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

Mateus de Oliveira Oliveira

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arXiv · 1609.09643

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

Mateus de Oliveira Oliveira

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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.

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Mateus de Oliveira Oliveira. 2016-09-30. A Near-Quadratic Lower Bound for the Size of Quantum Circuits of Constant Treewidth. https://arxiv.org/abs/1609.09643

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