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.

One of the essential differences between quantum information processing (QIP) and the classical counterpart is that QIP sometimes involves global operations on unknown input states, namely, arbitrary superpositions of quantum states where their superposition coefficients are unknown. Quantum teleportation and quantum error corrections are typical examples. We call such unknown states. Quantum information cannot be measured (i.e., estimating the unknown coefficients by finite measurements) perfectly and cannot be copied perfectly either. In contrast, classical information in QIP can be encoded in a set of known orthogonal states and can be perfectly measured. Classical information can be also obtained by the result of measurements in QIP. QIP can be analyzed by investigating how input quantum information is transformed to output quantum information due to global operations. In this talk, I present our recent results from investigating the globalness of unitary operations on quantum information in terms of delocalizing power, entanglement cost, and implementation over a butterfly network.

The graph state formalism is an elegant and powerful formalism for quantum information processing. We focus on the application of graph states in MBQC and in particular on the characterisation of graphs that can be used to perform quantum information processing in this context. We introduce a simpler but equivalent combinatorial characterisation using focused gflow and we provide a simple condition of existence of such a flow as the existence of a right inverse to the adjacency matrix of the graph. The main contribution of this work is the weakening of the determinism condition in order to consider the more general class of information preserving evolutions.

A special feature of the ground state of a topologically ordered phase is large-scale correlations depending on the topology of the regions. These correlations can be detected by the topological entanglement entropy, or by a measure called the irreducible correlation. In this work, we show that the two measures coincide for states obeying an area law and having zero-correlation length. Moreover, we provide an operational meaning for these measures by proving its equivalence to the optimal rate of a particular class of secret sharing protocols.