We argue that the natural way to generalise a tensor network variational class to a continuous quantum system is to use the Feynman path integral to implement a continuous tensor contraction. This approach is illustrated for the case of a recently in
troduced class of quantum field states known as continuous matrix-product states (cMPS). As an example of the utility of the path-integral representation we argue that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. An argument that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity is also provided.
In recent years weve seen the birth of a new field known as hamiltonian complexity lying at the crossroads between computer science and theoretical physics. Hamiltonian complexity is directly concerned with the question: how hard is it to simulate a
physical system? Here I review the foundational results, guiding problems, and future directions of this emergent field.
We show how continuous matrix product states of quantum field theories can be described in terms of the dissipative non-equilibrium dynamics of a lower-dimensional auxiliary boundary field theory. We demonstrate that the spatial correlation functions
of the bulk field can be brought into one-to-one correspondence with the temporal statistics of the quantum jumps of the boundary field. This equivalence: (1) illustrates an intimate connection between the theory of continuous quantum measurement and quantum field theory; (2) gives an explicit construction of the boundary field theory allowing the extension of real-space renormalization group methods to arbitrary dimensional quantum field theories without the introduction of a lattice parameter; and (3) yields a novel interpretation of recent cavity QED experiments in terms of quantum field theory, and hence paves the way toward observing genuine quantum phase transitions in such zero-dimensional driven quantum systems.
