# Quantum Compiler for Classical Dynamical Systems

We present a framework for simulating a measure-preserving, ergodic dynamical system by a finite-dimensional quantum system amenable to implementation on a quantum computer. The framework is based on a quantum feature map for representing classical states by density operators (quantum states) on a reproducing kernel Hilbert space (RKHS), H, of functions on classical state space. Simultaneously, a mapping is employed from classical observables into self-adjoint operators on H such that quantum mechanical expectation values are consistent with pointwise function evaluation. Meanwhile, quantum states and observables on H evolve under the action of a unitary group of Koopman operators in a consistent manner with classical dynamical evolution. To achieve quantum parallelism, the state of the quantum system is projected onto a finite-rank density operator on a 2^N-dimensional tensor product Hilbert space associated with N qubits. In this talk, we describe this "quantum compiler" framework, and illustrate it with applications to low-dimensional dynamical systems.

Dimitris Giannakis is an Associate Professor of Mathematics at the Courant Institute of Mathematical Sciences, New York University. He is also affiliated with Courant's Center for Atmosphere Ocean Science (CAOS). He received BA and MSci degrees in Natural Sciences from the University of Cambridge in 2001, and a PhD degree in Physics from the University of Chicago in 2009. Giannakis' current research focus is at the interface between operator-theoretic techniques for dynamical systems and machine learning. His recent work includes the development of techniques for coherent pattern extraction, statistical forecasting, and data assimilation based on data-driven approximations of Koopman operators of dynamical systems. He has worked on applications of these tools to atmosphere ocean science, fluid dynamics, and molecular dynamics.