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Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers [article]

Yen Ting Lin, Robert B. Lowrie, Denis Aslangil, Yiğit Subaşı, Andrew T. Sornborger
2022 arXiv   pre-print
Linear representations, such as the Koopman representation and Koopman von Neumann mechanics, have regained attention from the dynamical-systems research community.  ...  A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function  ...  This work was supported by the Beyond Moore's Law project of the Advanced Simulation and Computing Program at Los Alamos National Laboratory managed by Triad National Security, LLC, for the National Nuclear  ... 
arXiv:2202.02188v2 fatcat:4fm7bp6hfbe5lkxeumqa42wipu

Koopman wavefunctions and Clebsch variables in Vlasov-Maxwell kinetic theory [article]

Cesare Tronci, Ilon Joseph
2021 arXiv   pre-print
The celebrated Koopman-von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is  ...  Furthermore, the canonical structure is restored for a variant of the Koopman-von Neumann construction that carries a different phase dynamics.  ...  Both approaches can be used to develop a "quantum" representation of the classical Liouville equation and both can be simulated on a quantum computer.  ... 
arXiv:2105.00294v2 fatcat:hs7lrq6w7bdyvoxom5qtgyzxzm

Momentum maps for mixed states in quantum and classical mechanics [article]

Cesare Tronci
2018 arXiv   pre-print
Uhlmann's density matrix and Koopman-von Neumann wavefunctions are shown to be special cases of this construction.  ...  This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs.  ...  I also thank him for his suggestion to work in this direction, which led to considering quantum dynamics from a completely new perspective.  ... 
arXiv:1810.01332v1 fatcat:ufgtscsgcvfnvgjkkabrirreai

Momentum maps for mixed states in quantum and classical mechanics

Cesare Tronci, ,Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK, Mathematical Sciences Research Institute, Berkeley, CA 94720, USA
2019 Journal of Geometric Mechanics (JGM)  
Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical  ...  Both quantum and classical mechanics are shown to possess analogous momentum map pairs associated to left and right group actions.  ...  I also thank him for his suggestion to work in this direction, which led to considering quantum dynamics from a completely new perspective.  ... 
doi:10.3934/jgm.2019032 fatcat:qeymfwruy5c7net77qp757skiu

On deformations of classical mechanics due to Planck-scale physics [article]

O.I. Chashchina, A. Sen, Z.K. Silagadze
2019 arXiv   pre-print
Koopman-Von Neumann's Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system.  ...  This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale.  ...  Wilczek, Notes on Koopman von Neumann Mechanics, and A Step Beyond, http://docplayer.net/53985308-Notes-on-koopman-von-neumann-mechanics-and-a-step-beyond.html [84] U.  ... 
arXiv:1902.09728v2 fatcat:ygvk2ey2nzdqri5siim4sf5w3m

Modern Koopman Theory for Dynamical Systems [article]

Steven L. Brunton, Marko Budišić, Eurika Kaiser, J. Nathan Kutz
2021 arXiv   pre-print
Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of  ...  This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems  ...  We also thank Shervin Bagheri, Bing Brunton, Bethany Lusch, Ryan Mohr, Frank Noe, Josh Proctor, Clancy Rowley, and Peter Schmid for many fruitful discussions on DMD, Koopman theory, and control.  ... 
arXiv:2102.12086v2 fatcat:2oylyx25dbctvkjfnirfcgjuqu

Koopman wavefunctions and classical-quantum correlation dynamics [article]

Denys I. Bondar, François Gay-Balmaz, Cesare Tronci
2019 arXiv   pre-print
Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum  ...  The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.  ...  In Geometric Mechanics [43, 33] , momentum  ... 
arXiv:1802.04787v3 fatcat:25wloszoyzfz3k7etui5km4ipq

Koopman wavefunctions and classical–quantum correlation dynamics

Denys I. Bondar, François Gay-Balmaz, Cesare Tronci
2019 Proceedings of the Royal Society A  
Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum  ...  The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect-the quantum backreaction.  ...  After several computations, one obtains that the sum of (B 1)-(B 4) gives Hence the second equation in (3.10) follows if the following identity holds: for all H and Υ .  ... 
doi:10.1098/rspa.2018.0879 pmid:31611710 pmcid:PMC6784396 fatcat:k3qyeoweovfjfo7t76t55xeb7e

Probability representation of quantum mechanics and star product quantization [article]

S. N. Belolipetskiy, V. N. Chernega, O. V. Man'ko, V. I. Man'ko
2019 arXiv   pre-print
Examples of the Wigner-Weyl symbols (like Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified with the states of the quantum  ...  systems are discussed.  ...  In [66] the classical limit of the Wigner function was considered and was discussed its transform into a classical Koopman-von Neumann wave function rather than into a classical probability distribution  ... 
arXiv:1903.07932v1 fatcat:j7zcgzjrynhftey3mso6yzbkze

Koopman wavefunctions and classical states in hybrid quantum-classical dynamics [article]

François Gay-Balmaz, Cesare Tronci
2022 arXiv   pre-print
We deal with the reversible dynamics of coupled quantum and classical systems.  ...  Based on a recent proposal by the authors, we exploit the theory of hybrid quantum-classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and  ...  Also, we would like to thank Darryl Holm for his valuable feedback on these and related topics. This work was made possible through the support of Grant 62210 from the John Templeton Foundation.  ... 
arXiv:2108.01482v2 fatcat:pjp37ic5eveyfm5qeeutrbpwcy

Embedding classical dynamics in a quantum computer [article]

Dimitrios Giannakis, Abbas Ourmazd, Philipp Pfeffer, Joerg Schumacher, Joanna Slawinska
2021 arXiv   pre-print
We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer.  ...  To achieve an exponential quantum computational advantage, we project the state of the quantum system to a density matrix on a 2^n-dimensional tensor product Hilbert space associated with n qubits.  ...  We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.  ... 
arXiv:2012.06097v3 fatcat:6qcbjik62randgacgckybvu2pi

Duality and conditional expectations in the Nakajima-Mori-Zwanzig formulation

Jason M. Dominy, Daniele Venturi
2017 Journal of Mathematical Physics  
We show that a duality principle between the NMZ formulation in the space of observables and in the state space can be established, analogous to the Heisenberg and Schr\"odinger pictures in quantum mechanics  ...  The new theory is built upon rigorous mathematical foundations, and it can be applied to both classical and quantum systems.  ...  ACKNOWLEDGMENTS This work was supported by the Air Force Office of Scientific Research, grant number FA9550-16-1-0092.  ... 
doi:10.1063/1.4997015 fatcat:neqnyepijvhi5brupsaadxd5nu

John von Neumann, the Mathematician

Domokos Szász
2011 The Mathematical intelligencer  
My goal here is to concentrate on his development and achievements as a mathematician and the evo-  ...  Imagine a poll to choose the best-known mathematician of the twentieth century. No doubt the winner would be John von Neumann.  ...  Special thanks are due to Re ´ka Sza ´sz for polishing my English and to J. Fritz, I. Juha ´sz, D. Petz, L. Ro ´nyai, A. Simonovits, and A. Szenes for stimulating discussions.  ... 
doi:10.1007/s00283-011-9223-6 fatcat:kcgk2j6hhrhwpnyvroq6jog7si

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Benjamin Zanger, Christian B. Mendl, Martin Schulz, Martin Schreiber
2021 Quantum  
method on a D-Wave 2000Q system, showing good agreement with the reference solution.  ...  We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum  ...  operations on an "ibmq toronto" digital quantum computer via IBM Q cloud access.  ... 
doi:10.22331/q-2021-07-13-502 fatcat:etngvjyanncd3bttxucduwiesa

Foundations of quantum physics III. Measurement [article]

Arnold Neumaier
2019 arXiv   pre-print
-- the latter emerges approximately in non-isolated subsystems. * gives a valid interpretation of systems modeled by a quantum-classical dynamics. * explains the peculiar features of the Copenhagen interpretation  ...  are essentially free. * claims that the unmodeled environment influences the results enough to cause all randomness in quantum physics. * allows one to derive Born's rule for scattering and in the limit  ...  Earlier versions of this paper benefitted from discussions with Rahel Knöpfel.  ... 
arXiv:1902.10782v2 fatcat:skm6fyi6pbhnhcr5siyb4gkkp4
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