ASYMPTOTICS OF LARGE TRUNCATED HAAR UNITARY MATRICES

Author(s):  
J. RÉFFY
2019 ◽  
Vol 09 (02) ◽  
pp. 2050002
Author(s):  
Kartick Adhikari ◽  
Arup Bose

Let [Formula: see text], [Formula: see text], be [Formula: see text] probabilistically independent matrices of order [Formula: see text] (with [Formula: see text]) which are the left-uppermost blocks of [Formula: see text] Haar unitary matrices. Suppose that [Formula: see text] as [Formula: see text], with [Formula: see text]. Using free probability and Brown measure techniques, we find the limiting spectral distribution of [Formula: see text].


2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.


2002 ◽  
Vol 16 (30) ◽  
pp. 4593-4605 ◽  
Author(s):  
G. GIORGADZE

In this work, a gauge approach to quantum computing is considered. It is assumed that there exists a classical procedure for placing certain classical system in a state described by a holomorphic vector bundle with connection with logarithmic singularities. This bundle and its connection are constructed with the aid of unitary operators realizing the given algorithm using methods of the monodromic Riemann–Hilbert problem. Universality is understood in the sense that for any collection of unitary matrices there exists a connection with logarithmic singularities whose monodromy representation involves these matrices.


2017 ◽  
Vol 147 (6) ◽  
pp. 1279-1295
Author(s):  
Yicao Wang

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.


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