Geometric representations of braid and Yang–Baxter gates

Zhang, Kun; Hao, Kun; Yu, Kwangmin; Korepin, Vladimir; Yang, Wen-Li

doi:10.1088/1751-8121/ad85b2

Title: Geometric representations of braid and Yang–Baxter gates

Journal Article · 17 October 2024 · Journal of Physics. A, Mathematical and Theoretical

DOI: https://doi.org/10.1088/1751-8121/ad85b2 · OSTI ID:2478424

^[1];

^[2];

^[3];

^[4]; Yang, Wen-Li ^[5]

Northwestern University, Xi'an (China); Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an (China); Peng Huanwu Center for Fundamental Theory, Xi'an (China)
Northwestern University, Xi'an (China). Institute of Modern Physics; Peng Huanwu Center for Fundamental Theory, Xi'an (China)
Brookhaven National Laboratory (BNL), Upton, NY (United States)
Stony Brook Univ., NY (United States)
Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an (China); Peng Huanwu Center for Fundamental Theory, Xi'an (China); Northwestern University, Xi'an (China). Institute of Modern Physics

Brick-wall circuits composed of the Yang–Baxter gates are integrable. It becomes an important tool to study the quantum many-body system out of equilibrium. To put the Yang–Baxter gate on quantum computers, it has to be decomposed into the native gates of quantum computers. It is favorable to apply the least number of native two-qubit gates to construct the Yang–Baxter gate. We study the geometric representations of all X-type braid gates and their corresponding Yang–Baxter gates via the Yang–Baxterization. We find that the braid and Yang–Baxter gates can only exist on certain edges and faces of the two-qubit tetrahedron. We identify the parameters by which the braid and Yang–Baxter gates are the Clifford gate, the matchgate, and the dual-unitary gate. The geometric representations provide the optimal decompositions of the braid and Yang–Baxter gates in terms of other two-qubit gates. We also find that the entangling powers of the Yang–Baxter gates are determined by the spectral parameters. Our results provide the necessary conditions to construct the braid and Yang–Baxter gates on quantum computers.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Brookhaven National Laboratory (BNL), Upton, NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: SC0012704

OSTI ID:: 2478424

Report Number(s):: BNL--226332-2024-JAAM

Journal Information:: Journal of Physics. A, Mathematical and Theoretical, Journal Name: Journal of Physics. A, Mathematical and Theoretical Journal Issue: 44 Vol. 57; ISSN 1751-8113

Publisher:: IOP PublishingCopyright Statement

Country of Publication:: United States

Language:: English

References (68)

Optimal Realization of Yang–Baxter Gate on Quantum Computers Zhang, Kun; Yu, Kwangmin; Hao, Kun Advanced Quantum Technologies, Vol. 7, Issue 4 https://doi.org/10.1002/qute.202300345	journal	February 2024
Tetrahedron equations and the relativisticS-matrix of straight-strings in 2+1-Dimensions Zamolodchikov, A. B. Communications in Mathematical Physics, Vol. 79, Issue 4 https://doi.org/10.1007/BF01209309	journal	December 1981
Yang-Baxterization of braid group representations Cheng, Y.; Ge, M. L.; Xue, K. Communications in Mathematical Physics, Vol. 136, Issue 1 https://doi.org/10.1007/BF02096797	journal	February 1991
New R-matrices with non-additive spectral parameters and integrable models of strongly correlated fermions Zhang, Yao-Zhong; Werry, Jason L. Journal of High Energy Physics, Vol. 2020, Issue 3 https://doi.org/10.1007/JHEP03%282020%29141	journal	March 2020
Yang–Baxterizations, Universal Quantum Gates and Hamiltonians Zhang, Yong; Kauffman, Louis H.; Ge, Mo-Lin Quantum Information Processing, Vol. 4, Issue 3 https://doi.org/10.1007/s11128-005-7655-7	journal	August 2005
Quantum computing via the Bethe ansatz Zhang, Yong Quantum Information Processing, Vol. 11, Issue 2 https://doi.org/10.1007/s11128-011-0268-4	journal	August 2011
Integrable quantum computation Zhang, Yong Quantum Information Processing, Vol. 12, Issue 1 https://doi.org/10.1007/s11128-012-0409-4	journal	April 2012
Teleportation-based quantum computation, extended Temperley–Lieb diagrammatical approach and Yang–Baxter equation Zhang, Yong; Zhang, Kun; Pang, Jinglong Quantum Information Processing, Vol. 15, Issue 1 https://doi.org/10.1007/s11128-015-1158-y	journal	November 2015
Quantum teleportation and Birman–Murakami–Wenzl algebra Zhang, Kun; Zhang, Yong Quantum Information Processing, Vol. 16, Issue 2 https://doi.org/10.1007/s11128-016-1512-8	journal	January 2017
Topological aspects of quantum entanglement Kauffman, Louis H.; Mehrotra, Eshan Quantum Information Processing, Vol. 18, Issue 3 https://doi.org/10.1007/s11128-019-2191-z	journal	February 2019
Partition function of the Eight-Vertex lattice model Baxter, Rodney J. Annals of Physics, Vol. 70, Issue 1 https://doi.org/10.1016/0003-4916(72)90335-1	journal	March 1972
All solutions to the constant quantum Yang-Baxter equation in two dimensions Hietarinta, Jarmo Physics Letters A, Vol. 165, Issue 3 https://doi.org/10.1016/0375-9601(92)90044-M	journal	May 1992
Yang–Baxter Equations Perk, J. H. H.; Au-Yang, H. Encyclopedia of Mathematical Physics https://doi.org/10.1016/B0-12-512666-2/00191-7	book	January 2006
Bethe Ansatz solutions for certain Periodic Quantum Circuits Aleiner, Igor L. Annals of Physics, Vol. 433 https://doi.org/10.1016/j.aop.2021.168593	journal	October 2021
Quantum Computation and Quantum Information Nielsen, Michael A.; Chuang, Isaac L. Cambridge University Press https://doi.org/10.1017/CBO9780511976667	book	January 2011
Unitary Solutions to the Yang-Baxter Equation in Dimension Four Dye, H. A. Quantum Information Processing, Vol. 2, Issue 1/2, p. 117-152 https://doi.org/10.1023/A:1025843426102	journal	April 2003
Evidence of Kardar-Parisi-Zhang scaling on a digital quantum simulator Keenan, Nathan; Robertson, Niall F.; Murphy, Tara npj Quantum Information, Vol. 9, Issue 1 https://doi.org/10.1038/s41534-023-00742-4	journal	July 2023
Formation of robust bound states of interacting microwave photons Morvan, A.; Andersen, T. I.; Mi, X. Nature, Vol. 612, Issue 7939 https://doi.org/10.1038/s41586-022-05348-y	journal	December 2022
Evidence for the utility of quantum computing before fault tolerance Kim, Youngseok; Eddins, Andrew; Anand, Sajant Nature, Vol. 618, Issue 7965 https://doi.org/10.1038/s41586-023-06096-3	journal	June 2023
The Quantum Method of the Inverse Problem and the Heisenberg xyz Model Takhtadzhan, L. A.; Faddeev, Lyudvig D. Russian Mathematical Surveys, Vol. 34, Issue 5 https://doi.org/10.1070/RM1979v034n05ABEH003909	journal	October 1979
Modular Transformation and Twist Between Trigonometric Limits of sl ( n ) Elliptic R -Matrix Wen-Li, Yang; Yi, Zhen Communications in Theoretical Physics, Vol. 36, Issue 2 https://doi.org/10.1088/0253-6102/36/2/131	journal	August 2001
Towards a cladistics of double Yangians and elliptic algebras* Arnaudon, D.; Avan, J.; Frappat, L. Journal of Physics A: Mathematical and General, Vol. 33, Issue 36 https://doi.org/10.1088/0305-4470/33/36/302	journal	September 2000
Teleportation, braid group and Temperley–Lieb algebra Zhang, Yong Journal of Physics A: Mathematical and General, Vol. 39, Issue 37 https://doi.org/10.1088/0305-4470/39/37/017	journal	August 2006
Quantum entanglement and topological entanglement Kauffman, Louis H.; Lomonaco, Samuel J. New Journal of Physics, Vol. 4 https://doi.org/10.1088/1367-2630/4/1/373	journal	January 2002
Braiding operators are universal quantum gates Kauffman, Louis H.; Lomonaco, Samuel J. New Journal of Physics, Vol. 6 https://doi.org/10.1088/1367-2630/6/1/134	journal	October 2004
Yang–Baxter integrable models in experiments: from condensed matter to ultracold atoms Batchelor, Murray T.; Foerster, Angela Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 17 https://doi.org/10.1088/1751-8113/49/17/173001	journal	March 2016
Yang–Baxter operators need quantum entanglement to distinguish knots Alagic, Gorjan; Jarret, Michael; Jordan, Stephen P. Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 7 https://doi.org/10.1088/1751-8113/49/7/075203	journal	January 2016
Local invariants of braiding quantum gates—associated link polynomials and entangling power Padmanabhan, Pramod; Sugino, Fumihiko; Trancanelli, Diego Journal of Physics A: Mathematical and Theoretical, Vol. 54, Issue 13 https://doi.org/10.1088/1751-8121/abdfe9	journal	March 2021
A Yang–Baxter integrable cellular automaton with a four site update rule Pozsgay, Balázs Journal of Physics A: Mathematical and Theoretical, Vol. 54, Issue 38 https://doi.org/10.1088/1751-8121/ac1dbf	journal	September 2021
Conserved charges in the quantum simulation of integrable spin chains Maruyoshi, Kazunobu; Okuda, Takuya; Pedersen, Juan W. Journal of Physics A: Mathematical and Theoretical, Vol. 56, Issue 16 https://doi.org/10.1088/1751-8121/acc369	journal	March 2023
Braids, link polynomials and a new algebra Birman, Joan S.; Wenzl, Hans Transactions of the American Mathematical Society, Vol. 313, Issue 1 https://doi.org/10.1090/S0002-9947-1989-0992598-X	journal	January 1989
Matchgates and classical simulation of quantum circuits Jozsa, Richard; Miyake, Akimasa Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 464, Issue 2100 https://doi.org/10.1098/rspa.2008.0189	journal	July 2008
SMatrix for the One-DimensionalN-Body Problem with Repulsive or Attractiveδ-Function Interaction Yang, C. N. Physical Review, Vol. 168, Issue 5 https://doi.org/10.1103/PhysRev.168.1920	journal	April 1968
Two-bit gates are universal for quantum computation DiVincenzo, David P. Physical Review A, Vol. 51, Issue 2 https://doi.org/10.1103/PhysRevA.51.1015	journal	February 1995
Elementary gates for quantum computation Barenco, Adriano; Bennett, Charles H.; Cleve, Richard Physical Review A, Vol. 52, Issue 5 https://doi.org/10.1103/PhysRevA.52.3457	journal	November 1995
Theory of fault-tolerant quantum computation Gottesman, Daniel Physical Review A, Vol. 57, Issue 1 https://doi.org/10.1103/PhysRevA.57.127	journal	January 1998
Entangling power of quantum evolutions Zanardi, Paolo; Zalka, Christof; Faoro, Lara Physical Review A, Vol. 62, Issue 3 https://doi.org/10.1103/PhysRevA.62.030301	journal	August 2000
Optimal creation of entanglement using a two-qubit gate Kraus, B.; Cirac, J. I. Physical Review A, Vol. 63, Issue 6 https://doi.org/10.1103/PhysRevA.63.062309	journal	May 2001
Classical simulation of noninteracting-fermion quantum circuits Terhal, Barbara M.; DiVincenzo, David P. Physical Review A, Vol. 65, Issue 3 https://doi.org/10.1103/PhysRevA.65.032325	journal	March 2002
Geometric theory of nonlocal two-qubit operations Zhang, Jun; Vala, Jiri; Sastry, Shankar Physical Review A, Vol. 67, Issue 4 https://doi.org/10.1103/PhysRevA.67.042313	journal	April 2003
Universal quantum circuit for two-qubit transformations with three controlled-NOT gates Vidal, G.; Dawson, C. M. Physical Review A, Vol. 69, Issue 1 https://doi.org/10.1103/PhysRevA.69.010301	journal	January 2004
Optimal quantum circuits for general two-qubit gates Vatan, Farrokh; Williams, Colin Physical Review A, Vol. 69, Issue 3 https://doi.org/10.1103/PhysRevA.69.032315	journal	March 2004
Optimal quantum circuit synthesis from controlled-unitary gates Zhang, Jun; Vala, Jiri; Sastry, Shankar Physical Review A, Vol. 69, Issue 4 https://doi.org/10.1103/PhysRevA.69.042309	journal	April 2004
Minimal universal two-qubit controlled-NOT-based circuits Shende, Vivek V.; Markov, Igor L.; Bullock, Stephen S. Physical Review A, Vol. 69, Issue 6 https://doi.org/10.1103/PhysRevA.69.062321	journal	June 2004
Conditions for optimal construction of two-qubit nonlocal gates Zhang, Yong-Sheng; Ye, Ming-Yong; Guo, Guang-Can Physical Review A, Vol. 71, Issue 6 https://doi.org/10.1103/PhysRevA.71.062331	journal	June 2005
Braiding transformation, entanglement swapping, and Berry phase in entanglement space Chen, Jing-Ling; Xue, Kang; Ge, Mo-Lin Physical Review A, Vol. 76, Issue 4 https://doi.org/10.1103/PhysRevA.76.042324	journal	October 2007
Entangling power and local invariants of two-qubit gates Balakrishnan, S.; Sankaranarayanan, R. Physical Review A, Vol. 82, Issue 3 https://doi.org/10.1103/PhysRevA.82.034301	journal	September 2010
Extending matchgates into universal quantum computation Brod, Daniel J.; Galvão, Ernesto F. Physical Review A, Vol. 84, Issue 2 https://doi.org/10.1103/PhysRevA.84.022310	journal	August 2011
Classifying quantum entanglement through topological links Quinta, Gonçalo M.; André, Rui Physical Review A, Vol. 97, Issue 4 https://doi.org/10.1103/PhysRevA.97.042307	journal	April 2018
Absence of localization in weakly interacting Floquet circuits Hahn, Dominik; Colmenarez, Luis Physical Review B, Vol. 109, Issue 9 https://doi.org/10.1103/PhysRevB.109.094207	journal	March 2024
Integrable spin chains and cellular automata with medium-range interaction Gombor, Tamás; Pozsgay, Balázs Physical Review E, Vol. 104, Issue 5 https://doi.org/10.1103/PhysRevE.104.054123	journal	November 2021
Efficient Quantum Circuits for One-Way Quantum Computing Tanamoto, Tetsufumi; Liu, Yu-xi; Hu, Xuedong Physical Review Letters, Vol. 102, Issue 10 https://doi.org/10.1103/PhysRevLett.102.100501	journal	March 2009
Integrable Trotterization: Local Conservation Laws and Boundary Driving Vanicat, Matthieu; Zadnik, Lenart; Prosen, Tomaž Physical Review Letters, Vol. 121, Issue 3 https://doi.org/10.1103/PhysRevLett.121.030606	journal	July 2018
Ballistic Spin Transport in a Periodically Driven Integrable Quantum System Ljubotina, Marko; Zadnik, Lenart; Prosen, Tomaž Physical Review Letters, Vol. 122, Issue 15 https://doi.org/10.1103/PhysRevLett.122.150605	journal	April 2019
Exact Correlation Functions for Dual-Unitary Lattice Models in 1 + 1 Dimensions Bertini, Bruno; Kos, Pavel; Prosen, Tomaž Physical Review Letters, Vol. 123, Issue 21 https://doi.org/10.1103/PhysRevLett.123.210601	journal	November 2019
Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction Yang, C. N. Physical Review Letters, Vol. 19, Issue 23 https://doi.org/10.1103/PhysRevLett.19.1312	journal	December 1967
Practical Scheme for Quantum Computation with Any Two-Qubit Entangling Gate Bremner, Michael J.; Dawson, Christopher M.; Dodd, Jennifer L. Physical Review Letters, Vol. 89, Issue 24 https://doi.org/10.1103/PhysRevLett.89.247902	journal	November 2002
Introduction to Quantum Integrability Doikou, Anastasia; Evangelisti, Stefano; Feverati, Giovanni International Journal of Modern Physics A, Vol. 25, Issue 17 https://doi.org/10.1142/S0217751X10049803	journal	July 2010
Explicit Trigonometric Yang-Baxterization Ge, Mo-Lin; Wu, Yong-Shi; Xue, Kang International Journal of Modern Physics A, Vol. 06, Issue 21 https://doi.org/10.1142/S0217751X91001817	journal	September 1991
Baxterization Jones, V. F. R. International Journal of Modern Physics B, Vol. 04, Issue 05 https://doi.org/10.1142/S021797929000036X	journal	April 1990
Universal Quantum Gate, Yang–Baxterization and Hamiltonian Zhang, Yong; Kauffman, Louis H.; Ge, Mo-Lin International Journal of Quantum Information, Vol. 03, Issue 04 https://doi.org/10.1142/S0219749905001547	journal	December 2005
Correlations and commuting transfer matrices in integrable unitary circuits Claeys, Pieter W.; Herzog-Arbeitman, Jonah; Lamacraft, Austen SciPost Physics, Vol. 12, Issue 1 https://doi.org/10.21468/SciPostPhys.12.1.007	journal	January 2022
Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps Gombor, Tamás; Pozsgay, Balázs SciPost Physics, Vol. 12, Issue 3 https://doi.org/10.21468/SciPostPhys.12.3.102	journal	March 2022
The Floquet Baxterisation Miao, Yuan; Gritsev, Vladimir; Kurlov, Denis V. SciPost Physics, Vol. 16, Issue 3 https://doi.org/10.21468/SciPostPhys.16.3.078	journal	March 2024
Integrable deformations of superintegrable quantum circuits Gombor, Tamás; Pozsgay, Balázs SciPost Physics, Vol. 16, Issue 4 https://doi.org/10.21468/SciPostPhys.16.4.114	journal	April 2024
Quantum gates between distant qubits via spin-independent scattering Banchi, Leonardo; Compagno, Enrico; Korepin, Vladimir Quantum, Vol. 1 https://doi.org/10.22331/q-2017-11-30-36	journal	November 2017
Quantum Computing in the NISQ era and beyond Preskill, John Quantum, Vol. 2 https://doi.org/10.22331/q-2018-08-06-79	journal	August 2018
Braiding quantum gates from partition algebras Padmanabhan, Pramod; Sugino, Fumihiko; Trancanelli, Diego Quantum, Vol. 4 https://doi.org/10.22331/q-2020-08-27-311	journal	August 2020

Similar Records

Optimal Realization of Yang–Baxter Gate on Quantum Computers

Journal Article · 2024 · Advanced Quantum Technologies · OSTI ID:2336574

Zhang, Kun; Yu, Kwangmin; Hao, Kun; +1 more

Berry phase and entanglement of three qubits in a new Yang-Baxter system

Journal Article · 2009 · Journal of Mathematical Physics · OSTI ID:21294276

Taotao, Hu; Kang, Xue; Chunfeng, Wu

Braid group representation on quantum computation

Journal Article · 2015 · AIP Conference Proceedings · OSTI ID:22488902

Muchtadi-Alamsyah, Intan

Related Subjects

97 MATHEMATICS AND COMPUTING

Title: Geometric representations of braid and Yang–Baxter gates

Citation Formats

References (68)

Similar Records

Related Subjects