We study the problem of unbiased estimation of expectations with respect to
(w.r.t.) $\pi$ a given, general probability measure on
$(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ that is absolutely continuous with
respect to a standard Gaussian measure. We focus on simulation associated to a
particular class of diffusion processes, sometimes termed the
Schr\"odinger-F\"ollmer Sampler, which is a simulation technique that
approximates the law of a particular diffusion bridge process $\{X_t\}_{t\in
[0,1]}$ on $\mathbb{R}^d$, $d\in \mathbb{N}_0$. This latter process is
constructed such that, starting at $X_0=0$, one has $X_1\sim \pi$. Typically,
the drift of the diffusion is intractable and, even if it were not, exact
sampling of the associated diffusion is not possible. As a result,
\cite{sf_orig,jiao} consider a stochastic Euler-Maruyama scheme that allows the
development of biased estimators for expectations w.r.t.~$\pi$. We show that
for this methodology to achieve a mean square error of
$\mathcal{O}(\epsilon^2)$, for arbitrary $\epsilon>0$, the associated cost is
$\mathcal{O}(\epsilon^{-5})$. We then introduce an alternative approach that
provides unbiased estimates of expectations w.r.t.~$\pi$, that is, it does not
suffer from the time discretization bias or the bias related with the
approximation of the drift function. We prove that to achieve a mean square
error of $\mathcal{O}(\epsilon^2)$, the associated cost is, with high
probability, $\mathcal{O}(\epsilon^{-2}|\log(\epsilon)|^{2+\delta})$, for any
$\delta>0$. We implement our method on several examples including Bayesian
inverse problems.