  When a system is swept through a quantum critical point, the quantum
Kibble-Zurek mechanism makes universal predictions for quantities such as the
number and energy of excitations produced. This mechanism is now being used to
obtain critical exponents on emerging quantum computers and emulators, which in
some cases can be compared to Matrix Product State (MPS) numerical studies.
However, the mechanism is modified when the divergence of entanglement entropy
required for a faithful description of many quantum critical points is not
fully captured by the experiment or classical calculation. In this work, we
study how low-energy dynamics of quantum systems near criticality are modified
by finite entanglement, using conformally invariant critical points described
approximately by an MPS as an example. We derive that the effect of finite
entanglement on a Kibble-Zurek process is captured by a dimensionless scaling
function of the ratio of two length scales, one determined dynamically and one
by the entanglement restriction. Numerically we confirm first that dynamics at
finite bond dimension $\chi$ is independent of the algorithm chosen, then
obtain scaling collapses for sweeps in the transverse field Ising model and the
3-state Potts model. Our result establishes the precise role played by
entanglement in time-dependent critical phenomena and has direct implications
for quantum state preparation and classical simulation of quantum states.
