Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

Abstract

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

Improved quantum algorithms for linear and nonlinear differential equations

“…Other works that address aspects of differential equations but with an application to plasma physics are [19], where a quantum algorithm to solve a linearized Vlasov equation is given. In [20,21], a formulation of several interesting problems in plasma physics that could be amenable to quantum algorithms through linearization is given and in [22], a quantum algorithm to solve classical dynamics using the Koopman operator approach is given. In [23], a quantum algorithm to solve cold plasma waves is given.…”

Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation

“…First, it yields a quadratic speedup in the accuracy of Monte Carlo estimates [BHMT00, Mon15,HM18]. Achieving 'Heisenberg limited' accuracy scaling has numerous applications in physics, chemistry, machine learning, and finance [Wright&20,ESP20,Rall20,An&20]. Second, it allows quantum computers to diagonalize unitaries in a certain restricted sense: if U = j e 2πiλ j |ψ j ψ j | then phase estimation performs the transformation…”

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

“…In [43], a linearization technique of nonlinear classical dynamics based on Koopman-von Neumann method is proposed. [44] summarizes three classical linear embedding techniques, including Carleman embedding(Carleman linearization is also called Carleman embedding) [26,27], coherent states embedding [27,45] and position-space embedding [46], and then puts forward the prospects of these linear embedding techniques to construct effective quantum algorithms. An open question is whether there are other ways to induce nonlinearity in quantum computing.…”