In order to constrain beyond the Standard Model physics using
experiments with nuclear targets including neutrinoless double-beta
decay, nuclear electric dipole moment, and dark matter direct
detection searches, the responses of nuclei to electroweak forces and
possible new physics forces must be known. Similar nuclear responses
also determine nuclear physics quantities of interest including
nuclear effects on parton structure functions and input parameters for
effective theories of larger nuclei. Nuclear responses to interactions
with general spin and flavor structures can be difficult or impossible
to measure experimentally, but in principle they can be accurately
predicted by the theory of quantum chromodynamics (QCD). I will
present results from recent lattice QCD calculations of a full
spin-flavor decomposition of the static responses of nuclei with A=2-3
to external currents at unphysically heavy quark masses. Efforts to
extend these calculations to lighter quark masses and larger nuclei
are underway, but these calculations are made challenging in part by
an exponentially difficult signal-to-noise problem. I will also
discuss ongoing work to mitigate this signal-to-noise problem by
exploiting statistical random walk behavior in the phases of complex
path integrals and describe preliminary investigations of new tools
for complex scalar field theory and lattice QCD.