  Reconstructing the Hamiltonian of a quantum system is an essential task for
characterizing and certifying quantum processors and simulators. Existing
techniques either rely on projective measurements of the system before and
after coherent time evolution and do not explicitly reconstruct the full
time-dependent Hamiltonian or interrupt evolution for tomography. Here, we
experimentally demonstrate that an a priori unknown, time-dependent Hamiltonian
can be reconstructed from continuous weak measurements concurrent with coherent
time evolution in a system of two superconducting transmons coupled by a
flux-tunable coupler. In contrast to previous work, our technique does not
require interruptions, which would distort the recovered Hamiltonian. We
introduce an algorithm which recovers the Hamiltonian and density matrix from
an incomplete set of continuous measurements and demonstrate that it reliably
extracts amplitudes of a variety of single qubit and entangling two qubit
Hamiltonians. We further demonstrate how this technique reveals deviations from
a theoretical control Hamiltonian which would otherwise be missed by
conventional techniques. Our work opens up novel applications for continuous
weak measurements, such as studying non-idealities in gates, certifying analog
quantum simulators, and performing quantum metrology.
