Learning $k$-body Hamiltonians via compressed sensing (2410.18928v2)

Abstract: We study the problem of learning a $k$-body Hamiltonian with $M$ unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision $\epsilon$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/\epsilon)$ up to logarithmic factors, where the error is quantified by the $\ell^p$-distance between Pauli coefficients. Our learning protocol uses only single-qubit control operations and a GHZ state initial state, is non-adaptive, is robust against SPAM errors, and performs well even if $M$ and $k$ are not precisely known in advance or if the Hamiltonian is not exactly $M$-sparse. Methods from the classical theory of compressed sensing are used for efficiently identifying the $M$ terms in the Hamiltonian from among all possible $k$-body Pauli operators. We also provide a lower bound on the total evolution time needed in this learning task, and we discuss the operational interpretations of the $\ell^1$ and $\ell^2$ error metrics. In contrast to most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions.

• The paper presents an efficient protocol that reconstructs k-body Hamiltonians via compressed sensing, achieving near-Heisenberg-limited performance.
• It leverages single-qubit controls and GHZ state initialization to robustly mitigate SPAM errors without prior knowledge of the number of interaction terms.
• Numerical results demonstrate that the method overcomes geometric locality constraints, promising advances in quantum simulation and error correction.

Overview of "Learning $k$-body Hamiltonians via Compressed Sensing"

The paper "Learning $k$-body Hamiltonians via Compressed Sensing" addresses the challenge of efficiently learning Hamiltonians with $M$ unknown $k$-local Pauli terms, which are not necessarily geometrically local. The authors propose a protocol utilizing compressed sensing techniques to achieve precision $\epsilon$ in learned Hamiltonian coefficients, quantified by the $\ell^p$ distance.

Key Contributions

The protocol provides a framework that relies on single-qubit control operations and GHZ state initializations, ensuring non-adaptiveness and robustness to SPAM errors. The method maintains performance even without prior precise knowledge of $M$ and $k$. This approach diverges from conventional requirements of geometric locality by leveraging compressed sensing to effectively identify sparse Hamiltonian terms from a potentially vast basis of Pauli operators.

Numerical Results and Analysis

One of the critical results noted is the near-Heisenberg-limited scaling, $\mathcal{O}(1/\epsilon)$, in the total evolution time. The protocol shows that the total number of terms, $M$, impacts evolution time as $\mathcal{O}(M^{1/2+1/p}/\epsilon)$, which highlights efficiency in handling cases previously found challenging in quantum systems.

A specific numerical analysis reveals that for $\Gamma$ samples, with $\Gamma$ on the order of $CM\log^2(M)\log(M\log(D))\log(D)$, the compressed sensing techniques ensure a robust reconstruction of Hamiltonian coefficients. Moreover, the authors establish a theoretical lower bound for this learning task, presenting evidence that geometric locality constraints, often deemed necessary, can be bypassed under this framework.

Implications and Future Directions

Practically, the advancement allows for broader applications in quantum simulation and quantum error correction, where understanding Hamiltonian structures facilitates optimized control strategies. The advancement beyond prior methods equips quantum computing researchers with more potent tools for tackling non-local interaction models.

Theoretically, the framework opens prospects for exploring sparse signal reconstruction in broader quantum systems. Future developments could potentially focus on reducing the gap in total evolution time between lower and upper bounds or extending the protocols to bosonic and fermionic systems.

Moreover, the implications on the operational interpretation of error metrics provide insights into how learned Hamiltonians can predict observable properties in quantum systems, emphasizing worst-case and average-case scenarios.

Conclusion

This paper presents a significant step in Hamiltonian learning through compressed sensing, breaking away from traditional locality constraints. By handling $k$-body interactions non-locally, the authors shift the benchmark for quantum systems' model learning, providing clarity and novel methodologies for both theoretical explorations and practical applications in quantum information science.