  It would be very useful to find a way of reducing excited-state effects in
lattice QCD calculations of nucleon structure that has a low computational
cost. We explore the use of hybrid interpolators, which contain a nontrivial
gluonic excitation, in a variational basis together with the standard
interpolator with tuned smearing width. Using the clover discretization of the
field strength tensor, a calculation using a fixed linear combination of
standard and hybrid interpolators can be done using the same number of quark
propagators as a standard calculation, making this a cost-effective option. We
find that such an interpolator, optimized by solving a generalized eigenvalue
problem, reduces excited-state contributions in two-point correlators. However,
the effect in three-point correlators, which are needed for computing nucleon
matrix elements, is mixed: for some matrix elements such as the tensor charge,
excited-state effects are suppressed, whereas for others such as the axial
charge, they are enhanced. The results illustrate that the variational method
is not guaranteed to reduce the net contribution from excited states except in
its asymptotic regime, and suggest that it may be important to use a large
basis of interpolators capable of isolating all of the relevant low-lying
states.
