A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras
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AbstractIn 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} }► References [1] E. Artin. Theorie der Zöpfe (Theory of braids). Abh. Math. Semin. Univ. Hambg, 4: 47–72, 1925. 10.1007/BF02950718. https://doi.org/10.1007/BF02950718 [2] Emilio Cobanera and Gerardo Ortiz. Fock parafermions and self-dual representations of the braid group. Physical Review A - Atomic, Molecular, and Optical Physics, 89 (1), 2014. ISSN 10502947. 10.1103/PhysRevA.89.012328. https://doi.org/10.1103/PhysRevA.89.012328 [3] Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13 (4): 043016, April 2011. ISSN 1367-2630. 10.1088/1367-2630/13/4/043016. https://doi.org/10.1088/1367-2630/13/4/043016 [4] David M. Goldschmidt and V. F.R. Jones. Metaplectic link invariants. Geometriae Dedicata, 31: 165–191, 1989. 10.1007/BF00147477. https://doi.org/10.1007/BF00147477 [5] Daniel Gottesman and Isaac L. Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402 (6760): 390–393, Nov 1999. 10.1038/46503. https://doi.org/10.1038/46503 [6] E. R. Hansen. A table of series and products. Prentice Hall, 1975. [7] Arthur Jaffe and Zhengwei Liu. Planar para algebras, reflection positivity. Communications in Mathematical Physics, 352 (1), 2017. ISSN 14320916. 10.1007/s00220-016-2779-4. https://doi.org/10.1007/s00220-016-2779-4 [8] V. F. R. Jones. Planar algebras, I. 1999. 10.48550/arXiv.math/9909027. arXiv.math/9909027. https://doi.org/10.48550/arXiv.math/9909027 [9] V. F.R. Jones. On a certain value of the Kauffman polynomial. Communications in Mathematical Physics, 125, 1989. ISSN 00103616. 10.1007/BF01218412. https://doi.org/10.1007/BF01218412 [10] Louis H. Kauffman. Knot logic and topological quantum computing with Majorana fermions. 2013. 10.48550/arXiv.1301.6214. arXiv:1301.6214. https://doi.org/10.48550/arXiv.1301.6214 arXiv:1301.6214 [11] Evgeniy O. Kiktenko, Anastasiia S. Nikolaeva, and Aleksey K. Fedorov. Colloquium: Qudits for decomposing multiqubit gates and realizing quantum algorithms. Reviews of Modern Physics, 97 (2), June 2025. ISSN 1539-0756. 10.1103/revmodphys.97.021003. https://doi.org/10.1103/revmodphys.97.021003 [12] Robert Lin. An algebraic framework for multi-qudit computations with generalized Clifford algebras. March 2021. 10.48550/arXiv.2103.15324. arXiv:2103.15324. https://doi.org/10.48550/arXiv.2103.15324 arXiv:2103.15324 [13] Zhengwei Liu, Alex Wozniakowski, and Arthur M. Jaffe. Quon 3D language for quantum information. Proceedings of the National Academy of Sciences, 114 (10): 2497–2502, Feb 2017. 10.1073/pnas.1621345114. https://doi.org/10.1073/pnas.1621345114 [14] Kakutarō Morinaga and Takayuki Nōno. On the linearization of a form of higher degree and its representation.
Hiroshima Mathematical Journal, 16, 2019. 10.32917/hmj/1557367250. https://doi.org/10.32917/hmj/1557367250 [15] A. O. Morris. On a generalized Clifford algebra. Quarterly Journal of Mathematics, 18 (1), 1967. ISSN 00335606. 10.1093/qmath/18.1.7. https://doi.org/10.1093/qmath/18.1.7 [16] Michael Müger. From subfactors to categories and topology ii: The quantum double of tensor categories and subfactors. Journal of Pure and Applied Algebra, 180 (1-2), 2003. ISSN 00224049. 10.1016/S0022-4049(02)00248-7. https://doi.org/10.1016/S0022-4049(02)00248-7 [17] Sorin Popa. An axiomatization of the lattice of higher relative commutants of a subfactor. Inventiones Mathematicae, 120 (1), 1995. ISSN 00209910. 10.1007/BF01241137. https://doi.org/10.1007/BF01241137 [18] I. Popovici and C. Gheorghe. Algèbres de Clifford généralisées. C. R. Acad. Sci. Paris, 262: 682–685, 1966. [19] Boldizsár Poór, Robert I. Booth, Titouan Carette, John van de Wetering, and Lia Yeh. The qupit stabiliser zx-travaganza: Simplified axioms, normal forms and graph-theoretic simplification. Electronic Proceedings in Theoretical Computer Science, 384: 220–264, August 2023. ISSN 2075-2180. 10.4204/eptcs.384.13. https://doi.org/10.4204/eptcs.384.13 [20] H.N.V. Temperley and E.H. Lieb. Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 322 (1549), 1971. ISSN 0080-4630. 10.1098/rspa.1971.0067. https://doi.org/10.1098/rspa.1971.0067 [21] Chien-Cheng Tseng. Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices. IEEE Transactions on Signal Processing, 50 (4): 866–877, 2002. 10.1109/78.992133. https://doi.org/10.1109/78.992133 [22] K. Yamazaki. On projective representations and ring extensions of finite groups. J. Fat. Sci. University of Tokyo, Set I (10): 147–195, 1964. [23] C. N. Yang. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction.
Physical Review Letters, 19 (23), 1967. ISSN 00319007. 10.1103/PhysRevLett.19.1312. https://doi.org/10.1103/PhysRevLett.19.1312Cited byCould not fetch Crossref cited-by data during last attempt 2025-11-17 16:18:22: Could not fetch cited-by data for 10.22331/q-2025-11-17-1913 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-11-17 16:18:23: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractIn 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} }► References [1] E. Artin. Theorie der Zöpfe (Theory of braids). Abh. Math. Semin. Univ. Hambg, 4: 47–72, 1925. 10.1007/BF02950718. https://doi.org/10.1007/BF02950718 [2] Emilio Cobanera and Gerardo Ortiz. Fock parafermions and self-dual representations of the braid group. Physical Review A - Atomic, Molecular, and Optical Physics, 89 (1), 2014. ISSN 10502947. 10.1103/PhysRevA.89.012328. https://doi.org/10.1103/PhysRevA.89.012328 [3] Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13 (4): 043016, April 2011. ISSN 1367-2630. 10.1088/1367-2630/13/4/043016. https://doi.org/10.1088/1367-2630/13/4/043016 [4] David M. Goldschmidt and V. F.R. Jones. Metaplectic link invariants. Geometriae Dedicata, 31: 165–191, 1989. 10.1007/BF00147477. https://doi.org/10.1007/BF00147477 [5] Daniel Gottesman and Isaac L. Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402 (6760): 390–393, Nov 1999. 10.1038/46503. https://doi.org/10.1038/46503 [6] E. R. Hansen. A table of series and products. Prentice Hall, 1975. [7] Arthur Jaffe and Zhengwei Liu. Planar para algebras, reflection positivity. Communications in Mathematical Physics, 352 (1), 2017. ISSN 14320916. 10.1007/s00220-016-2779-4. https://doi.org/10.1007/s00220-016-2779-4 [8] V. F. R. Jones. Planar algebras, I. 1999. 10.48550/arXiv.math/9909027. arXiv.math/9909027. https://doi.org/10.48550/arXiv.math/9909027 [9] V. F.R. Jones. On a certain value of the Kauffman polynomial. Communications in Mathematical Physics, 125, 1989. ISSN 00103616. 10.1007/BF01218412. https://doi.org/10.1007/BF01218412 [10] Louis H. Kauffman. Knot logic and topological quantum computing with Majorana fermions. 2013. 10.48550/arXiv.1301.6214. arXiv:1301.6214. https://doi.org/10.48550/arXiv.1301.6214 arXiv:1301.6214 [11] Evgeniy O. Kiktenko, Anastasiia S. Nikolaeva, and Aleksey K. Fedorov. Colloquium: Qudits for decomposing multiqubit gates and realizing quantum algorithms. Reviews of Modern Physics, 97 (2), June 2025. ISSN 1539-0756. 10.1103/revmodphys.97.021003. https://doi.org/10.1103/revmodphys.97.021003 [12] Robert Lin. An algebraic framework for multi-qudit computations with generalized Clifford algebras. March 2021. 10.48550/arXiv.2103.15324. arXiv:2103.15324. https://doi.org/10.48550/arXiv.2103.15324 arXiv:2103.15324 [13] Zhengwei Liu, Alex Wozniakowski, and Arthur M. Jaffe. Quon 3D language for quantum information. Proceedings of the National Academy of Sciences, 114 (10): 2497–2502, Feb 2017. 10.1073/pnas.1621345114. https://doi.org/10.1073/pnas.1621345114 [14] Kakutarō Morinaga and Takayuki Nōno. On the linearization of a form of higher degree and its representation.
Hiroshima Mathematical Journal, 16, 2019. 10.32917/hmj/1557367250. https://doi.org/10.32917/hmj/1557367250 [15] A. O. Morris. On a generalized Clifford algebra. Quarterly Journal of Mathematics, 18 (1), 1967. ISSN 00335606. 10.1093/qmath/18.1.7. https://doi.org/10.1093/qmath/18.1.7 [16] Michael Müger. From subfactors to categories and topology ii: The quantum double of tensor categories and subfactors. Journal of Pure and Applied Algebra, 180 (1-2), 2003. ISSN 00224049. 10.1016/S0022-4049(02)00248-7. https://doi.org/10.1016/S0022-4049(02)00248-7 [17] Sorin Popa. An axiomatization of the lattice of higher relative commutants of a subfactor. Inventiones Mathematicae, 120 (1), 1995. ISSN 00209910. 10.1007/BF01241137. https://doi.org/10.1007/BF01241137 [18] I. Popovici and C. Gheorghe. Algèbres de Clifford généralisées. C. R. Acad. Sci. Paris, 262: 682–685, 1966. [19] Boldizsár Poór, Robert I. Booth, Titouan Carette, John van de Wetering, and Lia Yeh. The qupit stabiliser zx-travaganza: Simplified axioms, normal forms and graph-theoretic simplification. Electronic Proceedings in Theoretical Computer Science, 384: 220–264, August 2023. ISSN 2075-2180. 10.4204/eptcs.384.13. https://doi.org/10.4204/eptcs.384.13 [20] H.N.V. Temperley and E.H. Lieb. Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 322 (1549), 1971. ISSN 0080-4630. 10.1098/rspa.1971.0067. https://doi.org/10.1098/rspa.1971.0067 [21] Chien-Cheng Tseng. Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices. IEEE Transactions on Signal Processing, 50 (4): 866–877, 2002. 10.1109/78.992133. https://doi.org/10.1109/78.992133 [22] K. Yamazaki. On projective representations and ring extensions of finite groups. J. Fat. Sci. University of Tokyo, Set I (10): 147–195, 1964. [23] C. N. Yang. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction.
Physical Review Letters, 19 (23), 1967. ISSN 00319007. 10.1103/PhysRevLett.19.1312. https://doi.org/10.1103/PhysRevLett.19.1312Cited byCould not fetch Crossref cited-by data during last attempt 2025-11-17 16:18:22: Could not fetch cited-by data for 10.22331/q-2025-11-17-1913 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-11-17 16:18:23: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.