  Interacting many-body quantum systems and their dynamics, while fundamental
to modern science and technology, are formidable to simulate and understand.
However, by discovering their symmetries, conservation laws, and integrability
one can unravel their intricacies. Here, using up to 124 qubits of a fully
programmable quantum computer, we uncover local conservation laws and
integrability in one- and two-dimensional periodically-driven spin lattices in
a regime previously inaccessible to such detailed analysis. We focus on the
paradigmatic example of disorder-induced ergodicity breaking, where we first
benchmark the system crossover into a localized regime through anomalies in the
one-particle-density-matrix spectrum and other hallmark signatures. We then
demonstrate that this regime stems from hidden local integrals of motion by
faithfully reconstructing their quantum operators, thus providing a detailed
portrait of the system's integrable dynamics. Our results demonstrate a
versatile strategy for extracting hidden dynamical structure from noisy
experiments on large-scale quantum computers.
