On the complexity of quantum ACC

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, MODP is equivalent to MOD_t (up to constant depth). Based on the case q=2, C. Moore (1999) has shown that quantum analogs of AC⁽⁰⁾, ACC[q], and ACC, denoted QAC_wf ⁽⁰⁾ QACC[2], QACC respectively, define the same class of operators, leaving q>2 as an open question. Our result resolves this question, proving that QAC_wf⁽⁰⁾=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_Q. We define a notion of log-planar QACC operators and show the appropriately restricted versions of QACC and BQACC are contained in P/poly. We also define a notion of log-gate restricted QACC operators and show the appropriately restricted versions of QACC 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 we show that families of such graphs can encode the amplitudes resulting from applying an arbitrary QACC operator to an initial state