Quantum Finite Element Analysis

The quest for #quantum #computing is sweeping across many industries much before the quantum hardware are matured enough to solve realtime problems. Segments like #finance, life #sciences are closely rubbernecking its developments.

One of the most potential but widely ignored applications is the Finite Element Analysis (#FEM). In fact the FEM benefits a lot being a large systems of linear equations whose solutions can be found with the Quantum Linear Equation Algorithm introduced by Harrow, Hassidim and Lloyd (#HHL). This algorithm gives an exponential #quantum #speedup over classical algorithms for solving the LE (at the time of writing this).

The FEM is a tempting target for acceleration by the QLE algorithm, as put by Ashley Montanaro & Sam Pallister, for several reasons. One of the major & the most attractive reasons is, the large systems of linear equations of FEM that are produced algorithmically, need no a quantum RAM. A big headache is removed! The next attractive point is, the FEM naturally leads to sparse systems, meaning, they have nonzero entries in each row. These are sufficient to justify the drive for development of FEM (& #CFD) with quantum algorithms.

Of course Clader, Jacobs & Sprouse have used #QLE to the FEM while studying the #electromagnetic scattering & have argued that the quantum speedup had been achieved over the best classical algorithms known. The worst part of their claim is the authors didn’t consider the accuracy parameter in the quantum algorithm. With this approach, similar kind of speedup could be achieved even with classical computers!

Montanaro & Pallister have circumvented this problem to include the accuracy & managed to obtain the quantum speedup with their approach in “Quantum algorithms and the finite element method”, Physical Review A, 2016, vol 93, issue 3, pg 1–16. Will discuss this next.