Learning fermionic linear optics with Heisenberg scaling and physical operations

2026-02-04

AI Summary

This work addresses the efficient learning of unknown fermionic linear optical (FLO) transformations under the constraints imposed by fermionic superselection rules and physical realizability, significantly reducing dependence on system size and target precision. The authors propose a black-box learning protocol requiring at most one auxiliary mode, which integrates techniques from fermionic Gaussian unitary learning, Choi state construction, and Slater determinant state tomography. This approach yields the first FLO learning algorithm achieving Heisenberg-limited precision. The method reduces the query complexity for active FLO learning to ~O(n⁴/ε) and to O(n³/ε) in the passive setting; with n auxiliary modes, it can be further optimized to ~O(n³/ε). Additionally, the protocol improves the copy complexity of Slater determinant state tomography.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Quantum Machine Learning
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• fermionic linear optics
• Heisenberg scaling
• superselection rules
• query complexity
• Gaussian unitaries

Innovation

• fermionic linear optics
• Heisenberg scaling
• superselection rule
• query complexity
• physical operations