Publikationen

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, H. Furthermore, an embedding of classical observables into self-adjoint operators on H is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of H, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a 2n-dimensional tensor product Hilbert space associated with n qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an n-channel quantum circuit of size O(n) and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size O(n), and a quantum Fourier transform stage of size O(n2), which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, n&flech;∞. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension 2n using circuits of size O(n2). We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

We explore the transport mechanisms of heat in two- and three-dimensional turbulent convection flows by means of the long-term evolution of Lagrangian coherent sets. They are obtained from the spectral clustering of trajectories of massless fluid tracers that are advected in the flow. Coherent sets result from trajectories that stay closely together under the dynamics of the turbulent flow. For longer times, they are always destroyed by the intrinsic turbulent dispersion of material transport. Here, this constraint is overcome by the application of evolutionary clustering algorithms that add a time memory to the coherent set detection and allows individual trajectories to leak in or out of evolving clusters. Evolutionary clustering thus also opens the possibility to monitor the splits and mergers of coherent sets. These rare dynamic events leave clear footprints in the evolving eigenvalue spectrum of the Laplacian matrix of the trajectory network in both convection flows. The Lagrangian trajectories reveal the individual pathways of convective heat transfer across the fluid layer. We identify the long-term coherent sets as those fluid flow regions that contribute least to heat transfer. Thus, our evolutionary framework defines a complementary perspective on the slow dynamics of turbulent superstructure patterns in convection flows that were recently discussed in the Eulerian frame of reference. The presented framework might be well suited for studies in natural flows, which are typically based on sparse information from drifters and probes.

A duct flow generated by a planar jet at the inlet and affected by a magnetic field perpendicular to the jet's plane is analyzed in high-resolution numerical simulations. The case of very high Reynolds and Hartmann numbers is considered. It is found that the flow structure is drastically modified in the inlet area. It becomes determined by three new planar jets oriented along the magnetic field lines: two near the walls and one in the middle of the duct. The downstream evolution of the flow includes the Kelvin-Helmholtz instability of the jets and slow decay of the resulting quasi-two-dimensional turbulence.

A liquid metal flat jet immersed in a square duct under the influence of a transverse magnetic field is studied experimentally. Two cases are considered: when the applied magnetic field is oriented parallel (coplanar field) or perpendicularly (transverse field) to the initial plane of the jet. The main goal of the study is to investigate the mean flow characteristics and the stages of the jet's transformation. Signals of streamwise velocity at different locations are measured, which allows us to determine average velocity profiles and spatial-temporal characteristics of the velocity field. The two considered configurations are directly compared under the same flow regimes, with the same equipment.