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

• URL: http://arxiv.org/abs/2608.01424v1
• Date: Sun, 02 Aug 2026 18:12:46 GMT
• Title: Dense Hamiltonians at the Parseval Limit: The Noncommutative BH Constant is Exponential and the Quantum FEI Conjecture is False
• Abstract summary: For each $dgeq 1$ we construct a norm-1 Hermitian operator whose Pauli expansion contains $N(d)=exp((d2))$ terms.<n>This construction implies the noncommutative Bohnenblust-Hille (BH) constant satisfies $mathrmBH_M(d)geqexp((d))$.<n>Our lower bound also diagonalally separates $mathrmBH_M(d)$ from the hypercube BH constant $mathrmBH
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For each $d\geq 1$ we construct a norm-1 Hermitian operator whose Pauli expansion contains $N(d)=\exp(Ω(d^2))$ 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 $Ω(1/\sqrt{N(d)})$, then $N(d)\leq \exp(\widetilde{O}(d^{1.5}))$. This construction implies the noncommutative Bohnenblust--Hille (BH) constant satisfies $\mathrm{BH}_{M_2}(d)\geq\exp(Ω(d))$. 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).

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