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Characterising quantum processes is a key task in and constitutes a challengefor the development of quantum technologies, especially at the noisyintermediate scale of today's devices. One method for characterising processesis randomised benchmarking, which is robust against state preparation andmeasurement (SPAM) errors, and can be used to benchmark Clifford gates. Acomplementing approach asks for full tomographic knowledge. Compressed sensingtechniques achieve full tomography of quantum channels essentially at optimalresource efficiency. So far, guarantees for compressed sensing protocols relyon unstructured random measurements and can not be applied to the data acquiredfrom randomised benchmarking experiments. It has been an open question whetheror not the favourable features of both worlds can be combined. In this work, wegive a positive answer to this question. For the important case ofcharacterising multi-qubit unitary gates, we provide a rigorously guaranteedand practical reconstruction method that works with an essentially optimalnumber of average gate fidelities measured respect to random Cliffordunitaries. Moreover, for general unital quantum channels we provide an explicitexpansion into a unitary 2-design, allowing for a practical and guaranteedreconstruction also in that case. As a side result, we obtain a new statisticalinterpretation of the unitarity -- a figure of merit that characterises thecoherence of a process. In our proofs we exploit recent representationtheoretic insights on the Clifford group, develop a version of Collins'calculus with Weingarten functions for integration over the Clifford group, andcombine this with proof techniques from compressed sensing.

Recovering quantum gates from few average gate fidelities