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.
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