Dense Hamiltonians at the Parseval Limit: The Noncommutative BH Constant is Exponential and the Quantum FEI Conjecture is False

Abstract

For each $d\geq 1$ we construct a norm-1 Hermitian operator whose Pauli expansion contains $N(d)=\exp(\Omega(d^2))$ terms, each of degree $d$ and magnitude $1/\sqrt{N(d)}$ - the largest magnitude permitted by Parseval's identity. For comparison, if a bounded diagonal operator (or equivalently, a bounded degree-$d$ function on the Boolean cube) has $N(d)$ Pauli coefficients, all of magnitude $\Omega(1/\sqrt{N(d)})$, then $N(d)\leq \exp(\widetilde{O}(d^{1.5}))$. This construction implies the noncommutative Bohnenblust--Hille (BH) constant satisfies $\mathrm{BH}_{M_2}(d)\geq\exp(\Omega(d))$. Together with the upper bounds proved in prior work, this settles the asymptotic growth of $\mathrm{BH}_{M_2}(d)$ as exponential. Our lower bound also asymptotically separates $\mathrm{BH}_{M_2}(d)$ from the (classical) hypercube BH constant $\mathrm{BH}_{\{\pm 1\}}(d)$, which in turn is known to be subexponential: $\mathrm{BH}_{\{\pm 1\}}(d)\leq C^{\sqrt{d \log d}}$. Our Hamiltonians are also unitary and thus quantum Boolean functions in the sense of Montanaro and Osborne (2010). As such they refute the quantum Fourier Entropy-Influence conjecture of Bu et al. (2024), a natural generalization of the classical Fourier Entropy-Influence conjecture due to Friedgut and Kalai (1996).