Francesco A. Mele

The field of quantum learning theory has advanced rapidly in recent years, at the intersection of quantum information science, statistical learning, and computational complexity. A key task in this area is quantum process tomography, which seeks to learn unitary transformations of quantum states efficiently. Efficient process tomography would be highly valuable: for instance, learning an unknown natural process could enable its efficient implementation and simulation on a quantum computer. However, learning arbitrary unitary operators is generally prohibitively expensive, with several sample- and time-complexity lower bounds showing the task is intractable. Thus, work typically focuses on more structured classes of operators when computational efficiency is desired. Two especially important such classes are bosonic and fermionic Gaussian unitaries. These operators have compact parametrizations, rich algebraic structure, and enough expressiveness to capture many relevant physical processes. As a result, they are ubiquitous in quantum information theory. In this work, we advance the learning theory of bosonic and fermionic unitaries in two ways: (1) We give the first time-efficient algorithm to learn bosonic Gaussian unitaries. The complexity of the algorithm scales polynomially in the number of modes, a total photon number bound (which is critical in defining an energy-constrained distance measure), and a squeezing parameter which captures how much the operator increases the mean energy of a vacuum state. (2) We give a first-of-its-kind algorithm to learn fermionic unitaries prepared with at most t non-Gaussian gates. Our algorithm scales polynomially in the number of modes and exponentially in t, and we argue that this scaling is optimal up to polynomial factors. Both algorithms produce an output whose distance to the input unitary is small in the worst-case (diamond) distance. Our results are organized into two separate manuscripts: one is arXiv:2504.11318 (Mildly-Interacting Fermionic Unitaries are Efficiently Learnable), and the other will be released on arXiv within a month (Efficient Learning of Bosonic Gaussian Unitary Channels).