A paper co-authored by Matthias C. Caro has been featured on the cover of PRX Quantum. PRX Quantum is a premier journal for quantum information science and technology research. The work was a collaboration with Haimeng Zhao (Caltech & Tsinghua), Laura Lewis (Caltech & Google), Ishaan Kannan (Caltech), Yihui Quek (Harvard & MIT) and Hsin-Yuan Huang (Caltech, Google & MIT).

Characterizing a quantum system by learning its state or unitary evolution is a key tool in developing quantum devices, with applications in practical quantum machine learning, benchmarking, and error mitigation. However, in general, this task requires exponentially many resources. Prior knowledge is required to circumvent this exponential bottleneck. The paper pinpoints the complexity for learning states and unitaries that can be implemented by quantum circuits with a bounded number of gates, a broad setting that is topical for current quantum technologies. When measuring efficiency with respect to the number of accesses to the unknown quantum state or unitary, the paper presents and implements algorithms that are provably optimally efficient. Thereby, this work establishes the equivalence between the complexity of learning quantum states or unitaries and the complexity of creating them. However, it also shows that the data processing necessarily requires exponential computation time under reasonable cryptographic assumptions.