scholarly journals Eigenvalue rigidity for truncations of random unitary matrices

2020 ◽  
Vol 10 (01) ◽  
pp. 2150015
Author(s):  
Elizabeth Meckes ◽  
Kathryn Stewart
Keyword(s):  
Haar Measure ◽  
Spectral Measure ◽  
Unit Disc ◽  
Counting Function ◽  
Concentration Inequalities ◽  
Unitary Matrix ◽  
Unitary Matrices ◽  
Microscopic Scale ◽  
Principal Submatrix ◽  
Random Unitary Matrices

We consider the empirical eigenvalue distribution of an [Formula: see text] principal submatrix of an [Formula: see text] random unitary matrix distributed according to Haar measure. For [Formula: see text] and [Formula: see text] large with [Formula: see text], the empirical spectral measure is well approximated by a deterministic measure [Formula: see text] supported on the unit disc. In earlier work, we showed that for fixed [Formula: see text] and [Formula: see text], the bounded-Lipschitz distance [Formula: see text] between the empirical spectral measure and the corresponding [Formula: see text] is typically of order [Formula: see text] or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.

scholarly journals Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices

2015 ◽  
Vol 04 (03) ◽  
pp. 1550011 ◽  
Author(s):  
O. Marchal
Keyword(s):  
Return Time ◽  
Unitary Matrix ◽  
Unitary Matrices ◽  
Toeplitz Determinants ◽  
Recurrence Times ◽  
Toeplitz Determinant ◽  
Large N ◽  
Single Arc ◽  
Random Unitary Matrices ◽  
First Return Time

The purpose of this paper is to study the eigenvalues [Formula: see text] of Ut where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

scholarly journals Truncations of Random Unitary Matrices and Young Tableaux

10.37236/939 ◽  
2006 ◽  
Vol 14 (1) ◽  
Author(s):  
J. Novak
Keyword(s):  
Unitary Group ◽  
Lattice Gauge Theory ◽  
Main Tool ◽  
Unitary Matrix ◽  
Young Tableaux ◽  
Unitary Matrices ◽  
Average Value ◽  
Combinatorial Derivation ◽  
Single Entry ◽  
Random Unitary Matrices

Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ For any $k \leq d,$ and any $k \times k$ submatrix $U_k$ of $U,$ we express the average value of $|{\rm Tr}(U_k)|^{2n}$ as a sum over partitions of $n$ with at most $k$ rows whose terms count certain standard and semistandard Young tableaux. We combine our formula with a variant of the Colour-Flavour Transformation of lattice gauge theory to give a combinatorial expansion of an interesting family of unitary matrix integrals. In addition, we give a simple combinatorial derivation of the moments of a single entry of a random unitary matrix, and hence deduce that the rescaled entries converge in moments to standard complex Gaussians. Our main tool is the Weingarten function for the unitary group.

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

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.

scholarly journals Mixed moments of characteristic polynomials of random unitary matrices

2019 ◽  
Vol 60 (8) ◽  
pp. 083509 ◽  
Author(s):  
E. C. Bailey ◽  
S. Bettin ◽  
G. Blower ◽  
J. B. Conrey ◽  
A. Prokhorov ◽  
...  
Keyword(s):  
Characteristic Polynomials ◽  
Unitary Matrices ◽  
Random Unitary Matrices

scholarly journals On the distribution of the zeros of the derivative of the Riemann zeta-function

2014 ◽  
Vol 157 (3) ◽  
pp. 425-442 ◽  
Author(s):  
STEPHEN LESTER
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Horizontal Distribution ◽  
Characteristic Polynomials ◽  
Unitary Matrices ◽  
Number Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Random Unitary Matrices

AbstractWe establish an asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For ℜ(s) = σ satisfying (log T)−1/3+ε ⩽ (2σ − 1) ⩽ (log log T)−2, we show that the number of zeros of ζ′(s) with imaginary part between zero and T and real part larger than σ is asymptotic to T/(2π(σ−1/2)) as T → ∞. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for σ in this range the zeros of ζ′(s) are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

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