On the locality of qubit encodings of local fermionic modes

Tommaso Guaita

doi:10.22331/q-2025-02-25-1644

On the locality of qubit encodings of local fermionic modes

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany

Published:	2025-02-25, volume 9, page 1644
Editor:	Cyril Branciard
Eprint:	arXiv:2401.10077v3
Doi:	https://doi.org/10.22331/q-2025-02-25-1644
Citation:	Quantum 9, 1644 (2025).

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Abstract

Known mappings that encode fermionic modes into a bosonic qubit system are non-local transformations. In this paper we establish that this must necessarily be the case, if the locality graph is complex enough (for example for regular 2$d$ lattices). In particular we show that, in case of exact encodings, a fully local mapping is possible if and only if the locality graph is a tree. If instead we allow ourselves to also consider operators that only act fermionically on a subspace of the qubit Hilbert space, then we show that this subspace must be composed of long range entangled states, if the locality graph contains at least two overlapping cycles. This implies, for instance, that on 2$d$ lattices there exist states that are of low depth from the fermionic point of view, while in any encoding require a circuit of depth at least proportional to the system size to be prepared.

Featured image: A non-trivial graph (with two overlapping loops) where fermion-to-qubit encodings are non-local.

Popular summary

Fermions are fundamental particles appearing in many areas of quantum physics, including high-energy particle physics, atomic physics, solid state physics and quantum chemistry. Simulating systems of fermions using quantum computers is therefore a potentially very relevant application. However, most quantum computers are built from qubits, a fundamental component with different characteristics compared to a fermion. Therefore, fermionic computations need to be first encoded into equivalent computations that can be performed on qubits. In this paper we show that this encoding step necessarily requires doing some non-local operations, meaning that even simple fermionic computations may be mapped to more complex qubit ones.

► BibTeX data

@article{Guaita2025localityofqubit,
  doi = {10.22331/q-2025-02-25-1644},
  url = {https://doi.org/10.22331/q-2025-02-25-1644},
  title = {On the locality of qubit encodings of local fermionic modes},
  author = {Guaita, Tommaso},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1644},
  month = feb,
  year = {2025}
}

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Cited by

[1] Andreas Juul Bay-Smidt, Frederik Ravn Klausen, Christoph Sünderhauf, Róbert Izsák, Gemma C. Solomon, and Nick S. Blunt, "Fault-Tolerant Quantum Simulation of Generalized Hubbard Models", PRX Quantum 6 3, 030348 (2025).

[2] Riley W. Chien, Mitchell Chiew, Brent Harrison, Jason Necaise, Weishi Wang, Maryam Mudassar, Campbell McLauchlan, Thomas M. Henderson, Gustavo E. Scuseria, Sergii Strelchuk, and James D. Whitfield, "Simulating fermions with a digital quantum computer", Nature Reviews Physics 8 3, 131 (2026).

[3] Rodolfo Carobene, Andrea Giachero, Stefano Barison, and Jannes Nys, "Local fermion-to-qudit mappings: A practical recipe for four-level systems", Physical Review A 112 3, 032619 (2025).

[4] Fatemeh Moradi Kalarde, Xiangling Xu, and Marc-Olivier Renou, "Comment on "Quantum theory based on real numbers cannot be experimentally falsified": On the compatibility of physical principles with information theory for fermions", arXiv:2604.07425, (2026).

[5] Augustin Vanrietvelde, Octave Mestoudjian, and Pablo Arrighi, "Partitions in quantum theory", arXiv:2506.22218, (2025).

[6] Reinis Irmejs and J. Ignacio Cirac, "Efficient Simulation of Sparse, Non-Local Fermion Models", arXiv:2512.15843, (2025).

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