CS/YQI Quantum Computing Colloquium - Junyu Liu - University of Pittsburgh

 

It has long been believed that quantum computers can efficiently solve differential equations, even if those equations are from classical physics. Most existing algorithms for universal, digital, fault-tolerant quantum devices rely on Carleman linearization, which maps nonlinear systems into linear regimes and then applies quantum linear system solvers to simulate their time evolution. However, algorithms with provable quantum speedups typically require digital truncation and full fault tolerance, making them impractical for current quantum hardware. In this talk, I present a novel quantum algorithm that employs coupled bosonic modes and transmon qubits with adaptive measurements to efficiently solve nonlinear differential equations. This approach leverages the infinite-dimensional linear Hilbert space of bosonic systems via the von Neumann–Koopman formalism, thereby avoiding digital truncation inherent in Carleman linearization. We also establish theoretical conditions for quantum speedups: for highly dissipative equations, the algorithm can solve an L-site nonlinear PDE using the Euler method in only log (L) time. Numerical simulations demonstrate applications to fluid dynamics, and we further discuss the error resilience of this method in highly dissipative systems, as well as its implementation using circuit QED in real experiments. This talk is based on a series of works to appear, collaborating with mechanical engineering department of the University of Pittsburgh, NASA, and DOE-NNL.