  The unitary Fermi gas is a many-body system of two-component fermions with
zero-range interactions tuned to infinite scattering length. Despite much
activity and interest in unitary Fermi gases and its universal properties,
there have been great difficulties in performing accurate calculations of the
superfluid condensate fraction and pairing wave function. In this work we
present auxiliary-field lattice Monte Carlo simulations using a novel lattice
interaction which accelerates the approach to the continuum limit, thereby
allowing for robust calculations of these difficult observables. As a benchmark
test we compute the ground state energy of 33 spin-up and 33 spin-down
particles. As a fraction of the free Fermi gas energy $E_{FG}$, we find
$E_0/E_{FG}= 0.369(2), 0.372(2)$, using two different definitions of the
finite-system energy ratio, in agreement with the latest theoretical and
experimental results. We then determine the condensate fraction by measuring
off-diagonal long-range order in the two-body density matrix. We find that the
fraction of condensed pairs is $\alpha = 0.43(2)$. We also extract the pairing
wave function and find the pair correlation length to be $\zeta_pk_F = 1.8(3)
\hbar$, where $k_F$ is the Fermi momentum. Provided that the simulations can be
performed without severe sign oscillations, the methods we present here can be
applied to superfluid neutron matter as well as more exotic P-wave and D-wave
superfluids.
