  The nuclear matrix element determining the $pp\to d e^+ \nu$ fusion cross
section and the Gamow-Teller matrix element contributing to tritium
$\beta$-decay are calculated with lattice Quantum Chromodynamics (QCD) for the
first time. Using a new implementation of the background field method, these
quantities are calculated at the SU(3)-flavor-symmetric value of the quark
masses, corresponding to a pion mass of $m_\pi$ ~ 806 MeV. The Gamow-Teller
matrix element in tritium is found to be 0.979(03)(10) at these quark masses,
which is within $2\sigma$ of the experimental value. Assuming that the
short-distance correlated two-nucleon contributions to the matrix element
(meson-exchange currents) depend only mildly on the quark masses, as seen for
the analogous magnetic interactions, the calculated $pp\to d e^+ \nu$
transition matrix element leads to a fusion cross section at the physical quark
masses that is consistent with its currently accepted value. Moreover, the
leading two-nucleon axial counterterm of pionless effective field theory is
determined to be $L_{1,A}=3.9(0.1)(1.0)(0.3)(0.9)\ {\rm fm}^3$ at a
renormalization scale set by the physical pion mass, also in agreement with the
accepted phenomenological range. This work concretely demonstrates that weak
transition amplitudes in few-nucleon systems can be studied directly from the
fundamental quark and gluon degrees of freedom and opens the way for subsequent
investigations of many important quantities in nuclear physics.
