  Quantum ergodic restriction (QER) is the problem of finding conditions on a
hypersurface $H$ so that restrictions $\phi_j |_H$ to $H$ of
$\Delta$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic
flow are quantum ergodic on $H$. We prove two kinds of results: First (i) for
any smooth hypersurface $H$, the Cauchy data $(\phi_j|H, \partial \phi_j|H)$ is
quantum ergodic if the Dirichlet and Neumann data are weighted appropriately.
Secondly (ii) we give conditions on $H$ so that the Dirichlet (or Neumann) data
is individually quantum ergodic. The condition involves the almost nowhere
equality of left and right Poincar\'e maps for $H$. The proof involves two
further novel results: (iii) a local Weyl law for boundary traces of
eigenfunctions, and (iv) an 'almost-orthogonality' result for Fourier integral
operators whose canonical relations almost nowhere commute with the geodesic
flow.
