Geodesic flows on symmetric spaces of zero curvature and explicit solution of the generalized calogero model for the classical case

Ольшанецкий, Михаил Аронович; Perelomov, A. M.

doi:10.1007/bf01075536

Cited by 10 publications

(12 citation statements)

References 5 publications

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“…It is known [6] that the motion of this system is periodic with the same period T = π/ω. In two-particle case this reduces to the isochronicity of the potential (1.1).…”

Section: Copyright C 2005 By O a Chalykh And A P Veselovmentioning

confidence: 99%

A Remark on Rational Isochronous Potentials

Chalykh¹,

Веселов²

2005

JNMP

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“…It is known [6] that the motion of this system is periodic with the same period T = π/ω. In two-particle case this reduces to the isochronicity of the potential (1.1).…”

Section: Copyright C 2005 By O a Chalykh And A P Veselovmentioning

confidence: 99%

A Remark on Rational Isochronous Potentials

Chalykh¹,

Веселов²

2005

JNMP

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“…, α n as coordinates. The invariant symplectic form pulls back to ω This is the famous Hamiltonian function of the Calogero-Moser completely integrable system; see [167], [179], [103], and [192, 3.1 and 3.3]. The , [85] and [87].…”

Section: Poisson Manifoldsmentioning

confidence: 99%

Topics in Differential Geometry

Michor¹

2008

Graduate Studies in Mathematics

193

179

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“…Next, we study these dynamics and we show that they are governed by a Couloumb-like dynamics known as the classical Calogero-Moser model, a well-known Hamiltonian model in the field of integrable systems [36,52,53,54,55,56]. This model is ubiquitous across many disciplines [57] and captures the dynamics of poles of solutions of integrable differential equations.…”

Section: Single-mode Rank-preserving Evolutionsmentioning

confidence: 96%

“…These integrable models may be solved using, e.g., the Olshanetsky-Perelomov projection method [54], which consists in recovering the Calogero-Moser equations as the projection of simpler higher-dimensional equations. Then, the canonical variables q k are obtained as eigenvalues of an analytical matrix solution of these higher-dimensional equations of motion (see Appendix B for the case ω = 0 and [55] for the general case).…”

Section: The Calogero-moser Modelmentioning

confidence: 99%

Holomorphic representation of quantum computations

Chabaud¹,

Mehraban²

2021

Preprint

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We introduce a model of computing on holomorphic functions, which captures bosonic quantum computing through the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which not only gives a canonical definition of bosonic quantum computing using basic elements of complex analysis but also provides a unifying picture which delineates the boundary between discrete-and continuous-variable quantum information theory. Using this representation, we show that the evolution of a single bosonic mode under a Gaussian Hamiltonian can be described as an integrable dynamical system of classical Calogero-Moser particles corresponding to the zeros of the holomorphic function, together with a conformal evolution of Gaussian parameters. We explain that the Calogero-Moser dynamics is due to unique features of bosonic Hilbert spaces such as squeezing. We then generalize the properties of this holomorphic representation to the multimode case, deriving a non-Gaussian hierarchy of quantum states and relating entanglement to factorization properties of holomorphic functions. Finally, we apply this formalism to discrete-and continuous-variable quantum measurements and introduce formally the model of Holomorphic Quantum Computing. Non-adaptive circuits in this model allow for a classification of subuniversal models that are generalisations of Boson Sampling and Gaussian quantum computing, while adaptive circuits capture existing universal bosonic quantum computing models. Our results leave open the possibility that continuous-variable quantum computations may outperform their discrete-variable counterpart.

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Geodesic flows on symmetric spaces of zero curvature and explicit solution of the generalized calogero model for the classical case

Cited by 10 publications

References 5 publications

A Remark on Rational Isochronous Potentials

A Remark on Rational Isochronous Potentials

Topics in Differential Geometry

Holomorphic representation of quantum computations

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