Francisco Escudero Gutiérrez

We consider the problem of learning low-degree quantum objects up to ε-error in l_2-distance. We show the following results: (I) unknown n-qubit degree-d (in the Pauli basis) quantum channels and unitaries can be learned using O(1/ε^d) queries (which is independent of n), (II) polynomials p:-1,1^n -> [-1,1] arising from d-query quantum algorithms can be learned from O((1/ε)^d log n) many random examples (x,p(x)) (which implies learnability even for d=O(log n)), and (III) degree-d polynomials p:-1,1^n -> [-1,1] can be learned through O(1/ε^d) queries to a quantum unitary Up that block-encodes p. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded polynomials.

We give a new presentation of the main result of Arunachalam, Briët and Palazuelos (SICOMP'19) and show that quantum query algorithms are characterized by a new class of polynomials which we call Fourier completely bounded polynomials. We conjecture that all such polynomials have an influential variable. This conjecture is weaker than the famous Aaronson-Ambainis (AA) conjecture (Theory of Computing'14), but has the same implications for classical simulation of quantum query algorithms.
We prove a new case of the AA conjecture by showing that it holds for homogeneous Fourier completely bounded polynomials.
This implies that if the output of d-query quantum algorithm is a homogeneous polynomial p of degree 2d, then it has a variable with influence at least Var[p]^2.
In addition, we give an alternative proof of the results of Bansal, Sinha and de Wolf (CCC'22 and QIP'23) showing that block-multilinear completely bounded polynomials have influential variables. Our proof is simpler, obtains better constants and does not use randomness.