Approximating the operator norm of local Hamiltonians via few quantum states
Lars Becker, Joseph Slote, Alexander Volberg, Haonan Zhang
TL;DR
This work shows that the operator norm of a Hermitian, d-local Hamiltonian A on n qubits can be closely approximated, up to a constant depending only on d, by optimizing over a small, A-independent set of product states called a quantum norm design. The authors construct universal product-state designs X_n (and density designs Y_n) with |X_n| = |Y_n| = 6^n, and prove bounds ||A|| ≤ C(d) max_{ψ∈X_n} |<ψ|A|ψ>| with C(d) exponential in d but independent of n, extending prior homogeneous results to non-homogeneous d-local A. They further show that smaller designs are possible via sampling (|X_n| ≤ C(ε)(1+ε)^n) and that tensor powers of 1-qubit 2-designs also yield norm designs, highlighting a robust geometric structure. The paper connects these designs to classical inequality frameworks (Figiel, Bohnenblust–Hille) and analyzes random Hamiltonians, providing both upper and lower bounds on E||H(d,n)||, as well as large-deviation results via the BBoVH framework. It also discusses extensions to qudits and outlines open questions about regimes where d scales with n, offering a foundation for dimension-free norm estimation and potential applications in learning and verification of local Hamiltonians.
Abstract
Consider a Hermitian operator $A$ acting on a complex Hilbert space of dimension $2^n$. 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 $|\braket{ψ|A|ψ}|$ over a small collection $\mathbf{X}_n$ of product states $\ketψ\in (\mathbf{C}^{2})^{\otimes n}$. More precisely, we show that whenever $A$ is $d$-local, \textit{i.e.,} $°(A)\le d$, we have the following discretization-type inequality: \[ \|A\|\le C(d)\max_{ψ\in \mathbf{X}_n}|\braket{ψ|A|ψ}|. \] The constant $C(d)$ depends only on $d$. This collection $\mathbf{X}_n$ of $ψ$'s, termed a \emph{quantum norm design}, is independent of $A$, and consists of product states, and can have cardinality as small as $(1+\eps)^n$, which is essentially tight. Previously, norm designs were known only for homogeneous $d$-localHamiltonians $A$ \cite{L,BGKT,ACKK}, and for non-homogeneous $2$-local traceless $A$ \cite{BGKT}. Several other results, such as boundedness of Rademacher projections for all levels and estimates of operator norms of random Hamiltonians, are also given.