Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter

David T. Stephen1, Hendrik Poulsen Nautrup2, Juani Bermejo-Vega3, Jens Eisert3, and Robert Raussendorf4,5

1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
2Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
3Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
4Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
5Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada

Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.

Certain quantum phases of matter have computational power. This power is utilized by measurement-based quantum computation, where the process of computation is driven by local measurements on an entangled many-body quantum state. When every state within a phase of matter can be used as a resource for universal measurement-based quantum computation, that phase is deemed “computationally universal”. The search for computationally universal phases of matter has a long history, and has led to important insights into the properties of different types of quantum order. In fact, the first computationally universal phase of matter contained a brand-new type of quantum order that is protected by subsystem symmetries, which are symmetries that act on lower-dimensional subsystems of the whole system (in this case, lines across a 2D lattice). Since this discovery, the notion of “subsystem symmetry-protected topological order’’ has garnered notable interest both in the condensed matter and quantum information communities.

In this work, we continue the trend of pushing our understanding of quantum phases of matter in order to understand their computational properties. We present a general framework for constructing computationally universal phases of matter protected by subsystem symmetries. The essential new ingredient that we employ is the concept of quantum cellular automata. Quantum cellular automata can be defined as locality preserving unitaries and are, for the most part, equivalent to quantum circuits with constant depth. The use of quantum cellular automata in our framework is threefold. First, they are used to define subsystem symmetries, such as the fractal operator pictured above. Second, they characterize the type of entanglement that is ubiquitous among states exhibiting non-trivial order under the subsystem symmetries. Third, they form the backbone of our computational schemes. Bringing everything together, we get new phases of matter wherein both the physical and computational properties are described by quantum cellular automata.

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[1] A. Kitaev ``Fault-tolerant quantum computation by anyons'' Ann. Phys. 303, 2 - 30 (2003).
https://doi.org/10.1016/S0003-4916(02)00018-0

[2] M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, ``Topological quantum computation'' Bull. Amer. Math. Soc. 40, 31-38 (2003).
https://doi.org/10.1090/S0273-0979-02-00964-3

[3] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, ``Non-Abelian anyons and topological quantum computation'' Rev. Mod. Phys. 80, 1083-1159 (2008).
https://doi.org/10.1103/RevModPhys.80.1083

[4] A. Y. Kitaev ``Unpaired Majorana fermions in quantum wires'' Physics-Uspekhi 44, 131-136 (2001).
https://doi.org/10.1070/1063-7869/44/10s/s29

[5] J. Alicea, Y. Oreg, G. Refael, F. Oppen, and Fisher, ``Non-Abelian statistics and topological quantum information processing in 1D wire networks'' Nature Physics 7, 412-417 (2011).
https://doi.org/10.1038/nphys1915

[6] R. M. Lutchyn, Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y. Oreg, ``Majorana zero modes in superconductor-semiconductor heterostructures'' Nature Reviews Materials 3, 52-68 (2018).
https://doi.org/10.1038/s41578-018-0003-1

[7] 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).
https://doi.org/10.1103/PhysRevLett.103.020506

[8] A. Miyake ``Quantum Computation on the Edge of a Symmetry-Protected Topological Order'' Phys. Rev. Lett. 105, 040501 (2010).
https://doi.org/10.1103/PhysRevLett.105.040501

[9] S. D. Bartlett, G. K. Brennen, A. Miyake, and J. M. Renes, ``Quantum Computational Renormalization in the Haldane Phase'' Phys. Rev. Lett. 105, 110502 (2010).
https://doi.org/10.1103/PhysRevLett.105.110502

[10] 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).
https://doi.org/10.1103/PhysRevLett.108.240505

[11] 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).
https://doi.org/10.1088/1367-2630/14/11/113016

[12] J. Miller and A. Miyake ``Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation'' Phys. Rev. Lett. 114, 120506 (2015).
https://doi.org/10.1103/PhysRevLett.114.120506

[13] D.-S. Wang, D. T. Stephen, and R. Raussendorf, ``Qudit quantum computation on matrix product states with global symmetry'' Phys. Rev. A 95, 032312 (2017).
https://doi.org/10.1103/PhysRevA.95.032312

[14] 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).
https://doi.org/10.1103/PhysRevLett.119.010504

[15] 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).
https://doi.org/10.1103/PhysRevA.96.012302

[16] H. Poulsen Nautrup and T.-C. Wei ``Symmetry-protected topologically ordered states for universal quantum computation'' Phys. Rev. A 92, 052309 (2015).
https://doi.org/10.1103/PhysRevA.92.052309

[17] J. Miller and A. Miyake ``Hierarchy of universal entanglement in 2D measurement-based quantum computation'' npj Quantum Information 2, 16036 (2016).
https://doi.org/10.1038/npjqi.2016.36

[18] J. Miller and A. Miyake ``Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry'' Phys. Rev. Lett. 120, 170503 (2018).
https://doi.org/10.1103/PhysRevLett.120.170503

[19] 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).
https://doi.org/10.1103/PhysRevA.96.032317

[20] Y. Chen, A. Prakash, and T.-C. Wei, ``Universal quantum computing using $(Z_d)^3$ symmetry-protected topologically ordered states'' Phys. Rev. A 97, 022305 (2018).
https://doi.org/10.1103/PhysRevA.97.022305

[21] 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).
https://doi.org/10.1103/PhysRevLett.122.090501

[22] T. Devakul and D. J. Williamson ``Universal quantum computation using fractal symmetry-protected cluster phases'' Phys. Rev. A 98, 022332 (2018).
https://doi.org/10.1103/PhysRevA.98.022332

[23] S. Vijay, T. H. Hsieh, and L. Fu, ``Majorana Fermion Surface Code for Universal Quantum Computation'' Phys. Rev. X 5, 041038 (2015).
https://doi.org/10.1103/PhysRevX.5.041038

[24] A. Dua, B. Malomed, M. Cheng, and L. Jiang, ``Universal quantum computing with parafermions assisted by a half fluxon'' (2018).
arXiv:arXiv:1803.05886
https:/​/​arxiv.org/​abs/​1803.05886

[25] B. Bauer, T. Pereg-Barnea, T. Karzig, M.-T. Rieder, G. Refael, E. Berg, and Y. Oreg, ``Topologically protected braiding in a single wire using Floquet Majorana modes'' (2018).
arXiv:arXiv:1808.07066
https:/​/​arxiv.org/​abs/​1808.07066

[26] 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).
https://doi.org/10.1103/PhysRevA.76.052315

[27] H. Bombin and M. A. Martin-Delgado ``Family of non-Abelian Kitaev models on a lattice: Topological condensation and confinement'' Phys. Rev. B 78, 115421 (2008).
https://doi.org/10.1103/PhysRevB.78.115421

[28] B. Yoshida ``Gapped boundaries, group cohomology and fault-tolerant logical gates'' Ann. Phys. 377, 387 - 413 (2017).
https://doi.org/10.1016/j.aop.2016.12.014

[29] S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, ``Symmetry-protected topological order at nonzero temperature'' Phys. Rev. A 96, 022306 (2017).
https://doi.org/10.1103/PhysRevA.96.022306

[30] R. Raussendorf and H. J. Briegel ``A One-Way Quantum Computer'' Phys. Rev. Lett. 86, 5188-5191 (2001).
https://doi.org/10.1103/PhysRevLett.86.5188

[31] R. Raussendorf, D. E. Browne, and H. J. Briegel, ``Measurement-based quantum computation on cluster states'' Phys. Rev. A 68, 022312 (2003).
https://doi.org/10.1103/PhysRevA.68.022312

[32] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, ``Entanglement spectrum of a topological phase in one dimension'' Phys. Rev. B 81, 064439 (2010).
https://doi.org/10.1103/PhysRevB.81.064439

[33] 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).
https://doi.org/10.1103/PhysRevB.84.165139

[34] 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).
https://doi.org/10.1103/PhysRevB.83.035107

[35] 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).
https://doi.org/10.1103/PhysRevB.87.155114

[36] T. Senthil ``Symmetry-Protected Topological Phases of Quantum Matter'' Ann. Rev. Cond. Mat. Phys. 6, 299-324 (2015).
https://doi.org/10.1146/annurev-conmatphys-031214-014740

[37] A. Nietner, C. Krumnow, E. J. Bergholtz, and J. Eisert, ``Composite symmetry-protected topological order and effective models'' Phys. Rev. B 96, 235138 (2017).
https://doi.org/10.1103/PhysRevB.96.235138

[38] H. Song, S.-J. Huang, L. Fu, and M. Hermele, ``Topological Phases Protected by Point Group Symmetry'' Phys. Rev. X 7, 011020 (2017).
https://doi.org/10.1103/PhysRevX.7.011020

[39] R. Thorngren and D. V. Else ``Gauging Spatial Symmetries and the Classification of Topological Crystalline Phases'' Phys. Rev. X 8, 011040 (2018).
https://doi.org/10.1103/PhysRevX.8.011040

[40] S.-J. Huang, H. Song, Y.-P. Huang, and M. Hermele, ``Building crystalline topological phases from lower-dimensional states'' Phys. Rev. B 96, 205106 (2017).
https://doi.org/10.1103/PhysRevB.96.205106

[41] Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, ``Subsystem symmetry protected topological order'' Phys. Rev. B 98, 035112 (2018).
https://doi.org/10.1103/PhysRevB.98.035112

[42] T. Devakul, D. J. Williamson, and Y. You, ``Classification of subsystem symmetry-protected topological phases'' Phys. Rev. B 98, 235121 (2018).
https://doi.org/10.1103/PhysRevB.98.235121

[43] Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, ``Symmetric Fracton Matter: Twisted and Enriched'' (2018).
arXiv:arXiv:1805.09800
https:/​/​arxiv.org/​abs/​1805.09800

[44] D. J. Williamson ``Fractal symmetries: Ungauging the cubic code'' Phys. Rev. B 94, 155128 (2016).
https://doi.org/10.1103/PhysRevB.94.155128

[45] T. Devakul, Y. You, F. J. Burnell, and S. L. Sondhi, ``Fractal Symmetric Phases of Matter'' SciPost Phys. 6, 7 (2019).
https://doi.org/10.21468/SciPostPhys.6.1.007

[46] A. Kubica and B. Yoshida ``Ungauging quantum error-correcting codes'' (2018).
arXiv:arXiv:1805.01836
https:/​/​arxiv.org/​abs/​1805.01836

[47] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, ``Generalized global symmetries'' J. High En. Phys. 2015, 172 (2015).
https://doi.org/10.1007/JHEP02(2015)172

[48] B. Yoshida ``Topological phases with generalized global symmetries'' Phys. Rev. B 93, 155131 (2016).
https://doi.org/10.1103/PhysRevB.93.155131

[49] S. Roberts and S. D. Bartlett ``Symmetry-protected self-correcting quantum memories'' (2018).
arXiv:arXiv:1805.01474
https:/​/​arxiv.org/​abs/​1805.01474

[50] J. Haah ``Local stabilizer codes in three dimensions without string logical operators'' Phys. Rev. A 83, 042330 (2011).
https://doi.org/10.1103/PhysRevA.83.042330

[51] B. Yoshida ``Exotic topological order in fractal spin liquids'' Phys. Rev. B 88, 125122 (2013).
https://doi.org/10.1103/PhysRevB.88.125122

[52] S. Vijay, J. Haah, and L. Fu, ``A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations'' Phys. Rev. B 92, 235136 (2015).
https://doi.org/10.1103/PhysRevB.92.235136

[53] S. Vijay, J. Haah, and L. Fu, ``Fracton topological order, generalized lattice gauge theory, and duality'' Phys. Rev. B 94, 235157 (2016).
https://doi.org/10.1103/PhysRevB.94.235157

[54] Z. Nussinov and G. Ortiz ``A symmetry principle for topological quantum order'' Ann. Phys. 324, 977 - 1057 (2009).
https://doi.org/10.1016/j.aop.2008.11.002

[55] Z. Nussinov and G. Ortiz ``Sufficient symmetry conditions for Topological Quantum Order'' Proceedings of the National Academy of Sciences 106, 16944-16949 (2009).
https://doi.org/10.1073/pnas.0803726105

[56] W. Shirley, K. Slagle, and X. Chen, ``Foliated fracton order from gauging subsystem symmetries'' SciPost Phys. 6, 41 (2019).
https://doi.org/10.21468/SciPostPhys.6.4.041

[57] H. Song, A. Prem, S.-J. Huang, and M. A. Martin-Delgado, ``Twisted fracton models in three dimensions'' Phys. Rev. B 99, 155118 (2019).
https://doi.org/10.1103/PhysRevB.99.155118

[58] J. C. Bridgemanand C. T. Chubb ``Hand-waving and interpretive dance: an introductory course on tensor networks'' J. Phys. A 50, 223001 (2017).
https://doi.org/10.1088/1751-8121/aa6dc3

[59] N. Schuch, I. Cirac, and D. Perez-Garcia, ``PEPS as ground states: Degeneracy and topology'' Ann. Phys. 325, 2153 - 2192 (2010).
https://doi.org/10.1016/j.aop.2010.05.008

[60] F. Pollmann and A. M. Turner ``Detection of symmetry-protected topological phases in one dimension'' Phys. Rev. B 86, 125441 (2012).
https://doi.org/10.1103/PhysRevB.86.125441

[61] N. Bultinck, M. Mariën, D. Williamson, M. Şahinoğlu, J. Haegeman, and F. Verstraete, ``Anyons and matrix product operator algebras'' Ann. Phys. 378, 183 - 233 (2017).
https://doi.org/10.1016/j.aop.2017.01.004

[62] K. Duivenvoorden, M. Iqbal, J. Haegeman, F. Verstraete, and N. Schuch, ``Entanglement phases as holographic duals of anyon condensates'' Phys. Rev. B 95, 235119 (2017).
https://doi.org/10.1103/PhysRevB.95.235119

[63] 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).
https://doi.org/10.1103/PhysRevB.94.205150

[64] S. Jiang and Y. Ran ``Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases'' Phys. Rev. B 95, 125107 (2017).
https://doi.org/10.1103/PhysRevB.95.125107

[65] A. Molnar, Y. Ge, N. Schuch, and J. I. Cirac, ``A generalization of the injectivity condition for projected entangled pair states'' J. Math. Phys. 59, 021902 (2018).
https://doi.org/10.1063/1.5007017

[66] D. Poilblanc, J. I. Cirac, and N. Schuch, ``Chiral topological spin liquids with projected entangled pair states'' Phys. Rev. B 91, 224431 (2015).
https://doi.org/10.1103/PhysRevB.91.224431

[67] D. J. Williamson, N. Bultinck, and F. Verstraete, ``Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation'' (2017).
arXiv:arXiv:1711.07982
https:/​/​arxiv.org/​abs/​1711.07982

[68] N. Bultinck, D. J. Williamson, J. Haegeman, and F. Verstraete, ``Fermionic projected entangled-pair states and topological phases'' J. Phys. A 51, 025202 (2017).
https://doi.org/10.1088/1751-8121/aa99cc

[69] H. Dreyer, J. I. Cirac, and N. Schuch, ``Projected entangled pair states with continuous virtual symmetries'' Phys. Rev. B 98, 115120 (2018).
https://doi.org/10.1103/PhysRevB.98.115120

[70] J. F. Fitzsimons ``Private quantum computation: an introduction to blind quantum computing and related protocols'' npj Quantum Information 3, 23 (2017).
https://doi.org/10.1038/s41534-017-0025-3

[71] A. Mantri, T. F. Demarie, N. C. Menicucci, and J. F. Fitzsimons, ``Flow Ambiguity: A Path Towards Classically Driven Blind Quantum Computation'' Phys. Rev. X 7, 031004 (2017).
https://doi.org/10.1103/PhysRevX.7.031004

[72] R. Raussendorf ``Quantum computation via translation-invariant operations on a chain of qubits'' Phys. Rev. A 72, 052301 (2005).
https://doi.org/10.1103/PhysRevA.72.052301

[73] J. Fitzsimons and J. Twamley ``Globally Controlled Quantum Wires for Perfect Qubit Transport, Mirroring, and Computing'' Phys. Rev. Lett. 97, 090502 (2006).
https://doi.org/10.1103/PhysRevLett.97.090502

[74] J. Fitzsimons, L. Xiao, S. C. Benjamin, and J. A. Jones, ``Quantum Information Processing with Delocalized Qubits under Global Control'' Phys. Rev. Lett. 99, 030501 (2007).
https://doi.org/10.1103/PhysRevLett.99.030501

[75] J. Bermejo-Vega, D. Hangleiter, M. Schwarz, R. Raussendorf, and J. Eisert, ``Architectures for Quantum Simulation Showing a Quantum Speedup'' Phys. Rev. X 8, 021010 (2018).
https://doi.org/10.1103/PhysRevX.8.021010

[76] D. Hangleiter, J. Bermejo-Vega, M. Schwarz, and J. Eisert, ``Anticoncentration theorems for schemes showing a quantum speedup'' Quantum 2, 65 (2018).
https://doi.org/10.22331/q-2018-05-22-65

[77] B. Schumacher and R. F. Werner ``Reversible quantum cellular automata'' (2004).
arXiv:arXiv:quant-ph/0405174
https:/​/​arxiv.org/​abs/​quant-ph/​0405174

[78] D.-M. Schlingemann, H. Vogts, and R. F. Werner, ``On the structure of Clifford quantum cellular automata'' J. Math. Phys. 49, 112104 (2008).
https://doi.org/10.1063/1.3005565

[79] J. Gütschow, S. Uphoff, R. F. Werner, and Z. Zimboras, ``Time asymptotics and entanglement generation of Clifford quantum cellular automata'' J. Math. Phys. 51, 015203 (2010).
https://doi.org/10.1063/1.3278513

[80] J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, ``Matrix product unitaries: structure, symmetries, and topological invariants'' J. Stat. Mech. 2017, 083105 (2017).
https://doi.org/10.1088/1742-5468/aa7e55

[81] D. Gottesman ``Fault-Tolerant Quantum Computation with Higher-Dimensional Systems'' Selected papers from the First NASA International Conference on Quantum Computing and Quantum Communications (1998).
https://doi.org/10.1007/3-540-49208-9_27

[82] J. Bermejo-Vega and M. Van Den Nest ``Classical Simulations of Abelian-group Normalizer Circuits with Intermediate Measurements'' Quant. Inf. Comp. 14, 181-216 (2014).
https://doi.org/10.26421/QIC14.3-4

[83] A. Mantri, T. F. Demarie, and J. F. Fitzsimons, ``Universality of quantum computation with cluster states and (X, Y)-plane measurements'' Scientific Reports 7, 42861 (2017).
https://doi.org/10.1038/srep42861

[84] D. Gross, V. Nesme, H. Vogts, and R. F. Werner, ``Index Theory of One Dimensional Quantum Walks and Cellular Automata'' Commun. Math. Phys. 310, 419-454 (2012).
https://doi.org/10.1007/s00220-012-1423-1

[85] F. Verstraete, V. Murg, and J. Cirac, ``Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems'' Adv. Phys. 57, 143-224 (2008).
https://doi.org/10.1080/14789940801912366

[86] 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).
https://doi.org/10.1103/PhysRevLett.59.799

[87] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, ``Matrix Product State Representations'' Quantum Info. Comput. 7, 401-430 (2007).
http:/​/​dl.acm.org/​citation.cfm?id=2011832.2011833

[88] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, ``Entropy Scaling and Simulability by Matrix Product States'' Phys. Rev. Lett. 100, 030504 (2008).
https://doi.org/10.1103/PhysRevLett.100.030504

[89] F. Verstraete and J. I. Cirac ``Matrix product states represent ground states faithfully'' Phys. Rev. B 73, 094423 (2006).
https://doi.org/10.1103/PhysRevB.73.094423

[90] D. Sauerwein, A. Molnar, J. I. Cirac, and B. Kraus, ``Matrix Product States: Entanglement, symmetries, and state transformations'' (2019).
arXiv:arXiv:1901.07448
https:/​/​arxiv.org/​abs/​1901.07448

[91] D. Gottesman ``Stabilizer codes and quantum error correction'' (1997).
arXiv:arXiv:quant-ph/9705052
https:/​/​arxiv.org/​abs/​quant-ph/​9705052

[92] M. Hein, J. Eisert, and H. J. Briegel, ``Multiparty entanglement in graph states'' Phys. Rev. A 69, 062311 (2004).
https://doi.org/10.1103/PhysRevA.69.062311

[93] D. Pérez-García, 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).
https://doi.org/10.1103/PhysRevLett.100.167202

[94] I. Marvian ``Symmetry-protected topological entanglement'' Phys. Rev. B 95, 045111 (2017).
https://doi.org/10.1103/PhysRevB.95.045111

[95] I. G. Berkovich and E. Zhmud ``Characters of finite groups'' American Mathematical Soc. (1998).

[96] A. Molnar, J. Garre-Rubio, D. Pérez-García, N. Schuch, and J. I. Cirac, ``Normal projected entangled pair states generating the same state'' New Journal of Physics 20, 113017 (2018).
https://doi.org/10.1088/1367-2630/aae9fa

[97] X.-G. Wen ``Colloquium: Zoo of quantum-topological phases of matter'' Rev. Mod. Phys. 89, 041004 (2017).
https://doi.org/10.1103/RevModPhys.89.041004

[98] B. Nachtergaele ``The spectral gap for some spin chains with discrete symmetry breaking'' Communications in Mathematical Physics 175, 565-606 (1996).
https://doi.org/10.1007/BF02099509

[99] A. S. Darmawan and S. D. Bartlett ``Graph states as ground states of two-body frustration-free Hamiltonians'' New J. Phys. 16, 073013 (2014).
https://doi.org/10.1088/1367-2630/16/7/073013

[100] R. Verresen, R. Moessner, and F. Pollmann, ``One-dimensional symmetry protected topological phases and their transitions'' Phys. Rev. B 96, 165124 (2017).
https://doi.org/10.1103/PhysRevB.96.165124

[101] D. Grossand J. Eisert ``Novel Schemes for Measurement-Based Quantum Computation'' Phys. Rev. Lett. 98, 220503 (2007).
https://doi.org/10.1103/PhysRevLett.98.220503

[102] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, ``Elementary gates for quantum computation'' Phys. Rev. A 52, 3457-3467 (1995).
https://doi.org/10.1103/PhysRevA.52.3457

[103] D. Deutsch, A. Barenco, and A. Ekert, ``Universality in quantum computation'' Proc. R. Soc. London A 449, 669-677 (1995).
https://doi.org/10.1098/rspa.1995.0065

[104] M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, ``Practical Scheme for Quantum Computation with Any Two-Qubit Entangling Gate'' Phys. Rev. Lett. 89, 247902 (2002).
https://doi.org/10.1103/PhysRevLett.89.247902

[105] 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).
https://doi.org/10.1103/PhysRevA.99.052304

[106] R. Raussendorf, J. Harrington, and K. Goyal, ``A fault-tolerant one-way quantum computer'' Annals of Physics 321, 2242 - 2270 (2006).
https://doi.org/10.1016/j.aop.2006.01.012

[107] R Raussendorf, J Harrington, and K Goyal, ``Topological fault-tolerance in cluster state quantum computation'' New Journal of Physics 9, 199-199 (2007).
https://doi.org/10.1088/1367-2630/9/6/199

[108] B. Voorhees ``A note on injectivity of additive cellular automata'' Complex Systems 8, 151-160 (1994).

Cited by

[1] Trithep Devakul, "Classifying local fractal subsystem symmetry-protected topological phases", arXiv:1812.02721, Physical Review B 99 23, 235131 (2019).

[2] Trithep Devakul, Dominic J. Williamson, and Yizhi You, "Classification of subsystem symmetry-protected topological phases", Physical Review B 98 23, 235121 (2018).

[3] Kevin Slagle, David Aasen, and Dominic Williamson, "Foliated field theory and string-membrane-net condensation picture of fracton order", SciPost Physics 6 4, 043 (2019).

[4] Dong-Sheng Wang, "Quantum computation by teleportation and symmetry", arXiv:1808.06103.

[5] Albert T. Schmitz, Sheng-Jie Huang, and Abhinav Prem, "Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter", Physical Review B 99 20, 205109 (2019).

[6] Abhinav Prem and Dominic J. Williamson, "Gauging permutation symmetries as a route to non-Abelian fractons", arXiv:1905.06309.

[7] Dominic J. Williamson, Arpit Dua, and Meng Cheng, "Spurious Topological Entanglement Entropy from Subsystem Symmetries", Physical Review Letters 122 14, 140506 (2019).

[8] Trithep Devakul and Dominic J. Williamson, "Universal quantum computation using fractal symmetry-protected cluster phases", Physical Review A 98 2, 022332 (2018).

[9] Terry Farrelly, "A review of Quantum Cellular Automata", arXiv:1904.13318.

[10] David T. Stephen, Henrik Dreyer, Mohsin Iqbal, and Norbert Schuch, "Detecting subsystem symmetry-protected topological order via entanglement entropy", arXiv:1904.09450.

The above citations are from Crossref's cited-by service (last updated 2019-06-20 13:13:17) and SAO/NASA ADS (last updated 2019-06-20 13:13:18). The list may be incomplete as not all publishers provide suitable and complete citation data.