Preparing graph states forbidding a vertex-minor

• URL: http://arxiv.org/abs/2504.00291v1
• Date: Mon, 31 Mar 2025 23:25:35 GMT
• Title: Preparing graph states forbidding a vertex-minor
• Authors: James Davies, Andrew Jena,
• Abstract summary: Measurement based quantum computing is preformed by adding non-Clifford measurements to a prepared stabilizer states.<n>Every stabilizer state is local-Clifford equivalent to a graph state, so we may focus on graph states $leftvert G rightrangle$.<n>We obtain significantly improved bounds when $G$ is contained within certain proper classes of graphs.
• Score: 1.864621482724548
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Measurement based quantum computing is preformed by adding non-Clifford measurements to a prepared stabilizer states. Entangling gates like CZ are likely to have lower fidelities due to the nature of interacting qubits, so when preparing a stabilizer state, we wish to minimize the number of required entangling states. This naturally introduces the notion of CZ-distance. Every stabilizer state is local-Clifford equivalent to a graph state, so we may focus on graph states $\left\vert G \right\rangle$. As a lower bound for general graphs, there exist $n$-vertex graphs $G$ such that the CZ-distance of $\left\vert G \right\rangle$ is $\Omega(n^2 / \log n)$. We obtain significantly improved bounds when $G$ is contained within certain proper classes of graphs. For instance, we prove that if $G$ is a $n$-vertex circle graph with clique number $\omega$, then $\left\vert G \right\rangle$ has CZ-distance at most $4n \log \omega + 7n$. We prove that if $G$ is an $n$-vertex graph of rank-width at most $k$, then $\left\vert G \right\rangle$ has CZ-distance at most $(2^{2^{k+1}} + 1) n$. More generally, this is obtained via a bound of $(k+2)n$ that we prove for graphs of twin-width at most $k$. We also study how bounded-rank perturbations and low-rank cuts affect the CZ-distance. As a consequence, we prove that Geelen's Weak Structural Conjecture for vertex-minors implies that if $G$ is an $n$-vertex graph contained in some fixed proper vertex-minor-closed class of graphs, then $\left\vert G \right\rangle$ has CZ-distance at most $O(n\log n)$. Since graph states of locally equivalent graphs are local Clifford equivalent, proper vertex-minor-closed classes of graphs are natural and very general in this setting.

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