Approximating the operator norm of local Hamiltonians via few quantum states
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910114157428736 |
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| author | Becker, Lars Slote, Joseph Volberg, Alexander Zhang, Haonan |
| author_facet | Becker, Lars Slote, Joseph Volberg, Alexander Zhang, Haonan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximating the operator norm of local Hamiltonians via few quantum states Becker, Lars Slote, Joseph Volberg, Alexander Zhang, Haonan Quantum Physics Classical Analysis and ODEs Functional Analysis 41A17, 41A63, 42B05, 32A08 F.2.2 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. |
| title | Approximating the operator norm of local Hamiltonians via few quantum states |
| topic | Quantum Physics Classical Analysis and ODEs Functional Analysis 41A17, 41A63, 42B05, 32A08 F.2.2 |
| url | https://arxiv.org/abs/2509.11979 |