Tree-size complexity of multiqubit states

TL;DR

This paper investigates the tree size complexity of multiqubit quantum states, providing explicit characterizations of states with superpolynomial complexity and analyzing the complexity of matrix-product states.

Contribution

It improves upon previous results by explicitly characterizing states with superpolynomial complexity and analyzing the complexity of matrix-product states.

Findings

States with superpolynomial complexity have explicit characterizations.

Matrix-product states with tensors of dimension D have polynomial complexity.

Complexity bounds are established for different classes of multiqubit states.

Abstract

Complexity is often invoked alongside size and mass as a characteristic of macroscopic quantum objects. In 2004, Aaronson introduced the \textit{tree size} (TS) as a computable measure of complexity and studied its basic properties. In this paper, we improve and expand on those initial results. In particular, we give explicit characterizations of a family of states with superpolynomial complexity $n^{\Omega(\log n)}= \mathrm{TS} =O(\sqrt{n}!)$ in the number of qubits $n$; and we show that any matrix-product state whose tensors are of dimension $D\times D$ has polynomial complexity $\mathrm{TS}=O(n^{\log_2 2D})$.