quantum ACC
Abstract
We propose definitions of $\QAC^0$, the quantum analog of the classical class $\AC^0$ of constantdepth circuits with AND and OR gates of arbitrary fanin, 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 $¶/\poly$. We also define a notion of $\log$gate restricted $\QACC$ operators and show the appropriately restricted versions of $\EQACC$ and $\NQACC$ are contained in $\TC^0$.
 Publication:

arXiv eprints
 Pub Date:
 June 2001
 arXiv:
 arXiv:quantph/0106017
 Bibcode:
 2001quant.ph..6017G
 Keywords:

 Quantum Physics