Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

By Benoît Collins, Piotr Śniady. Communications in Mathematical Physics, 2004

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We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/61074782b2fc2c4f997665db26ca40ea74caa3b1

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Citations in the corpus

Asymptotics of unitary and orthogonal matrix integrals (2006, background): “It is interesting to see that a direct (yet difficult to describe) graphical reading of the Schwinger-Dyson equation, which is our main tool of investigation of unitarily invariant matrix models, yields non-crossing and could yield higher order moments related series and operations similar to convol”
Quantum variational algorithms are swamped with traps (2022, neutral): “Indeed, there have been promising preliminary numerical experiments on the performance of variational quantum algorithms in these regimes, but typically have relied on good initialization [18] or highly symmetric problem settings [54, 37, 38] to show convergence to a good approximation of the global”
Random Quantum Circuits (2022, neutral): “We will discuss these averages (87, 88) in Sec.”
Optimizing quantum process tomography with unitary 2-designs (2007, background): “1) can be integrated explicitly using group theoretical methods [31, 32].”
A Fourier view on the R-transform and related asymptotics of spherical integrals (2005, background): “2 in [7]) so that μ̂NAN+UBNU∗ converges towards μB ⊞ μA).”
The Random Matrix Theory of the Classical Compact Groups (2019, background): “A systematic approach called the Weingarten calculus for computing moments of all orders was developed by B. Collins [20] in the unitary case and extended to all the classical compact groups by Collins and Śniady [21].”
Free Probability and Random Matrices (2014, methodology): “However, we do have a replacement for the Wick formula in this context, which is known as the Weingarten convolution formula, see [57, 60].”
Emergent statistical mechanics of entanglement in random unitary circuits (2018, methodology): “The Weingarten function can be expanded perturbatively, and the leading order term for each permutation is [94]”
A semiclassical ramp in SYK and in gravity (2018, methodology): “Our discussion builds on the standard approach to Haar integrals over unitary matrices described in [66, 67, 68].”
Absence of Barren Plateaus in Quantum Convolutional Neural Networks (2020, methodology): “Speciﬁcally, following the elementwise formula of the Weingarten calculus [60, 61] we have that the following formulas are valid for the ﬁrst two moments: where w ij are the matrix elements of the unitary W ∈ U ( d ) , and Here we remark that the integrations in our calculations were performed by…”
Chaos, complexity, and random matrices (2017, methodology): “For the n-th moment, the Weingarten function is a function of an element σ of the permutation group Sn and presented as defined in [46],”
Entanglement Membrane in Chaotic Many-Body Systems (2019, neutral): “Unless q(2) = 0,±1,±2, · · · ,±(N − 1), the function Wg(τ−1σ), viewed as a matrix with τ and σ as its indices, is the inverse of the Gramian matrix 〈σ|τ〉 [45, 46]: ∑”
De Finetti theorems for easy quantum groups (2009, methodology): “We use a standard method from [11, 12], further developed in [2, 3, 13].”