Partial Unitarity Research Articles

Partially unitary learning.

The problem of an optimal mapping between Hilbert spaces IN of |ψ〉 and OUT of |ϕ〉 based on a set of wavefunction measurements (within a phase) ψ_{l}→ϕ_{l},l=1,⋯,M, is formulated as an optimization problem maximizing the total fidelity ∑_{l=1}^{M}ω^{(l)}|〈ϕ_{l}|U|ψ_{l}〉|^{2} subject to probability preservation constraints on U (partial unitarity). The constructed operator U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension dim(OUT)×dim(IN) transforming operators as A^{OUT}=UA^{IN}U^{†}. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.

Polyadic Braid Operators and Higher Braiding Gates

A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.