Improved Hamiltonian learning and sparsity testing through Bell sampling

Abstract: We consider the problem of learning an $M$-sparse Hamiltonian and the related problem of Hamiltonian sparsity testing. Through a detailed analysis of Bell sampling, we reduce the total evolution time required by the state-of-the-art algorithm for $M$-sparse Hamiltonian learning to $\widetilde{\mathcal{O}}(M/\epsilon)$, where $\epsilon$ denotes the $\ell^{\infty}$ error, achieving an improvement by a factor of $M$ (ignoring the logarithmic factor) while only requiring access to forward time-evolution. We then establish a connection between Hamiltonian learning and Hamiltonian sparsity testing through Bell sampling, which enables us to propose a Hamiltonian sparsity testing with state-of-the-art total evolution time scaling.

• The paper presents an algorithm that uses Bell sampling to significantly reduce the total evolution time for learning M-sparse Hamiltonians without requiring time-reversal dynamics.
• It provides tight probabilistic bounds and introduces emptiness testing algorithms employing Chernoff bounds to efficiently distinguish sparse from non-sparse Hamiltonians.
• The work establishes a strong connection between Hamiltonian learning and property testing, offering practical insights for experimental quantum systems and control tasks.

Improved Hamiltonian Learning and Sparsity Testing via Bell Sampling

Overview

This paper presents a refined analysis of Hamiltonian learning and sparsity testing in quantum systems, leveraging Bell sampling to achieve improved scaling in total evolution time. The authors focus on the regime of $M$-sparse Hamiltonians, where the Hamiltonian contains at most $M$ nonzero Pauli terms, and address both the learning of Hamiltonian coefficients and the property testing problem of determining sparsity. The main technical contribution is a reduction in the total evolution time required for learning, matching the best known scaling without requiring time-reversal dynamics, and establishing a rigorous connection between learning and property testing in the quantum setting.

Bell Sampling: Foundations and Bounds

Bell sampling is utilized to extract information about the Pauli decomposition of a unitary operator $U = \sum_x U_x \sigma_x$ by measuring in the Bell basis after applying $U$ to a maximally entangled state. The probability of sampling a particular Pauli operator $\sigma_x$ is $|U_x|^2$, which enables direct access to the distribution of Pauli coefficients in the time-evolution operator $U(t) = e^{-iHt}$.

The paper provides tight bounds on the probability of sampling the identity operator after time evolution under a traceless Hamiltonian $H$ with bounded spectral norm. For small $t$, the probability is close to $1 - \|H\|_F^2 t^2$, where $\|H\|_F$ is the normalized Frobenius norm of the Hamiltonian. These bounds are critical for designing efficient sampling protocols for both learning and testing tasks.

Emptiness Testing Algorithms

The authors formalize the problem of emptiness testing: determining whether a Hamiltonian is zero or has norm exceeding a threshold. Both intolerant (exact) and tolerant (approximate) versions are addressed. The algorithms rely on repeated Bell sampling and Chernoff bounds to distinguish between the two cases with high probability, achieving sample complexity and total evolution time scaling as $\mathcal{O}(L^2 \log(1/\delta)/\epsilon^2)$ and $\mathcal{O}(L \log(1/\delta)/\epsilon^2)$, respectively, where $L$ is the spectral norm bound and $\epsilon$ is the threshold.

The tolerant version generalizes to distinguishing between Hamiltonians with norm below $\epsilon_1$ and above $\epsilon_2$, with sample complexity scaling as $$\mathcal{O}(L^2 \log(1/\delta)/[\epsilon_2^2(1 - \epsilon_1^2/\epsilon_2^2)^3])$$.

Sparse Hamiltonian Learning: Algorithmic Improvements

The core learning algorithm is hierarchical, partitioning Pauli coefficients into buckets according to their magnitude and iteratively learning the support and coefficients. The support is identified via Bell sampling, and coefficients are estimated using Hamiltonian reshaping and robust frequency estimation. The reshaping procedure isolates individual Pauli terms through randomized Pauli twirling, enabling effective single-term evolution.

A key technical improvement is the refined analysis of the average-case sampling complexity. The previous worst-case bound of $\widetilde{\mathcal{O}}(M^2/\epsilon)$ for total evolution time is reduced to $\widetilde{\mathcal{O}}(M/\epsilon)$ by showing that the expected number of non-identity Pauli samples per bucket is $\mathcal{O}(M \log M)$, not $\mathcal{O}(M^2 \log M)$. This matches the scaling of algorithms that require time-reversal dynamics, but only uses forward evolution, which is experimentally more feasible.

Sparsity Testing via Learning

The paper establishes a rigorous connection between Hamiltonian learning and sparsity testing, analogous to classical property testing. The approach is to learn an $M$-sparse approximation $\hat{H}$ to the unknown Hamiltonian $H$, then estimate the norm $\|H - \hat{H}\|_F$ via Bell sampling after Trotterized evolution under $H - \hat{H}$. If $H$ is $M$-sparse, the norm is small; if $H$ is far from $M$-sparse, the norm is bounded away from zero.

The total evolution time for sparsity testing is shown to be $$\widetilde{\mathcal{O}}(LM^{1.5}/\epsilon + LM/\epsilon^2)$$, improving upon previous results by a factor of $\min\{1/\epsilon^2, \sqrt{M}/\epsilon\}$ in the non-tolerant setting. The analysis accounts for Trotterization errors and provides explicit bounds on the required number of steps.

Theoretical and Practical Implications

The results demonstrate that Bell sampling enables efficient Hamiltonian learning and property testing with Heisenberg-limited scaling in total evolution time, without the need for time-reversal dynamics. The connection between learning and testing is made explicit, suggesting that advances in quantum learning algorithms can directly translate to improved property testing protocols.

Practically, these algorithms are well-suited for experimental quantum systems where only forward time evolution is accessible. The reduction in total evolution time is significant for large-scale systems, where $M$ may scale polynomially with the number of qubits. The techniques for Hamiltonian reshaping and robust frequency estimation are broadly applicable to other quantum learning and control tasks.

Future Directions

The framework established in this paper opens several avenues for further research:

• Extending the connection between learning and property testing to other Hamiltonian properties, such as locality or symmetry.
• Investigating the robustness of the algorithms under experimental noise and imperfect control.
• Generalizing the Bell sampling approach to open quantum systems and Lindbladian dynamics.
• Exploring adaptive sampling strategies to further reduce sample complexity in practice.

Conclusion

This work refines the theoretical foundations and practical algorithms for Hamiltonian learning and sparsity testing in quantum systems, achieving improved scaling in total evolution time via Bell sampling. The explicit connection between learning and property testing provides a unified framework for quantum characterization tasks, with direct implications for experimental quantum information science and future algorithmic developments.