  In this paper, we propose an {\it exact holographic mapping} which is a
unitary mapping from the Hilbert space of a lattice system in flat space
(boundary) to that of another lattice system in one higher dimension (bulk). By
defining the distance in the bulk system from two-point correlation functions,
we obtain an emergent bulk space-time geometry that is determined by the
boundary state and the mapping. As a specific example, we study the exact
holographic mapping for $(1+1)$-dimensional lattice Dirac fermions and explore
the emergent bulk geometry corresponding to different boundary states including
massless and massive states at zero temperature, and the massless system at
finite temperature. We also study two entangled one-dimensional chains and show
that the corresponding bulk geometry consists of two asymptotic regions
connected by a worm-hole. The quantum quench of the coupled chains is mapped to
dynamics of the worm-hole. In the end we discuss the general procedure of
applying this approach to interacting systems, and other open questions.
