  We present a conceptual approach to quantum tomography based on first
expanding a quantum state across extra degrees of freedom and then exploiting
the introduced sparsity to perform reconstruction. We formulate its application
to photonic circuits, and show that measured spatial photon correlations at the
output of a specially tailored discrete-continuous quantum-walk can enable full
reconstruction of any two-photon spatially entangled and mixed state at the
input. This approach does not require any tunable elements, so is well suited
for integration with on-chip superconducting photon detectors.
