Kavli Affiliate: Long Zhang
| First 5 Authors: Yin Yang, Yin Yang, , ,
| Summary:
We develop a quantum algorithm for linear algebraic equations Ax=b from the
perspective of Schr"odingerization-form problems, which are characterized by a
system of linear convection equations in one higher dimension. When A is
positive definite, the solution x can be interpreted as the steady-state
solution of linear ODEs. This ODE can be solved by using the LCHS method in
[1], which serves as the continuous implementation of the Fourier transform in
the Schr"odingerization method from [2,3] Schr"odingerization transforms
linear PDEs and ODEs with non-unitary dynamics into Schr"odinger-type systems
via the warped phase transformation that maps the equation into one higher
dimension. Compared to the LCHS method, Schr"odingerization may be more
appealing to the PDE community, as it is better suited for leveraging
established computational PDE techniques to develop quantum algorithms. When A
is a general Hermitian matrix, the inverse matrix can still be represented in
the LCHS form in [1], but with a kernel function based on the Fourier approach
in [4]. Although this LCHS form provides the steady-state solution of linear
ODEs associated with the least-squares equation, applying Schr"odingerization
to this least-squares is not appropriate, as it results in a much larger
condition number. We demonstrate that in both cases, the solution x can be
expressed as the LCHS of Schr"odingerization-form problems, or equivalently,
as the steady-state solution to a Schr"odingerization-form problem. This
highlights the potential of Schr"odingerization in quantum scientific
computation. We provide a detailed, along with several numerical tests that
validate the correctness of our proposed method. Furthermore, we develop a
quantum preconditioning algorithm that combines the BPX multilevel
preconditioner with our method to address the finite element discretization of
the Poisson equation.
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