We consider the problem of learning low-degree quantum objects up to $\varepsilon$-error in $\ell_2$-distance. We show the following results: $(i)$ unknown $n$-qubit degree-$d$ (in the Pauli basis) quantum channels and unitaries can be learned using $O(1/\varepsilon^d)$ queries (independent of $n$), $(ii)$ polynomials $p:\{-1,1\}^n\rightarrow [-1,1]$ arising from $d$-query quantum algorithms can be classically learned from $O((1/\varepsilon)^d\cdot \log n)$ many random examples $(x,p(x))$ (which implies learnability even for $d=O(\log n)$), and $(iii)$ degree-$d$ polynomials $p:\{-1,1\}^n\to [-1,1]$ can be learned through $O(1/\varepsilon^d)$ queries to a quantum unitary $U_p$ that block-encodes $p$. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded~polynomials.

We prove a tight and close-to-optimal lower bound on the effectiveness of local quantum measurements (without classical communication) at discriminating any two bipartite quantum states. Our result implies, for example, that any two orthogonal quantum states of a $n_{A}\times n_{B}$ bipartite quantum system can be discriminated via local measurements with an error probability no larger than $\frac {1}2 \left ({1 - \frac {1}{c \min \{n_{A}, n_{B}\}} }\right)$ , where $1\leq c\leq 2\sqrt {2}$ is a universal constant, and our bound scales provably optimally with the local dimensions $n_{A},n_{B}$ . Mathematically, this is achieved by showing that the distinguishability norm $\|\cdot \|_{ \mathrm {LO}}$ associated with local measurements satisfies that $\|\cdot \|_{1}\leq 2\sqrt {2} \min \{n_{A},n_{B}\} \|\cdot \|_{ \mathrm {LO}}$ , where $\|\cdot \|_{1}$ is the trace norm.

We prove a characterization of $t$-query quantum algorithms in terms of the unit ball of a space of degree-$(2t)$ polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. [``Polynomials, Quantum Query Complexity, and Grothendieck's Inequality,” in Proceedings of the 31st Conference on Computational Complexity, CCC 2016, Schloss Dagstuh, 2016, pp. 25:1--25:19]. Our proof is based on a fundamental result of Christensen and Sinclair [ J. Funct. Anal., 72 (1987), pp. 151--181] that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.