It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition, or whether a given quantum bit has positive expectation value at the end of a quantum computation. This result is achieved by showing that the complexity class NQP of Adleman, Demarrais, and Huang [1], a quantum analog of NP, is equal to the counting class coC =

The correlation between two Boolean functions of n inputs is de ned as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper, we compute, in closed form, the correlation between any t wo symmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has an exponentially small correlation (in n) with the parity function. This improves a result of Smolensky 12] restricted to symmetric Boolean functions: the correlation between parity a n d a n y circuit consisting of a M o d q gate over AND-gates of small fan-in, where q is odd and the function computed by the sum of the AND-gates is symmetric, is bounded by 2 ; (n) .In addition, we nd that for a large class of symmetric functions the correlation with parity i s identically zero for in nitely many n. W e c haracterize exactly those symmetric Boolean functions having this property.

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.

We show that if a language is recognized within certain error bounds by constantdepth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, our results implywhere EQNC 0 is the constant-depth analog of the class EQP. On the other hand, we adapt and extend ideas of Terhal & DiVincenzo [TD02] to show that, for any family F of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depth-five circuits over F is just as hard as computing these probabilities for arbitrary quantum circuits over F. In particular, this implies thatwhere NQNC 0 is the constant-depth analog of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC ⊆ TC 0 [GHMP02].

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