  Three block encoding methods are evaluated for solving linear systems of
equations using QSVT (Quantum Singular Value Transformation). These are ARCSIN,
FABLE and PREPARE-SELECT. The performance of the encoders is evaluated using a
suite of 30 test cases including 1D, 2D and 3D Laplacians and 2D CFD matrices.
A subset of cases is used to characterise how the degree of the polynomial
approximation to $1/x$ influences the performance of QSVT. The results are used
to guide the evaluation of QSVT as the linear solver in hybrid non-linear
pressure correction and coupled implicit CFD solvers. The performance of QSVT
is shown to be resilient to polynomial approximation errors. For both CFD
solvers, error tolerances of $10^{-2}$ are more than sufficient in most cases
and in some cases $10^{-1}$ is sufficient. The pressure correction solver
allows subnormalised condition numbers, $\kappa_s$, as low as half the
theoretical values to be used, reducing the number of phase factors needed.
PREPARE-SELECT encoding relies on a unitary decomposition, e.g. Pauli strings,
that has significant classical preprocessing costs. Both ARCSIN and FABLE have
much lower costs, particularly for coupled solvers. However, their
subnormalisation factors, which are based on the rank of the matrix, can be
many times higher than PREPARE-SELECT leading to more phase factors being
needed. For both the pressure correction and coupled CFD calculations, QSVT is
more stable than previous HHL results due to the polynomial approximation
errors only affecting long wavelength CFD errors. Given that lowering
$\kappa_s$ increases the success probability, optimising the performance of
QSVT within a CFD code is a function of the number QSVT phase factors, the
number of non-linear iterations and the number of shots. Although phase factor
files can be reused, the time taken to generate them impedes scaling QSVT to
larger test cases.
