Mio Murao

Symmetry plays a crucial role in the design and analysis of quantum protocols. This result shows a canonical circuit decomposition of a quantum comb with $G\times H$ symmetry for compact groups $G$ and $H$ using the corresponding Clebsch--Gordan transforms. By using this circuit decomposition, we propose a parametrized quantum comb with group symmetry, and derive the optimal quantum comb which transforms an unknown unitary operation $U\in \SU(d)$ to its inverse $U^\dagger$ or transpose $U^\mathsf{T}$. From numerics, we find a deterministic and exact unitary transposition protocol for $d=3$ with $7$ queries to $U$, which is improved over the protocol shown in [Y.-A. Chen et al., arXiv:2403.04704], which requires $13$ queries to $U$. We also provide the simulation of random unitaries for any compact group $G$ using the compressed oracle, which can be implemented efficiently for the unitary group. The precision of our simulation for the unitary group is improved over the path-recording oracle introduced in [F. Ma and H.-Y. Huang, arXiv:2410.10116].

In this work, we report a deterministic and exact protocol to reverse any unknown qubit-unitary and qubit-encoding isometry operations. We present the semidefinite programming (SDP) to search the Choi matrix representing a quantum circuit reversing any unitary operation. We derive a quantum circuit transforming four calls of any qubit-unitary operation into its inverse operation by imposing the SU(2)×SU(2) symmetry on the Choi matrix. This protocol only applies only for qubit-unitary operations, but we extend this protocol to any qubit-encoding isometry operations. For that, we derive a subroutine to transform a unitary inversion protocol to an isometry inversion protocol by constructing a quantum circuit transforming finite sequential calls of any isometry operation into random unitary operations.