Quantum Lower Bounds for Fanout

TL;DR

This paper establishes new lower bounds on the depth of quantum circuits needed to implement fanout and parity, showing that certain fanout operations require logarithmic depth under specific constraints, advancing understanding of quantum circuit complexity.

Contribution

It introduces the first non-trivial lower bounds for fanout in constant depth quantum circuits with limited ancillae, highlighting the depth requirements for quantum fanout operations.

Findings

Fanout requires log depth in constant depth quantum circuits with limited ancillae.

Tradeoff between number of ancillae and circuit depth for fanout.

Lower bounds are close to optimal under the given constraints.

Abstract

We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of a non-constant number $a$ of ancillae, we give a tradeoff between $a$ and the required depth, that results in a non-trivial lower bound for fanout when $a = n^{1-o(1)}$.