  Using the estimate of the difference between the discrete harmonic function
and its corresponding continuous version we derive a rate of convergence of the
Loewner driving function for the harmonic explorer to the Brownian motion with
speed 4 on the real line. Based on this convergence rate, the derivative
estimate for chordal $\mbox{SLE}_4$, and the estimate of tip structure modulus
for harmonic explorer paths, we obtain an explicit power-law rate of
convergence of the harmonic explorer paths to the trace of chordal
$\mbox{SLE}_4$ in the supremum distance.
