Yuki Takeuchi

There exist two types of universality in quantum computation: strict and computational universalities. The former is known to be stronger than the latter. In this presentation, we give a method of transforming from the computational universality with an elementary gate set {H,CCZ} to the strict universality by using a maximally imaginary state |+i>, which is an eigenstate of the Pauli-Y operator. From the viewpoint of resource theory, it would be intriguing whether the maximum imaginarity is necessary for the universality transformation. We show that |+i> is a unique resource state up to free operations. More precisely, we obtain the stronger conclusion that if a given resource state cannot be used for the universality transformation, then the realizable quantum gates are restricted to only orthogonal operators. This implies that by rotating a single-qubit state from the eigenstate |+> of the Pauli-X operator to |+i>, we can observe a kind of phase transition. The first half of our results has been published in PRL 133, 050601 (2024).

Phase estimation serves as a unifying framework for precision measurements in physics. While protocols using Greenberger-Horne-Zeilinger (GHZ) states can achieve the Heisenberg limit, they require prior knowledge of the Hamiltonian to align the probe's sensitivity axis. This necessity for precise alignment imposes a burden on the state preparation stage, limiting practical applications where the field direction is unknown or fluctuating. In this work, we address this limitation by identifying the symmetric subspace as a source of universal metrological resources. We introduce the concept of universal resource states (URS), a class of probe states capable of achieving Heisenberg-limited precision for any linear collective Hamiltonian, independent of its orientation. Our contribution is the explicit analytical derivation of these states and proving that almost all symmetric states serve as URSs for all linear collective Hamiltonians simultaneously. This reveals the symmetric subspace as an inherent reservoir of universal metrological resources: a single, generic symmetric state suffices for high-precision metrology, enabling state preparation completely decoupled from the Hamiltonian's orientation.