Approximating the operator norm of local Hamiltonians via few quantum states

Consider a Hermitian operator A acting on a complex Hilbert space of dimension 2ⁿ. We show that when A has small degree in the Pauli expansion, or in other words, A is a local n-qubit Hamiltonian, its operator norm can be approximated independently of n by maximizing |ψ|A|ψ| over a small collection Xₙ of product states ψ∈ (C²)⊗ n. More precisely, we show that whenever A is d-local, i.e., °(A)≤ d, we have the following discretization-type inequality: \[ \|A\|≤ C(d)maxψ∈ X_n|{ψ|A|ψ}|. \] The constant $C(d)$ depends only on d. This collection Xₙ of $ψ$'s, termed a quantum norm design, is independent of A, and consists of product states, and can have cardinality as small as (1+)ⁿ, which is essentially tight. Previously, norm designs were known only for homogeneous d-localHamiltonians A {L,BGKT,ACKK}, and for non-homogeneous $2$-local traceless A {BGKT}. Several other results, such as boundedness of Rademacher projections for all levels and estimates of operator norms of random Hamiltonians, are also given.