Quantum Kernel Methods for Solving Differential Equations
- URL: http://arxiv.org/abs/2203.08884v1
- Date: Wed, 16 Mar 2022 18:56:35 GMT
- Title: Quantum Kernel Methods for Solving Differential Equations
- Authors: Annie E. Paine, Vincent E. Elfving, Oleksandr Kyriienko
- Abstract summary: We propose several approaches for solving differential equations (DEs) with quantum kernel methods.
We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented.
- Score: 21.24186888129542
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We propose several approaches for solving differential equations (DEs) with
quantum kernel methods. We compose quantum models as weighted sums of kernel
functions, where variables are encoded using feature maps and model derivatives
are represented using automatic differentiation of quantum circuits. While
previously quantum kernel methods primarily targeted classification tasks, here
we consider their applicability to regression tasks, based on available data
and differential constraints. We use two strategies to approach these problems.
First, we devise a mixed model regression with a trial solution represented by
kernel-based functions, which is trained to minimize a loss for specific
differential constraints or datasets. Second, we use support vector regression
that accounts for the structure of differential equations. The developed
methods are capable of solving both linear and nonlinear systems. Contrary to
prevailing hybrid variational approaches for parametrized quantum circuits, we
perform training of the weights of the model classically. Under certain
conditions this corresponds to a convex optimization problem, which can be
solved with provable convergence to global optimum of the model. The proposed
approaches also favor hardware implementations, as optimization only uses
evaluated Gram matrices, but require quadratic number of function evaluations.
We highlight trade-offs when comparing our methods to those based on
variational quantum circuits such as the recently proposed differentiable
quantum circuits (DQC) approach. The proposed methods offer potential quantum
enhancement through the rich kernel representations using the power of quantum
feature maps, and start the quest towards provably trainable quantum DE
solvers.
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