The quantum computing team at Institute of Natural Sciences, Shanghai Jiao Tong University, focuses on systematically developing quantum algorithms for partial differential equations and related problems on both qubit and continuous variable (qumode) platforms. These algorithms address complex challenges such as boundary and interface conditions, multiscale problems, and uncertainties in partial differential equations, demonstrating algebraic to exponential advantages compared to their classical counterparts.
We develop a general framework for quantum algorithms applied to linear dynamical systems through "Schrödingerisation" [ 4, 5 ], which transforms any linear ordinary and partial differential equations with non-unitary dynamics into a system characterized by unitary dynamics, making them suitable for quantum simulation. Additionally, this approach is particularly well-suited for continuous variables, making it an idea -- so far the only possible one -- for the analog quantum simulation of PDEs and ODEs.
We develop quantum algorithms specifically tailored to tackle a diverse range of PDEs, including open quantum systems within bounded domains with non-unitary artificial boundary conditions, problems entailing physical boundary or interface conditions, Maxwell's equations, Fokker-Planck equations, ill-posed scenarios such as the backward heat equations, etc.
We provide an explicit implementation of the corresponding quantum circuits designed to solve general PDEs using Schrödingerization techniques.
For several significant classes of nonlinear PDEs, including Hamilton-Jacobi equations and scalar nonlinear hyperbolic equations, we map the underlying nonlinear PDEs in (d+1)-dimensions to linear PDEs of at most (2d+1)-dimensions using the level set formalism. We can then solve these linear PDEs -- whose dimension is at most twice that of the original nonlinear PDE -- on a quantum computer without losing any physical information. This procedure is exact, meaning no approximations are made, and serves as an example of the linear representation method for nonlinear PDEs.
Analog information in the quantum regime is commonly referred to as continuous-variable (CV) quantum information. We apply Schrödingerisation to directly map d-dimensional linear PDEs onto a (d+1)-qumode quantum system, enabling analog or continuous-variable Hamiltonian simulation on (d+1) qumodes. This methodology eliminates the need to discretize the PDEs beforehand and is applicable not only to linear PDEs but also to certain nonlinear PDEs and systems of nonlinear ordinary differential equations (ODEs).
Chair Professor, SJTU.
Associate Professor, SJTU.
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Associate Professor, SJTU.
Associate Professor, XTU
Postdoctor, SJTU.
Postdoctor, SJTU.
Postdoctor, SJTU.
Ph.D. Student, SJTU.
Ph.D. Student, SJTU.
Ph.D. Student, SJTU.
Ph.D. Student, SJTU.
Visitor, SXU.