  We prove quantitative stability of isometries on the first Heisenberg group
with sub-Riemannian geometry: every $ (1+ \varepsilon)$-quasi-isometry of the
John domain of the Heisenberg group $ \mathbb {H} $ is close to some isometry
with order of closeness $ \sqrt{\varepsilon} + \varepsilon $ in the uniform
norm and with order of closeness $ \varepsilon $ in the Sobolev norm $L_2^1$.
Homogeneous dilations show the asymptotic sharpness of the results.
