  We develop an integrability-based framework to compute structure constants of
two sub-determinant operators and a single-trace non-BPS operator in ABJM
theory in the planar limit. In this first paper, we study them at weak coupling
using a relation to an integrable spin chain. We first develop a nested Bethe
ansatz for an alternating SU(4) spin chain that describes single-trace
operators made out of scalar fields. We then apply it to the computation of the
structure constants and show that they are given by overlaps between a Bethe
eigenstate and a matrix product state. We conjecture that the determinant
operator corresponds to an integrable matrix product state and present a
closed-form expression for the overlap, which resembles the so-called Gaudin
determinant. We also provide evidence for the integrability of general
sub-determinant operators. The techniques developed in this paper can be
applied to other quantities in ABJM theory including three-point functions of
single-trace operators.
