  Feynman's circuit-to-Hamiltonian construction enables the mapping of a
quantum circuit to a time-independent Hamiltonian. Here we investigate the
efficiency of Feynman's quantum computer by analysing the time evolution
operator $e^{-i\hat{H}t}$ for Feynman's clock Hamiltonian $\hat{H}$. A general
formula is established for the probability, $P_k(t)$, that the desired
computation is complete at time $t$ for a quantum computer which executes an
arbitrary number $k$ of operations. The optimal stopping time, denoted by
$\tau$, is defined as the time of the first local maximum of this probability.
We find numerically that there is a linear relationship between this optimal
stopping time and the number of operations, $\tau = 0.50 k + 2.37$.
Theoretically, we corroborate this linear behaviour by showing that at $\tau =
\frac{1}{2} k + 1$, $P_k(\tau)$ is approximately maximal. We also establish a
relationship between $\tau$ and $P_k(\tau)$ in the limit of a large number $k$
of operations. We show analytically that at the maximum, $P_k(\tau)$ behaves
like $k^{-2/3}$. This is further proven numerically where we find the inverse
cubic root relationship $P_k(\tau) = 6.76 \; k^{-2/3}$. This is significantly
more efficient than paradigmatic models of quantum computation.
