  The exponential growth in Hilbert space with increasing size of a quantum
system means that accurately characterising the system becomes significantly
harder with system dimension d. We show that self-guided tomography is a
practical, efficient, and robust technique of measuring higher-dimensional
quantum states. The achieved fidelities are over 99.9% for qutrits (d=3) and
ququints (d=5), and 99.1% for quvigints (d=20), the highest values ever
realised for qudits. We demonstrate robustness against experimental sources of
noise, both statistical and environmental. The technique is applicable to any
higher-dimensional system, from a collection of qubits through to individual
qudits, and any physical realisation, be it photonic, superconducting, ionic,
or spin.
