Quantum Bohnenblust–Hille inequality for low-degree operators

Let $M_2(\mathbb{C})$ denote the algebra of $2\times 2$ complex matrices, and let $\sigma_s$ be the Pauli-string basis indexed by $s\in\{0,1,2,3\}^n$. For a string $s$, let $|s|$ be the number of components different from $0$. An operator $A\in M_2(\mathbb{C})^{\otimes n}$ has degree at most $d$ when

$$A=\sum_{s\in\{0,1,2,3\}^n:|s|\le d}\widehat{A}_s\sigma_s.$$

Quantum Bohnenblust–Hille conjecture. Fix $d\ge 1$. There exists $C_d>0$, depending only on $d$, such that for all $n\ge 1$ and every degree-at-most-$d$ operator $A\in M_2(\mathbb{C})^{\otimes n}$,

$$\left(\sum_{s\in\{0,1,2,3\}^n:|s|\le d}|\widehat{A}_s|^{\frac{2d}{d+1}}\right)^{\frac{d+1}{2d}}\le C_d\|A\|.$$

This is proposed as a quantum analogue of the Bohnenblust–Hille inequality and is intended to support learning results for low-degree quantum Boolean functions. The source does not provide evidence resolving the conjecture.