  Adaptive quantum circuits, which combine local unitary gates, midcircuit
measurements, and feedforward operations, have recently emerged as a promising
avenue for efficient state preparation, particularly on near-term quantum
devices limited to shallow-depth circuits. Matrix product states (MPS) comprise
a significant class of many-body entangled states, efficiently describing the
ground states of one-dimensional gapped local Hamiltonians and finding
applications in a number of recent quantum algorithms. Recently, it was shown
that the AKLT state -- a paradigmatic example of an MPS -- can be exactly
prepared with an adaptive quantum circuit of constant-depth, an impossible feat
with local unitary gates due to its nonzero correlation length [Smith et al.,
PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this
approach and demonstrate that a diverse class of MPS can be exactly prepared
using constant-depth adaptive quantum circuits, outperforming optimal
preparation protocols that rely on unitary circuits alone. We show that this
class includes short- and long-ranged entangled MPS, symmetry-protected
topological (SPT) and symmetry-broken states, MPS with finite Abelian,
non-Abelian, and continuous symmetries, resource states for MBQC, and families
of states with tunable correlation length. Moreover, we illustrate the utility
of our framework for designing constant-depth sampling protocols, such as for
random MPS or for generating MPS in a particular SPT phase. We present
sufficient conditions for particular MPS to be preparable in constant time,
with global on-site symmetry playing a pivotal role. Altogether, this work
demonstrates the immense promise of adaptive quantum circuits for efficiently
preparing many-body entangled states and provides explicit algorithms that
outperform known protocols to prepare an essential class of states.
