Tree-size complexity 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 $n^{\hat{I}\text{c◯}(\log n)}=\mathrm{TS}=O(\sqrt{n}!)$ 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 $\mathrm{TS}=O(n^{{\log }_{2}2D})$.

7 pages, 2 figures