High order schemes for solving partial differential equations on a quantum computer

2024-12-26

AI Summary

Quantum algorithms for solving partial differential equations (PDEs) suffer from excessive quantum gate and qubit overhead, hindering practical implementation. Method: We propose a high-order finite-difference discretization framework and design an efficient decomposition of $d$-banded diagonal matrices into mutually commuting Pauli strings. Contribution/Results: We provide the first rigorous proof that high-order discretization substantially reduces required qubit count—paralleling classical computational advantages—while leaving Trotter step count unchanged, thereby correcting prior misconceptions about quantum evolution resource scaling. Numerical experiments on the one-dimensional wave equation demonstrate that our method significantly reduces qubit requirements at fixed accuracy and achieves superior asymptotic gate-count scaling with increasing precision. These results establish a critical resource benchmark and a scalable pathway toward practical quantum PDE solvers.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing $d$-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.

Problem

• Quantum Computing
• Partial Differential Equations
• Optimization

Innovation

• Quantum Computing
• Matrix Decomposition
• Partial Differential Equations