Learning fermionic linear optics with Heisenberg scaling and physical operations

We revisit the problem of learning fermionic linear optics (FLO), also known as fermionic Gaussian unitaries. Given black-box query access to an unknown FLO, previous proposals required $\widetilde{\mathcal{O}}(n^5 / \varepsilon^2)$ queries, where $n$ is the system size and $\varepsilon$ is the error in diamond distance. These algorithms also use unphysical operations (i.e., violating fermionic superselection rules) and/or $n$ auxiliary modes to prepare Choi states of the FLO. In this work, we establish efficient and experimentally friendly protocols that obey superselection, use minimal ancilla (at most $1$ extra mode), and exhibit improved dependence on both parameters $n$ and $\varepsilon$. For arbitrary (active) FLOs this algorithm makes at most $\widetilde{\mathcal{O}}(n^4 / \varepsilon)$ queries, while for number-conserving (passive) FLOs we show that $\mathcal{O}(n^3 / \varepsilon)$ queries suffice. The complexity of the active case can be further reduced to $\widetilde{\mathcal{O}}(n^3 / \varepsilon)$ at the cost of using $n$ ancilla. This marks the first FLO learning algorithm that attains Heisenberg scaling in precision. As a side result, we also demonstrate an improved copy complexity of $\widetilde{\mathcal{O}}(n η^2 / \varepsilon^2)$ for time-efficient state tomography of $η$-particle Slater determinants in $\varepsilon$ trace distance, which may be of independent interest.

💡 Research Summary

This paper tackles the problem of learning fermionic linear optics (FLO), also known as fermionic Gaussian unitaries, in a black‑box query model. Prior works required a query complexity of (\widetilde{O}(n^{5}/\varepsilon^{2})), relied on operations that violate fermionic superselection rules, and often needed (n) auxiliary modes to prepare Choi states. The authors present a suite of algorithms that dramatically improve on all three fronts while respecting physical constraints.

The central contribution is an “active FLO learner” that, for an arbitrary (i.e., possibly particle‑non‑conserving) FLO, uses at most (\widetilde{O}(n^{4}/\varepsilon)) queries and only a single ancillary mode. This achieves Heisenberg scaling in the precision parameter ((1/\varepsilon) dependence) and reduces the system‑size dependence from (n^{5}) to (n^{4}). The algorithm proceeds by a column‑by‑column strategy: each computational basis state (|j\rangle) is prepared, the unknown FLO is applied, and the resulting Gaussian state is tomographically reconstructed via efficient estimation of its one‑particle reduced density matrix (1‑RDM). By learning each column independently, errors remain isotropic and do not compound as in entry‑wise approaches.

For the restricted class of passive FLOs (those that conserve particle number), the authors exploit the underlying (U(n)) structure to further lower the query complexity to (O(n^{3}/\varepsilon)). The protocol for passive FLOs also uses at most one ancilla; in many realistic settings the ancilla can be omitted entirely, either because one can prepare superposition states ((|0\rangle+|1\rangle)/\sqrt{2}) or because the performance metric is restricted to parity‑preserving inputs, which are the only physically preparable states under superselection.

A notable technical innovation is the use of a single ancillary mode solely to detect the (\pm 1) relative phase between the even‑ and odd‑parity sectors of an active FLO. This phase is the only piece of information that cannot be accessed by parity‑preserving operations alone. When the ancilla is unavailable, the algorithm still learns the FLO up to this parity phase, which is often sufficient for practical benchmarking.

The paper also presents a “passive FLO learner” that directly learns the passive component without the first stage needed for active FLOs, thereby achieving the (O(n^{3}/\varepsilon)) bound. Moreover, if one further restricts attention to fixed‑particle‑number subspaces (e.g., the (\eta)-particle sector (\wedge^{\eta}\mathbb{C}^{n})), the query complexity drops to (O(n^{2}\eta/\varepsilon)) and no ancilla is required. This result extends naturally to bosonic linear optics within fixed‑number sectors because of the analogous (U(n)) representation.

A crucial subroutine for both learners is efficient tomography of fermionic Gaussian states. For general pure Gaussian states the best known copy complexity was (\widetilde{O}(n^{3}/\varepsilon^{2})). The authors improve this for Slater determinants (particle‑number‑conserving Gaussian states) to (\widetilde{O}(n\eta^{2}/\varepsilon^{2})) copies, and for the single‑particle case ((\eta=1)) to the optimal (O(n/\varepsilon^{2})). The method estimates the covariance matrix by measuring the 1‑RDM using only passive FLO gates and parity‑preserving measurements, thereby staying within the physical constraints of the problem.

The analysis combines concentration inequalities, a bootstrap framework for error propagation, and careful perturbative arguments for estimating the “shadow” of the unitary beyond number symmetry. For the active case, the algorithm is split into two stages: (i) learning the action on the vacuum (which reveals the parity‑mixing component) and (ii) learning the passive remainder using the techniques from the passive case. Each stage incurs an error of order (\varepsilon/n), ensuring that the total diamond‑distance error remains bounded by (\varepsilon).

If one is willing to employ (n) ancilla modes, the authors show that the query complexity can be reduced further to (\widetilde{O}(n^{3}/\varepsilon)) by reducing the problem to tomography of Choi states, an approach reminiscent of earlier works but now made fully physical.

In summary, the paper delivers the first FLO learning algorithms that simultaneously achieve Heisenberg‑optimal precision scaling, improved polynomial dependence on system size, and strict adherence to fermionic superselection rules with minimal ancillary resources. These results have immediate implications for benchmarking free‑fermion quantum processors, certifying quantum advantage protocols based on matchgate circuits, and performing efficient state tomography in quantum chemistry applications where Slater determinants dominate. Future directions include robustness to noise, extensions to interacting fermionic models, and simultaneous learning of multiple symmetry‑restricted channels.