Mildly-Interacting Fermionic Unitaries are Efficiently Learnable

• URL: http://arxiv.org/abs/2504.11318v1
• Date: Tue, 15 Apr 2025 15:59:32 GMT
• Title: Mildly-Interacting Fermionic Unitaries are Efficiently Learnable
• Authors: Vishnu Iyer,
• Abstract summary: We show that an algorithm can learn near-Gaussian fermionic unitaries of Gaussian dimension at least $2n - O(t)$ in time.<n>We also prove structural results about near-Gaussian fermionic unitaries that are likely to be of independent interest.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent work has shown that one can efficiently learn fermionic Gaussian unitaries, also commonly known as nearest-neighbor matchcircuits or non-interacting fermionic unitaries. However, one could ask a similar question about unitaries that are near Gaussian: for example, unitaries prepared with a small number of non-Gaussian circuit elements. These operators find significance in quantum chemistry and many-body physics, yet no algorithm exists to learn them. We give the first such result by devising an algorithm which makes queries to a $n$-mode fermionic unitary $U$ prepared by at most $O(t)$ non-Gaussian gates and returns a circuit approximating $U$ to diamond distance $\varepsilon$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. This resolves a central open question of Mele and Herasymenko under the strongest distance metric. In fact, our algorithm is much more general: we define a property of unitary Gaussianity known as unitary Gaussian dimension and show that our algorithm can learn $n$-mode unitaries of Gaussian dimension at least $2n - O(t)$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. Indeed, this class subsumes unitaries prepared by at most $O(t)$ non-Gaussian gates but also includes several unitaries that require up to $2^{O(t)}$ non-Gaussian gates to construct. In addition, we give a $\textrm{poly}(n,1/\varepsilon)$-time algorithm to distinguish whether an $n$-mode unitary is of Gaussian dimension at least $k$ or $\varepsilon$-far from all such unitaries in Frobenius distance, promised that one is the case. Along the way, we prove structural results about near-Gaussian fermionic unitaries that are likely to be of independent interest.

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