A Near-Quadratic Lower Bound for the Size of Quantum Circuits of Constant Treewidth

TL;DR

This paper establishes a near-quadratic lower bound on the size of quantum circuits with constant treewidth for computing the element distinctness function, extending previous results and employing advanced theoretical techniques.

Contribution

It introduces a novel lower bound for quantum circuit size based on treewidth, generalizing prior formula size bounds using an extended Neciporuk's method.

Findings

Quantum circuits of constant treewidth require near-quadratic size for element distinctness

Extension of Neciporuk's method to quantum circuits with structural graph theory techniques

Combines tensor-network theory and algebraic geometry tools for lower bound proofs

Abstract

We show that any quantum circuit of treewidth $t$, built from $r$-qubit gates, requires at least $\Omega(\frac{n^{2}}{2^{O(r\cdot t)}\cdot \log^4 n})$ gates to compute the element distinctness function. Our result generalizes a near-quadratic lower bound for quantum formula size obtained by Roychowdhury and Vatan [SIAM J. on Computing, 2001]. The proof of our lower bound follows by an extension of Ne\v{c}iporuk's method to the context of quantum circuits of constant treewidth. This extension is made via a combination of techniques from structural graph theory, tensor-network theory, and the connected-component counting method, which is a classic tool in algebraic geometry.