## Publication: Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states

Loading...

##### Official URL

##### Full text at PDC

##### Publication Date

2022-04-13

##### Advisors (or tutors)

##### Editors

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

Matrix Product Operators (MPOs) are tensor networks representing operators acting on 1D systems. They model a wide variety of situations, including communication channels with memory effects, quantum cellular automata, mixed states in 1D quantum systems, or holographic boundary models associated to 2D quantum systems. A scenario where MPOs have proven particularly useful is to represent algebras of non-trivial symmetries. Concretely, the boundary of both symmetry protected and topologically ordered phases in 2D quantum systems exhibit symmetries in the form of MPOs. In this paper, we develop a theory of MPOs as representations of algebraic structures. We establish a dictionary between algebra and MPO properties which allows to transfer results between both setups, covering the cases of pre-bialgebras, weak bialgebras, and weak Hopf algebras. We define the notion of pulling-through algebras, which abstracts the minimal requirements needed to define topologically ordered 2D tensor networks from MPO algebras. We show, as one of our main results, that any semisimple pivotal weak Hopf algebra is a pulling-trough algebra. We demonstrate the power of this framework by showing that they can be used to construct Kitaev’s quantum double models for Hopf algebras solely from an MPO representation of the Hopf algebra, in the exact same way as MPO symmetries obtained from fusion categories can be used to construct Levin-Wen string-net models, and to explain all their topological features; it thus allows to describe both Kitaev and string-net models on the same formal footing.

##### Description

##### UCM subjects

##### Unesco subjects

##### Keywords

##### Citation

[DM16] M. Dalmonte and S. Montangero, “Lattice gauge theory simulations in the quantum information era,”
Contemporary Physics, vol. 57, no. 3, pp. 388–412, Jul. 2, 2016.
[DS21] C. Delcamp and N. Schuch, “On tensor network representations of the (3+1)d toric code,” Quan�tum, vol. 5, p. 604, Dec. 16, 2021. arXiv: 2012.15631.
[DIH+17] K. Duivenvoorden, M. Iqbal, J. Haegeman, F. Verstraete, and N. Schuch, “Entanglement phases as holographic duals of anyon condensates,”
Physical Review B, vol. 95, no. 23, p. 235 119, Jun. 12, 2017. arXiv: 1702.08469.
[EGH+] P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, and E. Yudovina,
“Introduction to representation theory,” p. 108,
[ENO02] P. Etingof, D. Nikshych, and V. Ostrik, “On fusion categories,” Mar. 7, 2002. arXiv: math/0203060.
[FNW92] M. Fannes, B. Nachtergaele, and R. F. Werner, “Finitely correlated states on quantum spin chains,”
Commun. Math. Phys., vol. 144, no. 3, pp. 443–490, 1992.
[FTL+07] A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freed�man, “Interacting anyons in topological quantum liquids: The golden chain,” Physical Review
Letters, vol. 98, no. 16, p. 160 409, Apr. 20, 2007. arXiv: cond-mat/0612341.
[Gar21] J. J. Garc´ıa-Ripoll, “Quantum-inspired algorithms for multivariate analysis: From interpolation to partial differential equations,”
Quantum, vol. 5, p. 431, Apr. 15, 2021. arXiv: 1909.06619.
[GLM22] J. Garre-Rubio, L. Lootens, and A. Moln´ar, “Classifying phases protected by matrix product
operator symmetries using matrix product states,” Mar. 23, 2022. arXiv: 2203.12563.
[GAT+13] C. Gils, E. Ardonne, S. Trebst, D. A. Huse, A. W. W. Ludwig, M. Troyer, and Z. Wang,
“Anyonic quantum spin chains: Spin-1 generalizations and topological stability,” Physical Re�view B, vol. 87, no. 23, p. 235 120, Jun. 17, 2013. arXiv: 1303.4290.
[GSP+19] I. Glasser, R. Sweke, N. Pancotti, J. Eisert, and J. I. Cirac, “Expressive power of tensor-network
factorizations for probabilistic modeling, with applications from hidden Markov models to
quantum machine learning,” Nov. 29, 2019. arXiv: 1907.03741.
[GLSW09] Z.-C. Gu, M. Levin, B. Swingle, and X.-G. Wen, “Tensor-product representations for string-net condensed states,”
Physical Review B, vol. 79, no. 8, p. 085 118, Feb. 23, 2009. arXiv: 0809.2821.
[Hac19] W. Hackbusch, “Matrix Product Systems,” in Tensor Spaces and Numerical Tensor Calculus,
W. Hackbusch, Ed., Cham: Springer International Publishing, 2019, pp. 453–472.
[HV17] J. Haegeman and F. Verstraete, “Diagonalizing transfer matrices and matrix product operators: A medley of exact and computational methods, Annual Review of Condensed Matter Physics, vol. 8, no. 1, pp. 355–406, Mar. 31, 2017. arXiv:
1611.08519.
[HW20] A. Hahn and R. Wolf, “Generalized string-nets for unitary fusion categories without tetrahedral symmetry,”
Physical Review B, vol. 102, no. 11, p. 115 154, Sep. 24, 2020. arXiv: 2004.07045.
[Has07a] M. B. Hastings, “An Area Law for One Dimensional Quantum Systems,” Journal of Statistical
Mechanics: Theory and Experiment, vol. 2007, no. 08, P08024–P08024, Aug. 20, 2007. arXiv: 0705.2024.
[Has07b] M. B. Hastings, “Entropy and entanglement in quantum ground states,” Phys. Rev. B, vol. 76, no. 3, p. 035 114, Jul. 2007. arXiv: cond-mat/0701055.
[IOL07] S. Iblisdir, R. Orus, and J. I. Latorre, “Matrix Product States Algorithms and Continuous Systems,” Physical Review B, vol. 75, no. 10, p. 104 305, Mar. 23, 2007. arXiv: cond-mat/0610530.
[JE21] A. Jahn and J. Eisert, “Holographic tensor network models and quantum error correction: A topical review,” Quantum Science and Technology, vol. 6, no. 3, p. 033 002, Jul. 1, 2021. arXiv: 2102.02619.
[Kit03] A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics, vol. 303, no. 1, pp. 2–30, Jan. 2003. arXiv: quant-ph/9707021.
[KL17] A. Komar and O. Landon-Cardinal, “Anyons are not energy eigenspaces of quantum double Hamiltonians,” Physical Review B, vol. 96, no. 19, p. 195 150, Nov. 27, 2017. arXiv: 1701.04456.
[Laf16] N. Laflorencie, “Quantum entanglement in condensed matter systems,” Physics Reports, vol. 646,vpp. 1–59, Aug. 2016. arXiv: 1512.03388.
[Lat05] J. I. Latorre, “Image compression and entanglement,” Oct. 4, 2005. arXiv: quant-ph/0510031.
[LW05] M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases,”
Physical Review B, vol. 71, no. 4, 045110, p. 045 110, Jan. 12, 2005. arXiv: cond-mat/0404617.
[LH08] H. Li and F. D. M. Haldane, “Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States,”
Physical Review Letters, vol. 101, no. 1, p. 010 504, Jul. 3, 2008. arXiv: 0805.0332.
[LCSV21] L. Lootens, B. V.-D. Cuiper, N. Schuch, and F. Verstraete, “Mapping between Morita equiva�lent string-net states with a constant depth quantum circuit,” Dec. 23, 2021. arXiv: 2112.12757.
[LFH+20] L. Lootens, J. Fuchs, J. Haegeman, C. Schweigert, and F. Verstraete, “Matrix product operator
symmetries and intertwiners in string-nets with domain walls,” SciPost Physics, vol. 10, no. 3,vp. 053, Mar. 1, 2020. arXiv: 2008.11187.
[ME22] J. C. McCord and G. Evenbly, “Improved Wavelets for Image Compression from Unitary Circuits,” Mar. 4, 2022. arXiv: 2203.02556.
[MG02] I. P. McCulloch and M. Gul´acsi, “The non-Abelian density matrix renormalization group algorithm,” Europhysics Letters, vol. 57, no. 6, p. 852, Mar. 1, 2002.
[MGSC18] A. Molnar, Y. Ge, N. Schuch, and J. I. Cirac, “A generalization of the injectivity condition for Projected Entangled Pair States,” J. Math. Phys., vol. 59, no. 2, p. 021 902, Feb. 2018. arXiv: 1706.07329v1.
[MCPV10] V. Murg, J. I. Cirac, B. Pirvu, and F. Verstraete, “Matrix product operator representations,” New J. Phys., vol. 12, no. 2, 025012, pp. 1–9, Feb. 2010. arXiv: 0804.3976.
[Nik03] D. Nikshych, “Semisimple weak hopf algebras,” Apr. 2003. arXiv: math/0304098.
[NTO17] A. Novikov, M. Trofimov, and I. Oseledets, “Exponential Machines,” Dec. 8, 2017. arXiv: 1605.03795.
[ONU22] K. Okunishi, T. Nishino, and H. Ueda, “Developments in the Tensor Network – from Statistical
Mechanics to Quantum Entanglement,” Mar. 15, 2022. arXiv: 2111.12223.
[OHN+15] R. Olivares-Amaya, W. Hu, N. Nakatani, S. Sharma, J. Yang, and G. K.-L. Chan, “The ab-initio density matrix renormalization group in practice,”
The Journal of Chemical Physics, vol. 142, no. 3, p. 034 102, Jan. 21, 2015.
[Oru19] R. Orus, “Tensor networks for complex quantum systems,” Nature Reviews Physics, vol. 1, no. 9, pp. 538–550, Sep. 2019. arXiv: 1812.04011.
[Ose11] I. V. Oseledets, “Tensor-Train Decomposition,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2295–2317, Jan. 2011.
[PKS+19] S. Paeckel, T. K¨ohler, A. Swoboda, S. R. Manmana, U. Schollw¨ock, and C. Hubig, “Time-evolution methods for matrix-product states,” Annals of Physics, vol. 411, p. 167 998, Dec. 1, 2019.
[PYHP06] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence,” J. High Energy Phys., vol. 2015, no. 6, p. 149, 2015-03-20, 2015-06. arXiv: 1503.06237v2.
[PVWC08] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, “Matrix product state repre�sentations,” Quantum Inf. Comput. 7, 401 (2007), vol. 7, p. 401, 2007-05, 2007-08. arXiv: quant-ph/0608197.
[PWS+08] D. Perez-Garcia, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, “String order and symmetries in quantum spin lattices,”
Phys. Rev. Lett. 100, 167202 (2008), vol. 100, 167202, p. 167 202, Feb. 2008. arXiv: 0802.0447v1.
[PFM+16] H. C. Po, L. Fidkowski, T. Morimoto, A. C. Potter, and A. Vishwanath, “Chiral Floquet Phases of Many-Body Localized Bosons,”
Physical Review X, vol. 6, no. 4, p. 041 070, Dec. 30, 2016. arXiv: 1609.00006.
[RVHB19] G. Roose, L. Vanderstraeten, J. Haegeman, and N. Bultinck, “Anomalous domain wall condensation in a modified Ising chain,”
Physical Review B, vol. 99, no. 19, p. 195 132, May 20, 2019. arXiv: 1812.04656. [S¸SBC18] M. B. S¸ahino˘glu, S. K. Shukla, F. Bi, and X. Chen, “Matrix Product Representation of Locality Preserving Unitaries,” Physical Review B, vol. 98, no. 24, p. 245 122, Dec. 17, 2018. arXiv: 1704.01943.
[S¸WB+14] M. B. S¸ahino˘glu, D. Williamson, N. Bultinck, M. Mari¨en, J. Haegeman, N. Schuch, and F. Verstraete, “Characterizing Topological Order with Matrix Product Operators,” Sep. 7, 2014. arXiv: 1409.2150.
[Sch11] U. Schollwoeck, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics, vol. 326, no. 1, pp. 96–192, Jan. 2011. arXiv: 1008.3477.
[SSV+05] C. Sch¨on, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, “Sequential generation of entangled multiqubit states,”
Phys. Rev. Lett., vol. 95, no. 11, p. 110 503, Sep. 2005. arXiv: 0501096[quant-ph].
[SCP10] N. Schuch, I. Cirac, and D. Perez-Garcia, “PEPS as ground states: Degeneracy and topology,” Annals of Physics, vol. 325, no. 10, pp. 2153–2192, Oct. 2010. arXiv: 1001.3807.
[SPC10] N. Schuch, D. P´erez-Garc´ıa, and J. I. Cirac, “Classifying quantum phases using matrix product states and projected entangled pair states,” Phys. Rev. B, vol. 84, no. 16, p. 165 139, 2011-10-31, 2011-10. arXiv: 1010.3732.
[SC12] S. Sharma and G. K.-L. Chan, “Spin-adapted density matrix renormalization group algorithms for quantum chemistry,”
The Journal of Chemical Physics, vol. 136, no. 12, p. 124 121, Mar. 28, 2012.
[SS17] E. M. Stoudenmire and D. J. Schwab, “Supervised Learning with Quantum-Inspired Tensor
Networks,” May 18, 2017. arXiv: 1605.05775.
[SPM+15] S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, and O. Legeza, ¨ “Tensor product methods and entanglement optimization for ab initio quantum chemistry,” International Journal of Quantum Chemistry, vol. 115, no. 19, pp. 1342–1391, Oct. 5, 2015.
arXiv: 1412.5829.
[VCM03] F. Verstraete, J. I. Cirac, and V. Murg, “Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems,” Adv. Phys. 57,143 (2008), vol. 57, no. 2, pp. 143–224, 2009-07-16, 2008-03. arXiv: 0907.2796.
[VWPC06] F. Verstraete, M. M. Wolf, D. Pérez-García, and J. I. Cirac, “Criticality, the area law, and the computational power of projected entangled pair states,” Phys. Rev. Lett., vol. 96, no. 22, p. 220 601, Jun. 2006. arXiv: quant-ph/0601075.
[WBM+16] D. J. Williamson, N. Bultinck, M. Mari¨en, M. B. Sahinoglu, J. Haegeman, and F. Verstraete,
“Matrix product operators for symmetry-protected topological phases: Gauging and edge theories,” Phys. Rev. B 94, 205150 (2016), vol. 94, p. 205 150, Dec. 2016. arXiv: 1412.5604v3.
[WDVS21] D. J. Williamson, C. Delcamp, F. Verstraete, and N. Schuch, “On the stability of topological order in tensor network states,”
Physical Review B, vol. 104, no. 23, p. 235 151, Dec. 27, 2021. arXiv: 2012.15346.
[ZCC+17] B.-X. Zheng et al., “Stripe order in the underdoped region of the two-dimensional Hubbard model,”
Science, vol. 358, no. 6367, pp. 1155–1160, Dec. 2017. arXiv: 1701.00054.
[ZV04] M. Zwolak and G. Vidal, “Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm,” Phys. Rev. Lett., vol. 93, no. 20, p. 207 205, Nov. 2004. arXiv: cond-mat/0406426