Learning fermionic linear optics with Heisenberg scaling and physical operations
Aria Christensen, Andrew Zhao
TL;DR
This work delivers efficient, physically realistic protocols for learning fermionic linear optics (FLO) with Heisenberg-scaling precision while respecting fermionic superselection and using minimal ancilla. It distinguishes active FLOs (Bogoliubov transformations in O(2n)) from passive FLOs (U(n)) and achieves $ ilde{O}(n^4/\varepsilon)$ queries for active FLOs and $O(n^3/\varepsilon)$ for passive FLOs, with the latter optionally reduced to $\tilde{O}(n^3/\varepsilon)$ using $n$ ancilla via Choi-state tomography. A core technical advance is the use of fermionic shadows (U$(n)$-shadows and SO$(2n)$-shadows) to perform efficient, isotropic state tomography of Slater determinants, enabling a copy complexity of $\tilde{O}(n\eta^2/\varepsilon^2)$ for $\eta$-particle Slaters and tighter bounds in the $\eta=1$ case. The paper also develops a bootstrap framework that turns a base constant-error tomography into a diamond-distance accurate FLO, achieving Heisenberg scaling in precision and providing improved methods for Gaussian-state tomography and phase estimation under superselection. Collectively, these results advance practical FLO learning and Gaussian-state tomography with parity-conserving, resource-efficient protocols that are compatible with experimental limitations.
Abstract
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.