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 =

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.

This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomial-time reducibilities, it is shown that 1. a multivalued partial function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable via 2 k − 1 nonadaptive queries to NPMV;2. a characteristic function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable with k adaptive queries to NP;3. unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV are different than k + 1 adaptive (nonadaptive) queries to NPMV.Nondeterministic reducibilities, lowness, and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses.

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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].