Session: Alternative and Hybrid Algorithms in Quantum Computing for Optimization and Applications
Chair: Xiu Yang
Cluster: Optimization for Emerging Technologies (LLMs, Quantum Computing, ...)
Talk 1: Unleashed from Constrained Optimization: Quantum Computing for Quantum Chemistry Employing Generator Coordinate Method
Speaker: Bo Peng
Abstract: Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they also introduce challenges. These challenges include addressing the barren plateau and ensuring the accuracy of the ansatze, which often manifest as constrained optimization problems. In this work, we explore the interconnection between constrained optimization and generalized eigenvalue problems through a unique class of Givens rotations. These rotations frequently serve as disentangled unitary coupled cluster building blocks constituting the ansatze in variational quantum eigensolver (VQE) and adaptive derivative-assembled pseudo-Trotter VQE (ADAPT-VQE) simulations. Herein, we employ Givens rotations to construct non-orthogonal, overcomplete many-body generating functions, projecting the system Hamiltonian into a working subspace. The resulting generalized eigenvalue problem is proven to generate rigorous lower bounds to the VQE/ADAPT-VQE energies, effectively circumventing the barren plateau issue and the heuristic nature of numerical minimizers in standard VQE processes. For practical applications, we further propose an adaptive scheme for the robust construction of many-body basis sets using these Givens rotations, emphasizing a linear expansion that balances accuracy and efficiency. The effective Hamiltonian generated by our approach would also facilitate the computation of excited states and evolution, laying the groundwork for more sophisticated quantum simulations in chemistry.
Talk 2: Quantum DeepONet: Neural operators accelerated by quantum computing
Speaker: Lu Lu
Abstract: In the realm of mathematics, engineering, and science, constructing models that reflect real- world phenomena requires solving partial differential equations (PDEs) with different param- eters. Recent advancements in DeepONet, which learn mappings between infinite-dimensional function spaces, promise efficient evaluations of PDE solutions for new parameter sets in a single forward pass. However, classical DeepONet entails quadratic time complexity concerning input dimensions during evaluation. Given the progress in quantum algorithms and hardware, we propose utilizing quantum computing to accelerate DeepONet evaluations, resulting in time complexity that is linear in input dimensions. Our approach integrates unary encoding and orthogonal quantum layers to facilitate this process. We benchmark our Quantum DeepONet using a variety of equations, including the first-order linear ordinary differential equation, advection equation, and Burgers' equation, demonstrating the method’s efficacy in both ideal and noisy conditions. Furthermore, we show that our quantum DeepONet can also be informed by physics, minimizing its reliance on extensive data collection. We expect Quantum DeepONet to be particularly advantageous in applications in outer loop problems which require to explore parameter space and solving the corresponding PDEs, such as forward uncertainty propagation and optimal experimental design.
Talk 3: Koopman Linearization for Optimization in Quantum Computing
Speaker: Xiu Yang
Abstract: Nonlinearity presents a significant challenge in developing quantum algorithms involving differential equations, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. Instead, this paper introduces the Koopman Spectral Linearization method tailored for nonlinear autonomous ordinary differential equations. This innovative linearization approach harnesses the interpolation methods and the Koopman Operator Theory to yield a lifted linear system. It promises to serve as an alternative approach that can be employed in scenarios where Carleman Linearization is traditionally applied. Numerical experiments demonstrate the effectiveness of this linearization approach for several commonly used nonlinear ordinary differential equations. Hence, it enables a special design of gradient-descent type of method based on the technique called Schrodingerization that is used to solve linear differential equations on quantum computers.
