This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary oper...

A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie algebra u(n) associated with the Lie group U(n) of n× n unitary matrices. The e...

We construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering theory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrödinger operators on trees. We considerC∗-algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with r...

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space L(R). Our approach applies more generally to families of n arbitrary commuting unitary op...

let h be a finite dimensional complex hilbert space, b(h)+ be the set of all positive semi-definite operators on h and phi is a (not necessarily linear) unital map of b(h) + preserving the entropy-gibbs transformation. then there exists either a unitary or an anti-unitary operator u on h such that phi(a) = uau* for any b(h) +. thermodynamics, a branch of physics that is concerned with the study...

We provide, in an extremely simple way, an upper bound to the minimum number of unitary operators describing a general randomunitary channel.

We prove that any two unitary operators with simple singular spectrum which contains the whole circle are unitarily equivalent up to a rank one operator.

While the question “how many CNOT gates are needed to simulate an arbitrary twoqubit operator” has been conclusively answered – three are necessary and sufficient – previous work on this topic assumes that one wants to simulate a given unitary operator up to global phase. However, in many practical cases additional degrees of freedom are allowed. For example, if the computation is to be followe...

We describe Wasserstein isometries of the quantum bit state space with respect to distinguished cost operators. derive a Wigner-type result for operator involving all Pauli matrices: in this case, isometry group consists unitary or anti-unitary conjugations. In Bloch sphere model means that coincides classical symmetry O(3). On other hand, generated by qubit ‘‘clock” and ‘‘shift” operators, we ...