Feedback-Based Quantum Algorithms for the Poisson Equation
James Larsen, University of Michigan
Extensive scientific computing resources are devoted to the numerical simulation and solution of partial differential equations, but generic algorithms for accomplishing these tasks on quantum computers remain relatively unexplored. In this work, we explore several different feedback-based approaches for solving the Poisson equation on a quantum computer. We show how off-the-shelf variational ansätze lead to very deep circuits with questionable convergence properties. In an attempt to remedy this issue, we first switch to a randomized ansatz consisting of a series of unitary two-designs. Then, we explore how variational Trotter compression (VTC) can shorten these circuits, combining the feedback law from quantum Lyapunov control at each layer with an iteration of VTC to keep the circuit depths more manageable. While this work is only a first step in demonstrating how a feedback-based approach can be applied to problems from partial differential equations, we are optimistic that these methods can be generalized and improved upon in future work.