



@article {3064,
	title = {Efficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation},
	year = {2022},
	month = {5/2/2022},
	abstract = {<p>Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition R\&lt;1, where R measures the ratio of nonlinearity to dissipation using the l2 norm. Here we develop an efficient quantum algorithm based on [1] for reaction-diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in [1] to obtain a faster convergence rate under the condition RD\&lt;1, where RD measures the ratio of nonlinearity to dissipation using the l\&infin; norm. Since RD is independent of the number of spatial grid points n while R increases with n, the criterion RD\&lt;1 is significantly milder than R\&lt;1 for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.</p>},
	url = {https://arxiv.org/abs/2205.01141},
	author = {Dong An and Di Fang and Stephen Jordan and Jin-Peng Liu and Guang Hao Low and Jiasu Wang}
}
@article {2710,
	title = {Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance},
	journal = {Quantum 5, 481 (2021)},
	volume = {5},
	year = {2021},
	month = {6/22/2021},
	pages = {481},
	abstract = {<p>Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expection values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.</p>},
	doi = {https://doi.org/10.22331/q-2021-06-24-481},
	url = {https://arxiv.org/abs/2012.06283},
	author = {Dong An and Noah Linden and Jin-Peng Liu and Ashley Montanaro and Changpeng Shao and Jiasu Wang}
}
