Quantum Circuits: Fanout, Parity, and Counting

TL;DR

This paper defines quantum analogs of classical constant-depth circuit classes, demonstrating how quantum fanout and parity enable constant-depth construction of Mod-q gates, revealing quantum advantages over classical circuit classes.

Contribution

It introduces QAC^0 and QACC^0[q], establishing their properties and showing quantum fanout's power to construct Mod-q gates in constant depth, surpassing classical capabilities.

Findings

Quantum fanout enables constant-depth creation of cat states.

Parity and fanout allow constant-depth construction of Mod-q gates for any q.

QACC^0[2] is strictly more powerful than classical ACC^0[q].

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^0[q], where n-ary Mod-q gates are also allowed. We show that it is possible to make a `cat' state on n qubits in constant depth if and only if we can construct a parity or Mod-2 gate in constant depth; therefore, any circuit class that can fan out a qubit to n copies in constant depth also includes QACC^0[2]. In addition, we prove the somewhat surprising result that parity or fanout allows us to construct Mod-q gates in constant depth for any q, so QACC^0[2] = QACC^0. Since ACC^0[p] != ACC^0[q] whenever p and q are mutually prime, QACC^0[2] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed.