Abstract
In this work, we present novel concepts for quantum algorithms to solve transient, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the incompressible Navier-Stokes equations as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts solving nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. We propose a new framework based on matrix product states (MPSs) and matrix product operators (MPOs), in addition to the Krylov subspace methods. For example, the solution variables of the Navier-Stokes equations are represented by MPSs, and the linear and nonlinear terms are processed by MPOs. The time evolution of the operators is attained by a fast-forwarding algorithm using Krylov subspace methods. Furthermore, we discuss various techniques for efficient encoding of MPSs, measurement reduction for MPOs, and use of tensor operations to treat multi-variate, multi-physics characteristics of Navier-Stokes.
| Original language | English |
|---|---|
| Title of host publication | Technical Papers Program |
| Editors | Candace Culhane, Greg T. Byrd, Hausi Muller, Yuri Alexeev, Yuri Alexeev, Sarah Sheldon |
| Publisher | Institute of Electrical and Electronics Engineers Inc. |
| Pages | 662-668 |
| Number of pages | 7 |
| ISBN (Electronic) | 9798331541378 |
| DOIs | |
| State | Published - 2024 |
| Event | 5th IEEE International Conference on Quantum Computing and Engineering, QCE 2024 - Montreal, Canada Duration: Sep 15 2024 → Sep 20 2024 |
Publication series
| Name | Proceedings - IEEE Quantum Week 2024, QCE 2024 |
|---|---|
| Volume | 1 |
Conference
| Conference | 5th IEEE International Conference on Quantum Computing and Engineering, QCE 2024 |
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| Country/Territory | Canada |
| City | Montreal |
| Period | 09/15/24 → 09/20/24 |
Funding
This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the US Department of Energy (DOE) under Contract No. DE-AC05- 00OR22725. This work was partly supported by the National Science Foundation through awards OAC-2311949 and DMS- 2403552-Amendment-ID005. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52- 07NA27344, LLNL-PROC-867553. Y.Z. acknowledges the support from the US DOE, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under Triad National Security, LLC ("Triad") contract Grant 89233218CNA000001 (FWP: LANLECF7). This manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US DOE. The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide icense to publish or reproduce the published form of this manuscript or allow others to do so for US government urposes. DOE will provide public access to these results of ederally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Keywords
- Incompressible fluid dynamics
- Krylov subspace methods
- Matrix product operators
- Matrix product states
- Nonlinear partial differential equations
- Tensor networks