Invariant integration over the unitary group

Invariant Integration over the Unitary Group

Integrals for the product of unitary-matrix elements over the U (n) group will be discussed. A group-theoretical formula is available to convert them into a multiple sum, but unfortunately the sums are often tedious to compute. In this paper, we develop an alternative method in which these sums are avoided, and group theory is rendered unnecessary. Only unitarity and the invariance of the Haar ...

Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems

A diagrammatic method is presented for averaging over the circular ensemble of random-matrix theory. The method is applied to phase-coherent conduction through a chaotic cavity (a “quantum dot”) and through the interface between a normal metal and a superconductor.

Unitary-Group Invariant Kernels and Features from Transformed Unlabeled Data

The study of representations invariant to common transformations of the data is important to learning. Most techniques have focused on local approximate invariance implemented within expensive optimization frameworks lacking explicit theoretical guarantees. In this paper, we study kernels that are invariant to the unitary group while having theoretical guarantees in addressing practical issues ...

Integration over the Pauli Quantum Group

The notion of free quantum group appeared in Wang’s papers [17], [18]. The idea is that given a compact group G ⊂ Un, the matrix coordinates uij ∈ C(G) commute with each other, and satisfy certain relations R. One can define then the universal algebra A generated by abstract variables uij, subject to the relations R. In certain situations we get a Hopf algebra in the sense of Woronowicz [20], a...

Group Invariant Integration and the Fundamental Theorem of Algebra.