Antonio A. Mele

The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group—a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with $k$-fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of $k$, constrained by the number of qubits $n$. In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary $n$ and $k$. We also introduce an alternative and easy-to-manipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations— which generate the unitary group commutant— and at most three other operators. Additionally, we develop a \emph{graphical calculus} allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all \emph{measurable} magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension. This submission merges two of our recent works: one presenting a complete theory of the Clifford commutant with applications, and one focused on showcasing a major application to state $k$-design convergence.

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).