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 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.

Average-Case Quantum Advantage with Shallow Circuits

“…Besides being the first such unconditional separation in the circuit model, this separation is also especially important since shallow quantum circuits are likely to be the easiest quantum circuits to experimentally implement, due to their robustness to noise and decoherence. (Note that separations were already known when allowing gates with unbounded fanin or fanout [GHMP02,HS05,TT16]. The strength of Bravyi, Gosset and König's result is that it holds for the weaker model of quantum circuits with bounded fanin and fanout.…”

Parallelizing quantum circuits

“…These constructions usually require the use of auxiliary qubits. The depth complexity of quantum circuits has also been studied in [12,13].…”

Is entanglement monogamous?

“…Constant-depth quantum circuits, but with gates that have arbitrary fan-in, have been studied previously, see for example Ref. [GHMP02]. The constant depth circuits that we consider here have fan-in at most two.…”