Hamiltonian Property Testing

TL;DR

The paper establishes a fundamental separation between testing and learning for quantum Hamiltonians. By contrasting worst-case operator-norm locality testing—provably hard with exponential query/time requirements—with average-case Frobenius-distance locality testing, which admits efficient, stabilizer-based randomized measurements, the authors show that many Hamiltonian properties can be tested efficiently without full Hamiltonian reconstruction. The work introduces a general, scalable testing framework that applies to various Pauli-subset properties, and demonstrates both tight lower bounds and practical upper bounds, including tolerant and multi-property extensions. Overall, this work initiates Hamiltonian property testing as a distinct, resource-efficient paradigm in quantum information, with potential impact on quantum benchmarking, sensing, and learning pipelines that incorporate locality constraints. The results provide concrete guidance on the choice of distance measures and input states/measurements, clarifying when testing is feasible and when learning remains necessary.

Abstract

Locality is a fundamental feature of many physical time evolutions. Assumptions on locality and related structural properties also underlie recently proposed procedures for learning an unknown Hamiltonian from access to the induced time evolution. However, no protocols to rigorously test whether an unknown Hamiltonian is local were known. We investigate Hamiltonian locality testing as a property testing problem, where the task is to determine whether an unknown $n$-qubit Hamiltonian $H$ is $k$-local or $\varepsilon$-far from all $k$-local Hamiltonians, given access to the time evolution along $H$. First, we emphasize the importance of the chosen distance measure: With respect to the operator norm, a worst-case distance measure, incoherent quantum locality testers require $\tildeΩ(2^n)$ many time evolution queries and an expected total evolution time of $\tildeΩ(2^n / \varepsilon)$, and even coherent testers need $Ω(2^{n/2})$ many queries and $Ω(2^{n/2}/\varepsilon)$ total evolution time. In contrast, when distances are measured according to the normalized Frobenius norm, corresponding to an average-case distance, we give a sample-, time-, and computationally efficient incoherent Hamiltonian locality testing algorithm based on randomized measurements. In fact, our procedure can be used to simultaneously test a wide class of Hamiltonian properties beyond locality. Finally, we prove that learning a general Hamiltonian remains exponentially hard with this average-case distance, thereby establishing an exponential separation between Hamiltonian testing and learning. Our work initiates the study of property testing for quantum Hamiltonians, demonstrating that a broad class of Hamiltonian properties is efficiently testable even with limited quantum capabilities, and positioning Hamiltonian testing as an independent area of research alongside Hamiltonian learning.

Figures (3)

• Figure 1: Illustration of a non-adaptive incoherent strategy for learning/testing properties of a Hamiltonian $H$ from its time evolution channel $\mathcal{U}_t(\cdot)= \mathrm{e}^{-\mathrm{i}tH} (\cdot) \mathrm{e}^{\mathrm{i}tH}$. It is called ancilla-free if the auxiliary systems have dimension $1$, otherwise it is called ancilla-assisted. The classical computer processes the observations $(i_1, \dots, i_N)$ to distinguish between two hypotheses $H_0/H_1$ (in testing) or to produce an approximate Hamiltonian $\hat{H}$ (in learning).
• Figure 2: Illustration of an adaptive incoherent strategy for learning properties of a Hamiltonian $H$ from its time evolution channel $\mathcal{U}_t(\cdot)= \mathrm{e}^{-\mathrm{i}tH}(\cdot) \mathrm{e}^{\mathrm{i}tH}$. It is called ancilla-free if the auxiliary systems have dimension $1$, otherwise it is called ancilla-assisted. The classical computer processes the observations $(i_1, \dots, i_N)$ to distinguish between two hypotheses $H_0/H_1$ (in testing) or to produce an approximate Hamiltonian $\hat{H}$ (in learning).
• Figure 3: Illustration of a coherent strategy for learning properties of a Hamiltonian $H$ from its time evolution channel $\mathcal{U}_t(\cdot)= \mathrm{e}^{-\mathrm{i}tH} (\cdot) \mathrm{e}^{\mathrm{i}tH}$. The classical computer processes the observation $i$ to distinguish between two hypotheses $H_0/H_1$ (in testing) or to produce an approximate Hamiltonian $\hat{H}$ (in learning).

Theorems & Definitions (45)

• Definition 1.1: Hamiltonian locality testing
• Theorem 1.2: Hardness of Hamiltonian locality testing with respect to the operator norm -- Informal
• Theorem 1.3: Efficient Hamiltonian locality testing with respect to normalized Frobenius norm -- Informal
• Theorem 1.4: Hardness of Hamiltonian learning with respect to normalized Frobenius norm -- Informal
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3: Pauli group, MUB and $2$-design
• proof
• Lemma 2.4
• Remark 2.6