On the Complexity of Quantum ACC

• Authors: Frederic Green;Steven Homer;Christopher Pollett
• Venue: COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
• Year: 2000

For any \math, let MODq 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; \math, MODq is equivalent to MODt (up to constant depth). Based on the case q = 2, Moore [8] has shown that quantum analogs of AC(0) ,ACC[q], and ACC, denoted \math ,QACC[2], QACC respectively, define the same class of operators, leaving \math as an open question. Our result resolves this question, proving that \math = 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 BQACCQ. We define a notion of log-planar QACC operators and show the appropriately restricted versions of EQACC and NQACC are contained in P/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) . 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.