Design nearly optimal quantum algorithm for linear differential equations via Lindbladians

• URL: http://arxiv.org/abs/2410.19628v2
• Date: Tue, 29 Oct 2024 04:54:41 GMT
• Title: Design nearly optimal quantum algorithm for linear differential equations via Lindbladians
• Authors: Zhong-Xia Shang, Naixu Guo, Dong An, Qi Zhao,
• Abstract summary: We propose a new quantum algorithm for ODEs by harnessing open quantum systems. We use the natural non-unitary dynamics of Lindbladians with the aid of a new technique called the non-diagonal density matrix encoding. Our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters.
• Score: 11.53984890996377
• License:
• Abstract: Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary dynamics into intrinsically unitary quantum circuits. In this work, we propose a new quantum algorithm for ODEs by harnessing open quantum systems. Specifically, we utilize the natural non-unitary dynamics of Lindbladians with the aid of a new technique called the non-diagonal density matrix encoding to encode general linear ODEs into non-diagonal blocks of density matrices. This framework enables us to design a quantum algorithm that has both theoretical simplicity and good performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm, under a plausible input model, can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.

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