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

Quantum ergodic restriction theorems, I: interior hypersurfaces in domains with ergodic billiards