Certifying and learning local quantum Hamiltonians

Abstract: In this work, we study the problems of certifying and learning quantum $k$-local Hamiltonians, for a constant $k$. Our main contributions are as follows: - Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only $O(1/\varepsilon)$ evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of $Ω(1/\varepsilon)$. To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis. - Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature. - Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022).

• The paper achieves an optimal O(1/ε) time evolution protocol to certify whether a k-local Hamiltonian deviates from a reference Hamiltonian.
• It leverages quantum Bonami hypercontractivity to robustly bound eigenvalue differences, ensuring practical sample efficiency in learning Gibbs states.
• The study circumvents exponential lower bounds in thermal state estimation by applying classical shadow tomography to achieve trace-distance accuracy with polynomial sample complexity.

Certifying and Learning Local Quantum Hamiltonians

Introduction and Motivation

The precise characterization of quantum dynamics is central to the realization of robust quantum technologies. Two fundamental information-theoretic tasks in this context are Hamiltonian learning—reconstructing a local quantum Hamiltonian from accessible data—and Hamiltonian certification—determining whether an experimental device accurately implements a claimed local Hamiltonian, or if its actual generator is far from the nominal one. For $n$-qubit systems described by $k$-local Hamiltonians, these tasks have significant implications both for quantum simulation and verification, as well as for understanding thermodynamic properties via Gibbs states.

This paper addresses several open questions in the certification and learning of local quantum Hamiltonians, achieving multiple optimality results in terms of both sample and time complexity. The analysis applies to both the time-evolution (dynamics) and thermal-state (Gibbs state) access paradigms and leverages functional inequalities from quantum Fourier analysis, notably the quantum Bonami hypercontractivity lemma.

Optimal Certification from Time Evolution

The central technical contribution is an optimal and robust algorithm for certifying whether an unknown $k$-local Hamiltonian $H$ is close to, or $\varepsilon$-far in normalized Frobenius norm from, a reference Hamiltonian $H_0$, with access to their time-evolution operators.

Time Complexity Characterization

The authors establish that $O(1/\varepsilon)$ total evolution time suffices—and is also necessary up to constants—for certification in normalized Frobenius norm. This Heisenberg scaling is shown by designing a certification protocol that samples (approximations to) $\operatorname{Tr}(e^{-it(H-H_0)})$ ("Bell sampling") at random times $t\in[0,2/\varepsilon]$. The analysis links the probability of measuring the identity outcome to the empirical distribution of eigenvalue gaps in $H-H_0$.

Analysis via Hypercontractive Inequalities

A key technical component is the novel use of the Bonami hypercontractivity lemma for $k$0-local Hamiltonians. This result bounds the higher-order moments of eigenvalue differences, providing a lower bound on the fraction of eigenvalue pairs at separation at least $k$1 when $k$2 and thus guaranteeing that the Bell sampling probability deviates from unity by $k$3 in that regime. The protocol is thus both evolution-time optimal and robust to constant levels of SPAM errors and only requires efficient single-copy Clifford operations.

Relation to Prior and Concurrent Work

Previous results for similar property testing problems yielded at best an $k$4 time-evolution upper bound for locality testing [kallaugher2025hamiltonian], and $k$5 protocols for $k$6-local certification under sparsity assumptions [gao2025quantum]. This paper’s approach bypasses the need for sparsity or access to inverse time evolution and achieves the optimal $k$7 evolution time for all constant $k$8. The result is a quadratic improvement for locality certification and the first optimal algorithm for any Hamiltonian property testing task in this model (2603.29809).

Sample-Optimal Learning and Certification of Gibbs States

The paper also addresses the problem of learning and certifying the Gibbs states of local Hamiltonians (i.e., thermal states $k$9) with sample complexity polynomial in all relevant parameters, including inverse temperature $k$0, system size $k$1, and error parameter $k$2.

Learning: Circumventing Exponential Lower Bounds

Prior work established that any protocol which attempts to reconstruct the underlying Hamiltonian from thermal-state data incurs an exponential sample complexity in $k$3 [haah2022optimal]. The authors circumvent this by directly learning the Gibbs state rather than the Hamiltonian itself, using an $k$4-covering net over $k$5-local Hamiltonians and employing classical shadow tomography to estimate expectation values of all $k$6-local Pauli observables. The final output is the net element minimizing maximal deviation in energy differences, guaranteeing trace distance error $k$7 using only $k$8 samples. This breaks the exponential-in-$k$9 barrier if only the thermal state (not the Hamiltonian) is targeted.

Certification: Efficient and Practical Sample Complexity

For certification—deciding if an experimentally prepared Gibbs state matches a theoretical one (up to $H$0 in trace norm)—the protocol uses classical shadows to estimate all $H$1-local Pauli expectation values up to additive error $H$2. The total sample requirement is again $H$3, and the protocol is efficient in Clifford operations and classical post-processing. Notably, the certification algorithm is both sample- and time-efficient, resolving a question posed by Anshu [anshu2022some].

Technical and Practical Implications

The tools and results developed yield several key advances:

• Time-Optimality: Certification from dynamics achieves the Heisenberg scaling lower bound in total evolution time for all constant $H$4, matching theory and experiment.
• Sample-Optimal and Temperature-Efficient Gibbs State Tomography: The sample complexity for learning and certification is independent of the spectral gap or temperature, up to polynomial dependence, highlighting a separation between learning Hamiltonians and learning thermal states.
• Robustness and Feasibility: All procedures are robust to physically realistic imperfections (including SPAM errors) and require only minimal quantum memory.

The techniques also illustrate how advanced functional analysis (hypercontractivity, Paley-Zygmund inequalities) can be leveraged to obtain sharp, nonasymptotic bounds on quantum process discrimination.

Open Problems and Future Directions

The paper identifies several open problems of theoretical and practical interest:

• Certifying Non-Local Hamiltonians: Extending time-optimal certification to arbitrary Hamiltonians without locality assumptions remains unresolved in the absence of inverse time-evolution control.
• Sample and Time-Optimal Learning of Gibbs States: Achieving both time and sample optimality for Gibbs state learning, as done here for certification, is still open.
• Certification via Strictly Local Probes: Understanding the minimal experimental resources (probe size, measurement class) required for optimal certification is critical for bridging theory and experiment.
• Optimal Complexity of Gibbs State Certification: Establishing lower bounds for certification protocols and resolving the tightness of the sample complexity with classical analogs warrants further investigation.
• Extension to Quantum Channels and Lindbladian Dynamics: Adapting these techniques for certifying open-system evolutions or noise channels is a promising research avenue with significant implications for quantum control and error mitigation.

Conclusion

This paper demonstrates that both certification from dynamics of $H$5-local Hamiltonians and learning/certification of their Gibbs states admit optimal algorithms in natural complexity metrics, subject to information-theoretic lower bounds. The results delineate clear efficiency separations between reconstructing a full Hamiltonian versus certifying or learning a thermal state, and provide robust, practically viable protocols for next-generation quantum devices. The methods highlight the power of Fourier-analytic and hypercontractive inequalities in quantum algorithmic design and open new directions for adaptive, resource-aware quantum verification (2603.29809).