Bounds on the QAC$^0$ Complexity of Approximating Parity

TL;DR

This paper establishes bounds on the quantum circuit complexity needed to approximate the parity function, revealing depth and size limitations for QAC$^0$ circuits and introducing new circuit normal forms and state constructions.

Contribution

It provides the first bounds on sublogarithmic depth QAC circuits approximating parity, introduces a class of 'mostly classical' QAC circuits with tight lower bounds, and develops a new normal form for quantum circuits.

Findings

Depth-$d$ QAC circuits can approximate parity with size exponential in poly(n^{1/d})

Low-depth, mostly classical QAC circuits have tight size lower bounds for approximating certain components

Arbitrary depth-$d$ QAC circuits need at least Omega(n/d) multi-qubit gates to approximate parity

Abstract

QAC circuits are quantum circuits with one-qubit gates and Toffoli gates of arbitrary arity. QAC$^0$ circuits are QAC circuits of constant depth, and are quantum analogues of AC$^0$ circuits. We prove the following: $\bullet$ For all $d \ge 7$ and $\varepsilon>0$ there is a depth-$d$ QAC circuit of size $\exp(\mathrm{poly}(n^{1/d}) \log(n/\varepsilon))$ that approximates the $n$-qubit parity function to within error $\varepsilon$ on worst-case quantum inputs. Previously it was unknown whether QAC circuits of sublogarithmic depth could approximate parity regardless of size. $\bullet$ We introduce a class of "mostly classical" QAC circuits, including a major component of our circuit from the above upper bound, and prove a tight lower bound on the size of low-depth, mostly classical QAC circuits that approximate this component. $\bullet$ Arbitrary depth-$d$ QAC circuits require at least $\Omega(n/d)$ multi-qubit gates to achieve a $1/2 + \exp(-o(n/d))$ approximation of parity. When $d = \Theta(\log n)$ this nearly matches an easy $O(n)$ size upper bound for computing parity exactly. $\bullet$ QAC circuits with at most two layers of multi-qubit gates cannot achieve a $1/2 + \exp(-o(n))$ approximation of parity, even non-cleanly. Previously it was known only that such circuits could not cleanly compute parity exactly for sufficiently large $n$. The proofs use a new normal form for quantum circuits which may be of independent interest, and are based on reductions to the problem of constructing certain generalizations of the cat state which we name "nekomata" after an analogous cat y\=okai.