Complexity and Applications of Nearest Stabilizer Product State Problems

Abstract:Consider the following optimization problem over stabilizer product states: given an $n$-qubit stabilizer state $\left|\psi\right\rangle$ and a set of single-qubit stabilizer states $S$, maximize $\left|\langle \psi | \phi_1, \ldots, \phi_n \rangle\right|^2$ over single-qubit stabilizer states $\left|\phi_i\right\rangle \in S$. By varying the set $S$, we show that solutions to this problem can be useful in a variety of settings: tighter runtime bounds for certain classical simulation algorithms; measures of entanglement; and the complexity of low-rank matrix completion. Moreover, we give a complete complexity classification of this nearest stabilizer product state problem. After accounting for the symmetries in the Clifford group, there are $9$ distinct possible sets $S$, and we show that all but the two simplest of these are $\NP$-complete.

arXiv:2610.02037 [quant-ph]

(or arXiv:2610.02037v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.02037 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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