Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Austin K. Daniel, Rafael N. Alexander, and Akimasa Miyake

Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA

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What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all states within so-called SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. There are four subsystem symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the former three are fundamentally in one-to-one correspondence with three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices possess 1-form symmetries and have a different capability related to error correction.

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[1] R. Raussendorf and H. J. Briegel ``A One-Way Quantum Computer'' Phys. Rev. Lett. 86, 5188-5191 (2001).

[2] H. J. Briegel and R. Raussendorf ``Persistent Entanglement in Arrays of Interacting Particles'' Phys. Rev. Lett. 86, 910-913 (2001).

[3] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M Nest, and H.-J. Briegel, ``Entanglement in graph states and its applications'' arXiv preprint quant-ph/​0602096 (2006).

[4] F. Verstraete and J. I. Cirac ``Valence-bond states for quantum computation'' Phys. Rev. A 70, 060302 (2004).

[5] M. Nest, A. Miyake, W. Dür, and H. J. Briegel, ``Universal Resources for Measurement-Based Quantum Computation'' Phys. Rev. Lett. 97, 150504 (2006).

[6] M. V. Nest, W Dür, A Miyake, and H. J. Briegel, ``Fundamentals of universality in one-way quantum computation'' New Journal of Physics 9, 204-204 (2007).

[7] D. Gross and J. Eisert ``Novel Schemes for Measurement-Based Quantum Computation'' Phys. Rev. Lett. 98, 220503 (2007).

[8] D. Gross, J. Eisert, N. Schuch, and D. Perez-Garcia, ``Measurement-based quantum computation beyond the one-way model'' Phys. Rev. A 76, 052315 (2007).

[9] A. C. Doherty and S. D. Bartlett ``Identifying Phases of Quantum Many-Body Systems That Are Universal for Quantum Computation'' Phys. Rev. Lett. 103, 020506 (2009).

[10] A. Miyake ``Quantum computational capability of a 2D valence bond solid phase'' Annals of Physics 326, 1656-1671 (2011).

[11] T.-C. Wei, I. Affleck, and R. Raussendorf, ``Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource'' Phys. Rev. Lett. 106, 070501 (2011).

[12] T.-C. Wei, I. Affleck, and R. Raussendorf, ``Two-dimensional Affleck-Kennedy-Lieb-Tasaki state on the honeycomb lattice is a universal resource for quantum computation'' Phys. Rev. A 86, 032328 (2012).

[13] T.-C. Wei ``Quantum computational universality of Affleck-Kennedy-Lieb-Tasaki states beyond the honeycomb lattice'' Phys. Rev. A 88, 062307 (2013).

[14] T.-C. Wei, P. Haghnegahdar, and R. Raussendorf, ``Hybrid valence-bond states for universal quantum computation'' Phys. Rev. A 90, 042333 (2014).

[15] T.-C. Wei and R. Raussendorf ``Universal measurement-based quantum computation with spin-2 Affleck-Kennedy-Lieb-Tasaki states'' Phys. Rev. A 92, 012310 (2015).

[16] J. Miller and A. Miyake ``Hierarchy of universal entanglement in 2D measurement-based quantum computation'' npj Quantum Information 2, 16036 (2016).

[17] T.-C. Wei and C.-Y. Huang ``Universal measurement-based quantum computation in two-dimensional symmetry-protected topological phases'' Phys. Rev. A 96, 032317 (2017).

[18] J. Miller and A. Miyake ``Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry'' Phys. Rev. Lett. 120, 170503 (2018).

[19] M. Gachechiladze, O. Gühne, and A. Miyake, ``Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states'' Phys. Rev. A 99, 052304 (2019).

[20] Z.-C. Guand X.-G. Wen ``Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order'' Phys. Rev. B 80, 155131 (2009).

[21] X. Chen, Z.-C. Gu, and X.-G. Wen, ``Classification of gapped symmetric phases in one-dimensional spin systems'' Phys. Rev. B 83, 035107 (2011).

[22] N. Schuch, D. Pérez-García, and I. Cirac, ``Classifying quantum phases using matrix product states and projected entangled pair states'' Phys. Rev. B 84, 165139 (2011).

[23] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, ``Symmetry protection of topological phases in one-dimensional quantum spin systems'' Phys. Rev. B 85, 075125 (2012).

[24] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, ``Symmetry protected topological orders and the group cohomology of their symmetry group'' Phys. Rev. B 87, 155114 (2013).

[25] A. Miyake ``Quantum Computation on the Edge of a Symmetry-Protected Topological Order'' Phys. Rev. Lett. 105, 040501 (2010).

[26] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, ``Rigorous results on valence-bond ground states in antiferromagnets'' Phys. Rev. Lett. 59, 799-802 (1987).

[27] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, ``Valence bond ground states in isotropic quantum antiferromagnets'' Communications in Mathematical Physics 115, 477-528 (1988).

[28] D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, ``Symmetry-Protected Phases for Measurement-Based Quantum Computation'' Phys. Rev. Lett. 108, 240505 (2012).

[29] D. V. Else, S. D. Bartlett, and A. C. Doherty, ``Symmetry protection of measurement-based quantum computation in ground states'' New Journal of Physics 14, 113016 (2012).

[30] J. Miller and A. Miyake ``Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation'' Phys. Rev. Lett. 114, 120506 (2015).

[31] D. T. Stephen, D.-S. Wang, A. Prakash, T.-C. Wei, and R. Raussendorf, ``Computational Power of Symmetry-Protected Topological Phases'' Phys. Rev. Lett. 119, 010504 (2017).

[32] R. Raussendorf, D.-S. Wang, A. Prakash, T.-C. Wei, and D. T. Stephen, ``Symmetry-protected topological phases with uniform computational power in one dimension'' Phys. Rev. A 96, 012302 (2017).

[33] R. Raussendorf, C. Okay, D.-S. Wang, D. T. Stephen, and H. P. Nautrup, ``Computationally Universal Phase of Quantum Matter'' Phys. Rev. Lett. 122, 090501 (2019).

[34] T. Devakul and D. J. Williamson ``Universal quantum computation using fractal symmetry-protected cluster phases'' Phys. Rev. A 98, 022332 (2018).

[35] D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eisert, and R. Raussendorf, ``Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter'' Quantum 3, 142 (2019).

[36] D. E. Browne, E. Kashefi, M. Mhalla, and S. Perdrix, ``Generalized flow and determinism in measurement-based quantum computation'' New Journal of Physics 9, 250-250 (2007).

[37] M. Mhalla, M. Murao, S. Perdrix, M. Someya, and P. S. Turner, ``Which graph states are useful for quantum information processing?'' Conference on Quantum Computation, Communication, and Cryptography 174-187 (2011).

[38] J. Richter, J. Schulenburg, and A. Honecker, ``Quantum magnetism in two dimensions: From semi-classical Né'' Springer Berlin Heidelberg (2004).

[39] D.-M. Schlingemann, H. Vogts, and R. F. Werner, ``On the structure of Clifford quantum cellular automata'' Journal of Mathematical Physics 49, 112104 (2008).

[40] J. Gütschow, S. Uphoff, R. F. Werner, and Z. Zimborás, ``Time asymptotics and entanglement generation of Clifford quantum cellular automata'' Journal of Mathematical Physics 51, 015203 (2010).

[41] D. Gottesman ``The Heisenberg representation of quantum computers'' arXiv preprint quant-ph/​9807006 (1998).

[42] X. Chen, Z.-C. Gu, and X.-G. Wen, ``Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order'' Phys. Rev. B 82, 155138 (2010).

[43] X. Chen, Z.-X. Liu, and X.-G. Wen, ``Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations'' Phys. Rev. B 84, 235141 (2011).

[44] Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, ``Subsystem symmetry protected topological order'' Phys. Rev. B 98, 035112 (2018).

[45] T. Devakul, D. J. Williamson, and Y. You, ``Classification of subsystem symmetry-protected topological phases'' Phys. Rev. B 98, 235121 (2018).

[46] M. B. Hastings and X.-G. Wen ``Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance'' Phys. Rev. B 72, 045141 (2005).

[47] J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, ``Matrix product unitaries: structure, symmetries, and topological invariants'' Journal of Statistical Mechanics: Theory and Experiment 2017, 083105 (2017).

[48] M. B. Şahinoğlu, S. K. Shukla, F. Bi, and X. Chen, ``Matrix product representation of locality preserving unitaries'' Phys. Rev. B 98, 245122 (2018).

[49] D. J. Williamson, N. Bultinck, M. Mariën, M. B. Şahinoğlu, J. Haegeman, and F. Verstraete, ``Matrix product operators for symmetry-protected topological phases: Gauging and edge theories'' Phys. Rev. B 94, 205150 (2016).

[50] R. Raussendorf, J. Harrington, and K. Goyal, ``A fault-tolerant one-way quantum computer'' Annals of Physics 321, 2242-2270 (2006).

[51] S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, ``Symmetry-protected topological order at nonzero temperature'' Phys. Rev. A 96, 022306 (2017).

[52] S. Roberts and S. D. Bartlett ``Symmetry-protected self-correcting quantum memories'' arXiv preprint arXiv:1805.01474 (2018).

[53] A. Kubica and B. Yoshida ``Ungauging quantum error-correcting codes'' arXiv preprint arXiv:1805.01836 (2018).

[54] R. Silva, E. F. Galvão, and E. Kashefi, ``Closed timelike curves in measurement-based quantum computation'' Phys. Rev. A 83, 012316 (2011).

[55] B. Coecke and R. Duncan ``Interacting quantum observables: categorical algebra and diagrammatics'' New Journal of Physics 13, 043016 (2011).

[56] E. Jeandel, S. Perdrix, and R. Vilmart, ``A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics'' Proceedings of the 33rd Annual ACM/​IEEE Symposium on Logic in Computer Science 559-568 (2018).

[57] N. Nickerson and H. Bombín ``Measurement based fault tolerance beyond foliation'' arXiv preprint arXiv:1810.09621 (2018).

[58] H. Bombin ``2D quantum computation with 3D topological codes'' arXiv preprint arXiv:1810.09571 (2018).

[59] J. C. Bridgeman and C. T. Chubb ``Hand-waving and interpretive dance: an introductory course on tensor networks'' Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017).

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[2] Arpit Dua, Pratyush Sarkar, Dominic J. Williamson, and Meng Cheng, "Bifurcating entanglement-renormalization group flows of fracton stabilizer models", arXiv:1909.12304.

[3] Nathanan Tantivasadakarn and Sagar Vijay, "Searching for Fracton Orders via Symmetry Defect Condensation", arXiv:1912.02826.

[4] Michael Newman, Leonardo Andreta de Castro, and Kenneth R. Brown, "Generating Fault-Tolerant Cluster States from Crystal Structures", arXiv:1909.11817.

[5] Patrik Knopf and Kei Fong Lam, "Convergence of a Robin boundary approximation for a Cahn--Hilliard system with dynamic boundary conditions", arXiv:1908.06124.

[6] Daniel Azses, Rafael Haenel, Yehuda Naveh, Robert Raussendorf, Eran Sela, and Emanuele G. Dalla Torre, "Identification of symmetry-protected topological states on noisy quantum computers", arXiv:2002.04620.

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