A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras

Robert Lin

doi:10.22331/q-2025-11-17-1913

A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras

Department of Physics, Harvard University, Cambridge, MA 02138

Published:	2025-11-17, volume 9, page 1913
Editor:	Ion Nechita
Eprint:	arXiv:2103.16081v3
Doi:	https://doi.org/10.22331/q-2025-11-17-1913
Citation:	Quantum 9, 1913 (2025).

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Abstract

In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor $\zeta$, which is an appropriate square root of a primitive root of unity.

► BibTeX data

@article{Lin2025graphicalcalculus,
  doi = {10.22331/q-2025-11-17-1913},
  url = {https://doi.org/10.22331/q-2025-11-17-1913},
  title = {A {G}raphical {C}alculus for {Q}uantum {C}omputing with {M}ultiple {Q}udits using {G}eneralized {C}lifford {A}lgebras},
  author = {Lin, Robert},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1913},
  month = nov,
  year = {2025}
}

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Cited by

[1] Priyaa Varshinee Srinivasan, "Dagger linear logic and categorical quantum mechanics", arXiv:2303.14231, (2023).

[2] Colin Blake, "A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs", arXiv:2602.09873, (2026).

The above citations are from SAO/NASA ADS (last updated successfully 2026-08-17 17:47:49). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-17 17:47:48).

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