Complexity of graph-state preparation by Clifford circuits

• URL: http://arxiv.org/abs/2402.05874v1
• Date: Thu, 8 Feb 2024 18:08:09 GMT
• Title: Complexity of graph-state preparation by Clifford circuits
• Authors: Soh Kumabe, Ryuhei Mori, Yusei Yoshimura
• Abstract summary: We show a connection between the CZ-complexity of graph state $|Grangle$ and the rank-width of the graph $G$. We present quantum algorithms preparing $|Grangle$ with $O(n)$ CZ-complexity when $G$ is included in special classes of graphs.
• Score: 2.7010154811483167
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we study a complexity of graph-state preparation. We consider general quantum algorithms consisting of the Clifford operations on at most two qubits for graph-state preparations. We define the CZ-complexity of graph state $|G\rangle$ as the minimum number of two-qubit Clifford operations (excluding single-qubit Clifford operations) for generating $|G\rangle$ from a trivial state $|0\rangle^{\otimes n}$. We first prove that a graph state $|G\rangle$ is generated by at most $t$ two-qubit Clifford operations if and only if $|G\rangle$ is generated by at most $t$ controlled-Z (CZ) operations. We next prove that a graph state $|G\rangle$ is generated from another graph state $|H\rangle$ by $t$ CZ operations if and only if the graph $G$ is generated from $H$ by some combinatorial graph transformation with cost $t$. As the main results, we show a connection between the CZ-complexity of graph state $|G\rangle$ and the rank-width of the graph $G$. Indeed, we prove that for any graph $G$ with $n$ vertices and rank-width $r$, 1. The CZ-complexity of $|G\rangle$ is $O(rn\log n)$. 2. If $G$ is connected, the CZ-complexity of $|G\rangle$ is at least $n + r - 2$. We also show the existence of graph states whose CZ-complexities are close to the upper and lower bounds. Finally, we present quantum algorithms preparing $|G\rangle$ with $O(n)$ CZ-complexity when $G$ is included in special classes of graphs, namely, cographs, interval graphs, permutation graphs and circle graphs.

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