Uniqueness of imaginarity-assisted exact transformation from real orthogonal operations to arbitrary unitary operations

Abstract:The computational universality with an elementary gate set $\{H,CCZ\}$ can be transformed to the strict universality by using a maximally imaginary state $|+i \rangle$ and some non-imaginary ancillary qubits. From the viewpoint of operational resource theory, it would be intriguing to elucidate a resource for the universality transformation. In this paper, we consider the exact universality transformation in which arbitrary real orthogonal matrices can be applied and a supplied resource state is used to simulate unitary operations exactly and deterministically. Within this operational model, we explore a necessary and sufficient condition for resource states to realize the universality transformation under free real operations. We show that $|+i \rangle$ is a unique resource state up to the free operations. Moreover, we obtain a stronger conclusion. If a given resource state cannot be used for the universality transformation, then realizable quantum gates are restricted to unitary matrices proportional to real orthogonal matrices. Therefore, we can tell that $|+i \rangle$ is unique (up to the free operations) not only as a state whose resource measure of imaginarity is maximal, but also as a state which empowers real operations with the ability to apply at least one non-real quantum gate (regardless of the magnitudes of its imaginary parts).

8 pages, 1 figure. Revised and retitled in response to referee comments

arXiv:2603.11812 [quant-ph]

(or arXiv:2603.11812v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2603.11812 Focus to learn more arXiv-issued DOI via DataCite

Sci. Rep. (2026)

https://doi.org/10.1038/s41598-026-70782-1 Focus to learn more DOI(s) linking to related resources

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