Weingarten calculus via orthogonality
relations: new applications

Collins, Benoı̂t; Matsumoto, Sho

doi:10.30757/alea.v14-31

Cited by 25 publications

(18 citation statements)

References 21 publications

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“…In the second step we have used trðA s σÞ ≤ 1. Carrying the sum over s and using known properties of the Weingarten function 47 , for k < 2 ' = ffiffi ffi 6 p À Á 4=7 we obtain…”

mentioning

confidence: 99%

Quantum algorithmic measurement

Aharonov

Cotler

2022

Nat Commun

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There has been recent promising experimental and theoretical evidence that quantum computational tools might enhance the precision and efficiency of physical experiments. However, a systematic treatment and comprehensive framework are missing. Here we initiate the systematic study of experimental quantum physics from the perspective of computational complexity. To this end, we define the framework of quantum algorithmic measurements (QUALMs), a hybrid of black box quantum algorithms and interactive protocols. We use the QUALM framework to study two important experimental problems in quantum many-body physics: determining whether a system’s Hamiltonian is time-independent or time-dependent, and determining the symmetry class of the dynamics of the system. We study abstractions of these problems and show for both cases that if the experimentalist can use her experimental samples coherently (in both space and time), a provable exponential speedup is achieved compared to the standard situation in which each experimental sample is accessed separately. Our work suggests that quantum computers can provide a new type of exponential advantage: exponential savings in resources in quantum experiments.

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“…In the second step we have used trðA s σÞ ≤ 1. Carrying the sum over s and using known properties of the Weingarten function 47 , for k < 2 ' = ffiffi ffi 6 p À Á 4=7 we obtain…”

mentioning

confidence: 99%

Quantum algorithmic measurement

Aharonov

Cotler

2022

Nat Commun

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“…and for U(N ) gauge theory q = 0. Invariant integration over compact groups have been studied extensively in the last decades [48,49,[52][53][54][55][56][57][58][59][60][61][62][63]. Although many results concerning the U(N ) group are known since many years, only recently the SU(N ) generalization has been found [50,51].…”

Section: Strong Coupling Expansion and Link Integrationmentioning

confidence: 99%

New dual representation for staggered lattice QCD

Gagliardi

Unger

2020

Phys. Rev. D

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We propose a new strategy to evaluate the partition function of lattice QCD with Wilson gauge action coupled to staggered fermions, based on a strong coupling expansion in the inverse bare gauge coupling β = 2N/g 2 . Our method makes use of the recently developed formalism to evaluate the SU(N ) 1−link integrals and consists in an exact rewriting of the partition function in terms of a set of additional dual degrees of freedom which we call "Decoupling Operator Indices" (DOI). The method is not limited to any particular number of dimensions or gauge group U(N ), SU(N ). In terms of the DOI the system takes the form of a Tensor Network which can be simulated using Wormlike algorithms. Higher order β-corrections to strong coupling lattice QCD can be, in principle, systematically evaluated, helping to answer the question whether the finite density sign problem remains mild when plaquette contributions are included. Issues related to the complexity of the description and strategies for the stochastic evaluation of the partition function are discussed.

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“…A beautiful interpretation of asymptotic coefficients of the Weingarten function as the number of monotone factorizations by Matsumoto and Novak [MN13] allowed one of the present authors with Kuipers [BK13a] to put the use of random matrix theory in quantum chaotic transport on a more solid mathematical basis. Notation in the present section is kept in line with the mathematical sources such as [Mat13,CM17].…”

Section: Appendix a Weingarten Calculusmentioning

confidence: 99%

“…5.4]. Uniform bound in terms of k obtained in [CM17] could be crucial to considering the simultaneous limit k, N → ∞ and accessing the spectral form factor.…”

Section: From Invariance Properties Of Coe One Deduces That Wg Coementioning

confidence: 99%

Convergence of moments of twisted COE matrices

Berkolaiko,

Booton

2020

Preprint

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We investigate eigenvalue moments of matrices from Circular Orthogonal Ensemble multiplicatively perturbed by a permutation matrix. More precisely we investigate variance of the sum of the eigenvalues raised to power k, for arbitrary but fixed k and in the limit of large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the Circular Unitary Ensemble with a particularly fast rate: the error is of order 1/N 3 and the terms of orders 1/N and 1/N 2 disappear due to cancellations. We prove this rate of convergence using Weingarten calculus and classifying the contributing Weingarten functions first in terms of a graph model and then algebraically.Date: June 12, 2020. 1 More precisely, the probability measure we consider is supported on a high codimension submanifold of U (N ), the compact manifold which is the support of the uniform probability measure defining the CUE.

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Weingarten calculus via orthogonality relations: new applications

Cited by 25 publications

References 21 publications

Quantum algorithmic measurement

Quantum algorithmic measurement

New dual representation for staggered lattice QCD

Convergence of moments of twisted COE matrices

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