In his paper in 1979, Polyakov envisaged a possibility of reformulating the gauge theory as a Principal Chiral Model defined on a space of loops and discussed "the loop-space integrability". This idea, together with a closely related idea of the loop equation, led to numerous important results in matrix models and 2d gauge theories, but its application to four-dimensional gauge theories had only limited success. Now, after 50 years, we have a concrete example of integrable four-dimensional gauge theory, N=4 SYM. However integrability in N=4 SYM is formulated mostly in terms of local operators, although important progress has been made in constructing the Yangian for the Wilson loops. In this talk, I will present a framework which would bridge these two distant notions of integrabililty. The key player in the story is a correlation function of a local operator and the Wilson loop. I reformulate the gauge-theory computation of this observable as an overlap between an energy eigenstate of a spin chain and a matrix product state (MPS). Unlike standard MPS's discussed in the literature, our MPS has infinite bond dimensions in order to accommodate infinite dimensionality of the space of loops. It provides an "intertwiner" between integrable structures of the local operators and the Wilson loops, and in particular implies the existence of a special set of deformations of the Wilson loop which satisfy the QQ-relation. I will also explain how to formulate a nonperturbative bootstrap program based on the results obtained in this framework and compute the correlator of the circular BPS Wilson loop and general non-BPS operators at finite coupling, emphasizing the relation to and the difference from other observables that were computed by a similar approach. ----------------- Part of London Integrability Journal Club. Please register using the form at integrability-london.weebly.com