Abstract
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity poly(1/ϵ), where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be poly(d,log(1/ϵ)), where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.
Highlights
Many scientific problems involve partial differential equations (PDEs)
We describe our first approach to quantum algorithms for linear PDEs, based on the finite difference method (FDM)
Since interpolation facilitates constructing a straightforward linear system, we develop a quantum algorithm based on the pseudo-spectral method [15, 28, 34] for second-order elliptic equations with global strict diagonal dominance, under various boundary conditions
Summary
Many scientific problems involve partial differential equations (PDEs). Prominent examples include Maxwell’s equations for electromagnetism, Boltzmann’s equation and the Fokker-Planck equation in thermodynamics, and Schrodinger’s equation in continuum quantum mechanics. Montanaro and Pallister [22] use QLSAs to implement the FEM for d-dimensional boundary value problems and evaluate the quantum speedup that can be achieved when estimating a function of the solution within precision This involves a careful analysis of how different algorithmic parameters (such as the dimension and condition number of the FEM linear system and the number of post-processing measurements) scale with respect to input variables (such as the spatial dimension d and desired precision ), since all of these affect the complexity. We state our result in Theorem 2, which (informally) gives a complexity of d2 poly(log(1/ )) for producing a quantum state approximating the solution of general second-order elliptic PDEs with Dirichlet boundary conditions Both of these approaches have complexity poly(d, log(1/ )), providing optimal dependence on and an exponential improvement over classical methods as a function of the spatial dimension d.
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