How to generate random matrices from the classical compact groups

By Francesco Mezzadri. 2006

599 citations recorded by the source corpus.

We discuss how to generate random unitary matrices from the classical compact groups U(N), O(N) and USp(N) with probability distributions given by the respective invariant measures. The algorithm is straightforward to implement using standard linear algebra packages. This approach extends to the Dyson circular ensembles too. This article is based on a lecture given by the author at the summer school on Number Theory and Random Matrix Theory held at the University of Rochester in June 2006. The exposition is addressed to a general mathematical audience.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

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Citations in the corpus

Superposition of many models into one (2019, background): “Let w ∈ R M be a vector and the context vector C k be a random orthogonal matrix drawn from the O ( M ) Haar distribution (Mezzadri, 2006).”
Nonlinear Propagation in Multimode and Multicore Fibers: Generalization of the Manakov Equations (2012, unknown): “matrices [27]: pick a matrix whose all elements are normally distributed (with zero mean) and apply a QR decomposition, , such that is an upper triangular matrix with positive diagonal entries.”
Variable selection using pseudo-variables (2018, methodology): “Subsequently, V2 is a random orthonormal matrix distributed with Haar measure (Mezzadri, 2006).”
Randomized numerical linear algebra: Foundations and algorithms (2020, methodology): “To achieve this guarantee, use the algorithms from Mezzadri (2007).)”
Unifying Orthogonal Monte Carlo Methods (2019, methodology): “Several methods are known for solving this problem exactly (Genz, 1998; Mezzadri, 2007”
Multivariate quantile mapping bias correction: an N-dimensional probability density function transform for climate model simulations of multiple variables (2017, methodology): “In step (a) of the jth iteration of the N-pdft algorithm, denoted by the superscript [j], construct an N × N uniformly distributed random orthogonal rotation matrix , for example via QR decomposition of normally distributed random values (Mezzadri 2007).”
Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, methodology): “The Haar random unitaries are generated using Mezzadri’s algorithm [140].”
Sampling From A Manifold (2012, background): “Mezzadri (2007); Diaconis and Shahshahani (1986) develop this.”
Geometry of Quantum States: Index (2007, background): “The articles by Diaconis [260] and by Mezzadri [657] provide entry points.”
Realization of a programmable two-qubit quantum processor (2009, methodology): “To demonstrate the ability of the processor to generate arbitrary unitary transformations, we program it with 160 different randomly chosen operations distributed in SU(4) according to the Haar measure[23].”
The Random Matrix Theory of the Classical Compact Groups (2019, methodology): “A survey of methods of generating random matrices from a more computational viewpoint is given in [83].”
Measurement-driven entanglement transition in hybrid quantum circuits (2019, methodology): “1, where each rectangle now represents a two qubit gate which is a 4× 4 matrix chosen randomly and independently from the Haar measure of the unitary group [15, 16, 51].”