Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables

Daniel McNulty; Susane Calegari; Michał Oszmaniec

doi:10.22331/q-2025-11-17-1914

Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables

Daniel McNulty^1,2, Susane Calegari², and Michał Oszmaniec^2,3

¹Dipartimento di Fisica, Università di Bari, 70126 Bari, Italy
²Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
³NASK National Research Institute, ul. Kolska 12, 01-045 Warszawa, Poland

Published:	2025-11-17, volume 9, page 1914
Editor:	Ion Nechita
Eprint:	arXiv:2402.19349v2
Doi:	https://doi.org/10.22331/q-2025-11-17-1914
Citation:	Quantum 9, 1914 (2025).

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Abstract

An important class of fermionic observables, relevant in tasks such as fermionic partial tomography and estimating energy levels of chemical Hamiltonians, are the binary measurements obtained from the product of anti-commuting Majorana operators. In this work, we investigate efficient estimation strategies of these observables based on a joint measurement which, after classical post-processing, yields all sufficiently unsharp (noisy) Majorana observables of even-degree. By exploiting the symmetry properties of the Majorana observables, as described by the braid group, we show that the incompatibility robustness, i.e., the minimal classical noise necessary for joint measurability, relates to the spectral properties of the Sachdev-Ye-Kitaev (SYK) model. In particular, we show that for an $n$ mode fermionic system, the incompatibility robustness of all degree-$2k$ Majorana observables satisfies $\Theta(n^{-k/2})$ for $k\leq 5$. Furthermore, we present a joint measurement scheme achieving the asymptotically optimal noise, implemented by a small number of fermionic Gaussian unitaries and sampling from the set of all Majorana monomials. Our joint measurement, which can be performed via a randomization over projective measurements, provides rigorous performance guarantees for estimating fermionic observables comparable with fermionic classical shadows.

Featured image: Circuit implementation of the joint measurement, together with a perfect matching of the Majorana modes used to construct a suitable fermionic Gaussian unitary.

► BibTeX data

@article{McNulty2025optimalfermionic,
  doi = {10.22331/q-2025-11-17-1914},
  url = {https://doi.org/10.22331/q-2025-11-17-1914},
  title = {Optimal {F}ermionic {J}oint {M}easurements for {E}stimating {N}on-{C}ommuting {M}ajorana {O}bservables},
  author = {McNulty, Daniel and Calegari, Susane and Oszmaniec, Micha{\l{}}},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1914},
  month = nov,
  year = {2025}
}

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Cited by

[1] Michał Kotowski and Michał Oszmaniec, "Pretty Good Simulation of All Quantum Measurements by Projective Measurements", IEEE Transactions on Information Theory 72 5, 3156 (2026).

[2] Michał Kotowski and Michał Oszmaniec, "Pretty-good simulation of all quantum measurements by projective measurements", arXiv:2501.09339, (2025).

[3] Joanna Majsak, Daniel McNulty, and Michał Oszmaniec, "A simple and efficient joint measurement strategy for estimating fermionic observables and Hamiltonians", npj Quantum Information 11 1, 61 (2025).

[4] Oliver Reardon-Smith, "The fermionic linear optical extent is multiplicative for 4 qubit parity eigenstates", arXiv:2407.20934, (2024).

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