Bounds on the $\mathrm{QAC}^0$ Complexity of Approximating Parity

Abstract

$\mathrm{QAC}$ circuits are quantum circuits with one-qubit gates and Toffoli gates of arbitrary arity. $\mathrm{QAC}^0$ circuits are $\mathrm{QAC}$ circuits of constant depth, and are quantum analogues of $\mathrm{AC}^0$ circuits. We prove the following: $\bullet$ For all $d \ge 7$ and $\varepsilon>0$ there is a depth-$d$ $\mathrm{QAC}$ 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 $\mathrm{QAC}$ circuits of sublogarithmic depth could approximate parity regardless of size. $\bullet$ We introduce a class of "mostly classical" $\mathrm{QAC}$ 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 $\mathrm{QAC}$ circuits that approximate this component. $\bullet$ Arbitrary depth-$d$ $\mathrm{QAC}$ 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. $\bullet$ $\mathrm{QAC}$ 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 yōkai.