Learning fermionic linear optics with Heisenberg scaling and physical operations

TL;DR

This paper introduces efficient, physically feasible algorithms for learning fermionic linear optics with Heisenberg scaling, improving query complexity and respecting superselection rules, advancing quantum tomography and simulation.

Contribution

It presents the first FLO learning algorithms achieving Heisenberg scaling, with minimal ancilla and superselection compliance, improving upon prior query complexities.

Findings

Active FLO learning requires $ ilde{O}(n^4/\varepsilon)$ queries.

Passive FLO learning requires $O(n^3/\varepsilon)$ queries.

New tomography method for Slater determinants with $\tilde{O}(n \eta^2/\varepsilon^2)$ complexity.

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 $\widetilde{\mathcal{O}}(n \eta^2 / \varepsilon^2)$ for time-efficient state tomography of $\eta$-particle Slater determinants in $\varepsilon$ trace distance, which may be of independent interest.