Kavli Affiliate: Birgitta Whaley
| First 5 Authors: Ahmad M. Alkadri, Ahmad M. Alkadri, , ,
| Summary:
The finite element method (FEM) is a cornerstone numerical technique for
solving partial differential equations (PDEs). Here, we present
$textbfQu-FEM$, a fault-tolerant era quantum algorithm for the finite
element method. In contrast to other quantum PDE solvers, Qu-FEM preserves the
geometric flexibility of FEM by introducing two new primitives, the unit of
interaction and the local-to-global indicator matrix, which enable the assembly
of global finite element arrays with a constant-size linear combination of
unitaries. We study the modified Poisson equation as an elliptic problem of
interest, and provide explicit circuits for Qu-FEM in Cartesian domains. For
problems with constant coefficients, our algorithm admits block-encodings of
global arrays that require only $tildemathcalOleft(d^2 p^2 nright)$
Clifford+$T$ gates for $d$-dimensional, order-$p$ tensor product elements on
grids with $2^n$ degrees of freedom in each dimension, where $n$ is the number
of qubits representing the $N=2^n$ discrete grid points. For problems with
spatially varying coefficients, we perform numerical integration directly on
the quantum computer to assemble global arrays and force vectors. Dirichlet
boundary conditions are enforced via the method of Lagrange multipliers,
eliminating the need to modify the block-encodings that emerge from the
assembly procedure. This work presents a framework for extending the geometric
flexibility of quantum PDE solvers while preserving the possibility of a
quantum advantage.
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