  In this paper, we construct a tensor network representation of quantum causal
histories, as a step towards directly representing states in quantum gravity
via bulk tensor networks. Quantum causal histories are quantum extensions of
causal sets in the sense that on each event in a causal set is assigned a
Hilbert space of quantum states, and the local causal evolutions between events
are modeled by completely positive and trace-preserving maps. Here we utilize
the channel-state duality of completely positive and trace-preserving maps to
transform the causal evolutions to bipartite entangled states. We construct the
matrix product state for a single quantum causal history by projecting the
obtained bipartite states onto the physical states on the events. We also
construct the two dimensional tensor network states for entangled quantum
causal histories in a restricted case with compatible causal orders. The
possible holographic tensor networks are explored by mapping the quantum causal
histories in a way analogous to the exact holographic mapping. The constructed
tensor networks for quantum causal histories are exemplified by the non-unitary
local time evolution moves in a quantum system on temporally varying
discretizations, and these non-unitary evolution moves are shown to be
necessary for defining a bulk causal structure and a quantum black hole.
Finally, we comment on the limitations of the constructed tensor networks, and
discuss some directions for further studies aiming at applications in quantum
gravity.
