Quantum algorithms for simulating dissipative linear and nonlinear dynamics of plasmas*

Quantum computing (QC) has the potential to speed up classical simulations of plasma dynamics, which requires dealing with large amounts of high-dimensional data, by leveraging quantum superposition and entanglement. Yet, due to the intrinsically linear and unitary nature of quantum mechanics, modeling dissipative nonlinear (NL) dynamics has required significant development. We have derived explicit QAs for modeling plasma physics including the linear dynamics of cold fluid waves in plasmas [1], electromagnetic waves [2], and diffusion, and have begun to develop QAs for nonlinear dynamics. In this talk, we discuss the new QAs, various techniques for encoding classical systems into quantum circuits [3], and their potential for quantum speedup. 

QAs for dissipative dynamics are usually based on the transformation of nonunitary initial-value problems into a system of linear equations which then can be solved by a quantum linear solver such as the quantum singular value transformation. Another recent method that we have explored, the Linear Combination of Hamiltonian Simulations algorithm, provides an explicit encoding in terms of multiple unitary evolution operators and thereby avoids matrix inversion.  

Nonlinear dynamics is especially challenging for quantum computers due to the no-cloning theorem which forbids copying unknown quantum states. This results in an exponential growth of resources when any iterative QA is applied to a NL problem. The Koopman—von Neumann (KvN) formulation [4] allows one to embed a NL system into a linear problem that describes the linear evolution of the wavefunction and, hence, the probability distribution function.  We present the first explicit KvN-based QA for simulating NL dynamics and the initial application of this method to representative test cases.