Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
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
| Published: | 2019-05-20, volume 3, page 142 |
| Eprint: | arXiv:1806.08780v3 |
| Doi: | https://doi.org/10.22331/q-2019-05-20-142 |
| Citation: | Quantum 3, 142 (2019). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
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.
Popular summary
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.
► BibTeX data
► References
[1] A.Yu. Kitaev ``Fault-tolerant quantum computation by anyons'' Ann. Phys. 303, 2 - 30 (2003).
https://doi.org/10.1016/S0003-4916(02)00018-0
[2] Michael Freedman, Alexei Kitaev, Michael Larsen, and Zhenghan Wang, ``Topological quantum computation'' Bull. Amer. Math. Soc. 40, 31–38 (2003).
https://doi.org/10.1090/S0273-0979-02-00964-3
[3] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar 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 Yu Kitaev ``Unpaired Majorana fermions in quantum wires'' Physics-Uspekhi 44, 131–136 (2001).
https://doi.org/10.1070/1063-7869/44/10s/s29
[5] Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, and Matthew P. A. 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, E. P. A. M. 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] Andrew C. Doherty and Stephen 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] Akimasa 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] Stephen D. Bartlett, Gavin K. Brennen, Akimasa Miyake, and Joseph M. Renes, ``Quantum Computational Renormalization in the Haldane Phase'' Phys. Rev. Lett. 105, 110502 (2010).
https://doi.org/10.1103/PhysRevLett.105.110502
[10] Dominic V. Else, Ilai Schwarz, Stephen D. Bartlett, and Andrew 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] Dominic V Else, Stephen D Bartlett, and Andrew 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] Jacob Miller and Akimasa 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] Dong-Sheng Wang, David T. Stephen, and Robert 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] David T. Stephen, Dong-Sheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, and Robert Raussendorf, ``Computational Power of Symmetry-Protected Topological Phases'' Phys. Rev. Lett. 119, 010504 (2017).
https://doi.org/10.1103/PhysRevLett.119.010504
[15] Robert Raussendorf, Dong-Sheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, and David 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] Hendrik Poulsen Nautrup and Tzu-Chieh 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] Jacob Miller and Akimasa 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] Jacob Miller and Akimasa 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] Tzu-Chieh Wei and Ching-Yu 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] Yanzhu Chen, Abhishodh Prakash, and Tzu-Chieh 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] Robert Raussendorf, Cihan Okay, Dong-Sheng Wang, David T. Stephen, and Hendrik Poulsen Nautrup, ``Computationally Universal Phase of Quantum Matter'' Phys. Rev. Lett. 122, 090501 (2019).
https://doi.org/10.1103/PhysRevLett.122.090501
[22] Trithep Devakul and Dominic 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] Sagar Vijay, Timothy H. Hsieh, and Liang Fu, ``Majorana Fermion Surface Code for Universal Quantum Computation'' Phys. Rev. X 5, 041038 (2015).
https://doi.org/10.1103/PhysRevX.5.041038
[24] Arpit Dua, Boris Malomed, Meng Cheng, and Liang Jiang, ``Universal quantum computing with parafermions assisted by a half fluxon'' (2018).
https://arxiv.org/abs/1803.05886
[25] Bela Bauer, T. Pereg-Barnea, Torsten Karzig, Maria-Theresa Rieder, Gil Refael, Erez Berg, and Yuval Oreg, ``Topologically protected braiding in a single wire using Floquet Majorana modes'' (2018).
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] Beni 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] Sam Roberts, Beni Yoshida, Aleksander Kubica, and Stephen D. Bartlett, ``Symmetry-protected topological order at nonzero temperature'' Phys. Rev. A 96, 022306 (2017).
https://doi.org/10.1103/PhysRevA.96.022306
[30] Robert Raussendorf and Hans J. Briegel ``A One-Way Quantum Computer'' Phys. Rev. Lett. 86, 5188–5191 (2001).
https://doi.org/10.1103/PhysRevLett.86.5188
[31] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, ``Measurement-based quantum computation on cluster states'' Phys. Rev. A 68, 022312 (2003).
https://doi.org/10.1103/PhysRevA.68.022312
[32] Frank Pollmann, Ari M. Turner, Erez Berg, and Masaki 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] Norbert Schuch, David Pérez-García, and Ignacio 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] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang 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] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang 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] Hao Song, Sheng-Jie Huang, Liang Fu, and Michael Hermele, ``Topological Phases Protected by Point Group Symmetry'' Phys. Rev. X 7, 011020 (2017).
https://doi.org/10.1103/PhysRevX.7.011020
[39] Ryan Thorngren and Dominic 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] Sheng-Jie Huang, Hao Song, Yi-Ping Huang, and Michael Hermele, ``Building crystalline topological phases from lower-dimensional states'' Phys. Rev. B 96, 205106 (2017).
https://doi.org/10.1103/PhysRevB.96.205106
[41] Yizhi You, Trithep 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] Trithep Devakul, Dominic J. Williamson, and Yizhi You, ``Classification of subsystem symmetry-protected topological phases'' Phys. Rev. B 98, 235121 (2018).
https://doi.org/10.1103/PhysRevB.98.235121
[43] Yizhi You, Trithep Devakul, F. J. Burnell, and S. L. Sondhi, ``Symmetric Fracton Matter: Twisted and Enriched'' (2018).
https://arxiv.org/abs/1805.09800
[44] Dominic J. Williamson ``Fractal symmetries: Ungauging the cubic code'' Phys. Rev. B 94, 155128 (2016).
https://doi.org/10.1103/PhysRevB.94.155128
[45] Trithep Devakul, Yizhi 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] Aleksander Kubica and Beni Yoshida ``Ungauging quantum error-correcting codes'' (2018).
https://arxiv.org/abs/1805.01836
[47] Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett, ``Generalized global symmetries'' J. High En. Phys. 2015, 172 (2015).
https://doi.org/10.1007/JHEP02(2015)172
[48] Beni Yoshida ``Topological phases with generalized global symmetries'' Phys. Rev. B 93, 155131 (2016).
https://doi.org/10.1103/PhysRevB.93.155131
[49] Sam Roberts and Stephen D. Bartlett ``Symmetry-protected self-correcting quantum memories'' (2018).
https://arxiv.org/abs/1805.01474
[50] Jeongwan 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] Beni Yoshida ``Exotic topological order in fractal spin liquids'' Phys. Rev. B 88, 125122 (2013).
https://doi.org/10.1103/PhysRevB.88.125122
[52] Sagar Vijay, Jeongwan Haah, and Liang 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] Sagar Vijay, Jeongwan Haah, and Liang 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] Zohar Nussinov and Gerardo 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] Zohar Nussinov and Gerardo 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] Wilbur Shirley, Kevin Slagle, and Xie Chen, ``Foliated fracton order from gauging subsystem symmetries'' SciPost Phys. 6, 41 (2019).
https://doi.org/10.21468/SciPostPhys.6.4.041
[57] Hao Song, Abhinav Prem, Sheng-Jie 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] Jacob C Bridgemanand Christopher 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] Norbert Schuch, Ignacio Cirac, and David 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] Frank Pollmann and Ari 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.J. Williamson, M.B. 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] Kasper Duivenvoorden, Mohsin Iqbal, Jutho Haegeman, Frank Verstraete, and Norbert Schuch, ``Entanglement phases as holographic duals of anyon condensates'' Phys. Rev. B 95, 235119 (2017).
https://doi.org/10.1103/PhysRevB.95.235119
[63] Dominic J. Williamson, Nick Bultinck, Michael Mariën, Mehmet B. Şahinoğlu, Jutho Haegeman, and Frank 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] Shenghan Jiang and Ying 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] Andras Molnar, Yimin Ge, Norbert Schuch, and J. Ignacio 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] Didier Poilblanc, J. Ignacio Cirac, and Norbert 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] Dominic J. Williamson, Nick Bultinck, and Frank Verstraete, ``Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation'' (2017).
https://arxiv.org/abs/1711.07982
[68] Nick Bultinck, Dominic J Williamson, Jutho Haegeman, and Frank Verstraete, ``Fermionic projected entangled-pair states and topological phases'' J. Phys. A 51, 025202 (2017).
https://doi.org/10.1088/1751-8121/aa99cc
[69] Henrik Dreyer, J. Ignacio Cirac, and Norbert Schuch, ``Projected entangled pair states with continuous virtual symmetries'' Phys. Rev. B 98, 115120 (2018).
https://doi.org/10.1103/PhysRevB.98.115120
[70] Joseph 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] Atul Mantri, Tommaso F. Demarie, Nicolas C. Menicucci, and Joseph 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] Robert 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] Joseph Fitzsimons and Jason 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] Joseph Fitzsimons, Li Xiao, Simon C. Benjamin, and Jonathan 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] Juan Bermejo-Vega, Dominik Hangleiter, Martin Schwarz, Robert Raussendorf, and Jens Eisert, ``Architectures for Quantum Simulation Showing a Quantum Speedup'' Phys. Rev. X 8, 021010 (2018).
https://doi.org/10.1103/PhysRevX.8.021010
[76] Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens 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).
https://arxiv.org/abs/quant-ph/0405174
[78] Dirk-M. Schlingemann, Holger Vogts, and Reinhard F. Werner, ``On the structure of Clifford quantum cellular automata'' J. Math. Phys. 49, 112104 (2008).
https://doi.org/10.1063/1.3005565
[79] Johannes Gütschow, Sonja Uphoff, Reinhard F. Werner, and Zoltan 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 Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete, ``Matrix product unitaries: structure, symmetries, and topological invariants'' J. Stat. Mech. 2017, 083105 (2017).
https://doi.org/10.1088/1742-5468/aa7e55
[81] Daniel 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] Juan Bermejo-Vega and Maarten 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] Atul Mantri, Tommaso F. Demarie, and Joseph 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.I. 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] Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal 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] Norbert Schuch, Michael M. Wolf, Frank Verstraete, and J. Ignacio 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] David Sauerwein, Andras Molnar, J. Ignacio Cirac, and Barbara Kraus, ``Matrix Product States: Entanglement, symmetries, and state transformations'' (2019).
https://arxiv.org/abs/1901.07448
[91] Daniel Gottesman ``Stabilizer codes and quantum error correction'' (1997).
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] Iman Marvian ``Symmetry-protected topological entanglement'' Phys. Rev. B 95, 045111 (2017).
https://doi.org/10.1103/PhysRevB.95.045111
[95] I_A G Berkovich and EM Zhmud ``Characters of finite groups'' American Mathematical Soc. (1998).
[96] Andras Molnar, José Garre-Rubio, David Pérez-García, Norbert Schuch, and J Ignacio 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] Xiao-Gang Wen ``Colloquium: Zoo of quantum-topological phases of matter'' Rev. Mod. Phys. 89, 041004 (2017).
https://doi.org/10.1103/RevModPhys.89.041004
[98] Bruno 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] Ruben Verresen, Roderich Moessner, and Frank 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] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter, ``Elementary gates for quantum computation'' Phys. Rev. A 52, 3457–3467 (1995).
https://doi.org/10.1103/PhysRevA.52.3457
[103] David Deutsch, Adriano Barenco, and Artur Ekert, ``Universality in quantum computation'' Proc. R. Soc. London A 449, 669–677 (1995).
https://doi.org/10.1098/rspa.1995.0065
[104] Michael J. Bremner, Christopher M. Dawson, Jennifer L. Dodd, Alexei Gilchrist, Aram W. Harrow, Duncan Mortimer, Michael A. Nielsen, and Tobias 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] Mariami Gachechiladze, Otfried Gühne, and Akimasa 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] Burton 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", Physical Review B 99 23, 235131 (2019).
[2] Giulio Camillo, Filipa C.R. Peres, Markus Heinrich, and Juani Bermejo-Vega, "Symmetry-Accelerated Classical Simulation of Clifford-Dominated Circuits", PRX Quantum 7 2, 020356 (2026).
[3] Aniruddha Bapat, Eddie Schoute, Alexey V. Gorshkov, and Andrew M. Childs, "Nearly optimal time-independent reversal of a spin chain", Physical Review Research 4 1, L012023 (2022).
[4] Tianyi Chen, Yunting Li, and Huangjun Zhu, "Efficient verification of Affleck-Kennedy-Lieb-Tasaki states", Physical Review A 107 2, 022616 (2023).
[5] Mengyao Hu, Eloïc Vallée, Tim Seynnaeve, Patrick Emonts, Fatemeh Mohammadi, and Jordi Tura, "Characterizing translation-invariant Bell inequalities using tropical algebra and graph polytopes", Physical Review A 113 3, 032421 (2026).
[6] Weiguang Cao, Linhao Li, Masahito Yamazaki, and Yunqin Zheng, "Subsystem non-invertible symmetry operators and defects", SciPost Physics 15 4, 155 (2023).
[7] Fiona J. Burnell, Trithep Devakul, Pranay Gorantla, Ho Tat Lam, and Shu-Heng Shao, "Anomaly inflow for subsystem symmetries", Physical Review B 106 8, 085113 (2022).
[8] Jonathan Francisco San Miguel, Arpit Dua, and Dominic J. Williamson, "Bifurcating subsystem symmetric entanglement renormalization in two dimensions", Physical Review B 103 3, 035148 (2021).
[9] Sam Roberts and Dominic J. Williamson, "3-Fermion Topological Quantum Computation", PRX Quantum 5 1, 010315 (2024).
[10] David T. Stephen, Henrik Dreyer, Mohsin Iqbal, and Norbert Schuch, "Detecting subsystem symmetry protected topological order via entanglement entropy", Physical Review B 100 11, 115112 (2019).
[11] Hendrik Poulsen Nautrup and Hans J. Briegel, "Measurement-based quantum computation from Clifford quantum cellular automata", Physical Review A 110 6, 062617 (2024).
[12] Zhangjie Qin, Daniel Azses, Eran Sela, Robert Raussendorf, and V. W. Scarola, "Redundant string symmetry-based error correction: Demonstrations on quantum devices", Physical Review A 110 1, 012465 (2024).
[13] Nathanan Tantivasadakarn and Ashvin Vishwanath, "Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps", Physical Review Letters 129 9, 090501 (2022).
[14] Abhinav Prem and Dominic Williamson, "Gauging permutation symmetries as a route to non-Abelian fractons", SciPost Physics 7 5, 068 (2019).
[15] J. P. Ibieta-Jimenez, L. N. Queiroz Xavier, M. Petrucci, and P. Teotonio-Sobrinho, "Fractonlike phases from subsystem symmetries", Physical Review B 102 4, 045104 (2020).
[16] Nathanan Tantivasadakarn and Sagar Vijay, "Searching for fracton orders via symmetry defect condensation", Physical Review B 101 16, 165143 (2020).
[17] Arpit Dua, Pratyush Sarkar, Dominic J. Williamson, and Meng Cheng, "Bifurcating entanglement-renormalization group flows of fracton stabilizer models", Physical Review Research 2 3, 033021 (2020).
[18] Weslei B. Fontana, Fabrizio G. Oliviero, Rodrigo G. Pereira, and Willian M. H. Natori, "Spin-orbital Kitaev model: From kagome spin ice to classical fractons", Physical Review B 111 19, 195112 (2025).
[19] Ho Tat Lam, "Classification of dipolar symmetry-protected topological phases: Matrix product states, stabilizer Hamiltonians, and finite tensor gauge theories", Physical Review B 109 11, 115142 (2024).
[20] Huangjun Zhu, Yunting Li, and Tianyi Chen, "Efficient Verification of Ground States of Frustration-Free Hamiltonians", Quantum 8, 1221 (2024).
[21] Christopher Fechisin, Nathanan Tantivasadakarn, and Victor V. Albert, "Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State", Physical Review X 15 1, 011058 (2025).
[22] Tzu-Chieh Wei, Robert Raussendorf, and Ian Affleck, Quantum Science and Technology 89 (2022) ISBN:978-3-031-03997-3.
[23] Huan He, Yizhi You, and Abhinav Prem, "Lieb-Schultz-Mattis–type constraints on fractonic matter", Physical Review B 101 16, 165145 (2020).
[24] Yuchen Guo, Jian-Hao Zhang, Zhen Bi, and Shuo Yang, "Triggering boundary phase transitions through bulk measurements in two-dimensional cluster states", Physical Review Research 5 4, 043069 (2023).
[25] Salvatore D. Pace, Guilherme Delfino, Ho Tat Lam, and Ömer Mert Aksoy, "Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections", SciPost Physics 18 1, 021 (2025).
[26] Austin K. Daniel and Akimasa Miyake, "Quantum Computational Advantage with String Order Parameters of One-Dimensional Symmetry-Protected Topological Order", Physical Review Letters 126 9, 090505 (2021).
[27] Arnab Adhikary, Wang Yang, and Robert Raussendorf, "Counterintuitive Yet Efficient Regimes for Measurement-Based Quantum Computation on Symmetry-Protected Spin Chains", Physical Review Letters 133 16, 160601 (2024).
[28] Wang Yang, Arnab Adhikary, and Robert Raussendorf, "Measurement-based quantum computation in symmetry-protected topological states of one-dimensional integer spin systems", Physical Review A 113 3, 032605 (2026).
[29] Paul Herringer, Vir B. Bulchandani, Younes Javanmard, David T. Stephen, and Robert Raussendorf, "Measurement-Based Quantum Computation in Symmetry-Enriched Topological Phases", PRX Quantum 6 4, 040314 (2025).
[30] Daniel Ranard, Michael Walter, and Freek Witteveen, "A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory", Annales Henri Poincaré 23 11, 3905 (2022).
[31] Lorenzo Piroli and J. Ignacio Cirac, "Quantum Cellular Automata, Tensor Networks, and Area Laws", Physical Review Letters 125 19, 190402 (2020).
[32] David T. Stephen, "Non-onsite symmetries and quantum teleportation in split-index matrix product states", Quantum 9, 1738 (2025).
[33] Lukas Hantzko, Arnab Adhikary, and Robert Raussendorf, "Sub-Riemannian geometry of measurement-based quantum computation", Physical Review A 113 1, 012444 (2026).
[34] Ke Ding, Hao-Ran Zhang, Bai-Ting Liu, and Shuo Yang, "Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases", Physical Review B 111 20, 205125 (2025).
[35] Trithep Devakul, Wilbur Shirley, and Juven Wang, "Strong planar subsystem symmetry-protected topological phases and their dual fracton orders", Physical Review Research 2 1, 012059 (2020).
[36] Wilbur Shirley, Kevin Slagle, and Xie Chen, "Twisted foliated fracton phases", Physical Review B 102 11, 115103 (2020).
[37] David T. Stephen, Arpit Dua, José Garre-Rubio, Dominic J. Williamson, and Michael Hermele, "Fractionalization of subsystem symmetries in two dimensions", Physical Review B 106 8, 085104 (2022).
[38] Trithep Devakul and Dominic J. Williamson, "Fractalizing quantum codes", Quantum 5, 438 (2021).
[39] Rahul Sahay and Ruben Verresen, "Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements", PRX Quantum 6 1, 010329 (2025).
[40] Meng Sun, Bowen Yang, Zongyuan Wang, Nathanan Tantivasadakarn, and Yu-An Chen, "Clifford Quantum Cellular Automata from Topological Quantum Field Theories and Invertible Subalgebras", PRX Quantum 7 1, 010362 (2026).
[41] Gabriel Wong, Robert Raussendorf, and Bartlomiej Czech, "The Gauge Theory of Measurement-Based Quantum Computation", Quantum 8, 1397 (2024).
[42] Tyler D. Ellison, Kohtaro Kato, Zi-Wen Liu, and Timothy H. Hsieh, "Symmetry-protected sign problem and magic in quantum phases of matter", Quantum 5, 612 (2021).
[43] Robert Raussendorf, Outstanding Contributions to Logic 25, 595 (2023) ISBN:978-3-031-24116-1.
[44] David T. Stephen, Arpit Dua, Ali Lavasani, and Rahul Nandkishore, "Nonlocal Finite-Depth Circuits for Constructing Symmetry-Protected Topological States and Quantum Cellular Automata", PRX Quantum 5 1, 010304 (2024).
[45] Qiang Jia and Zhian Jia, "Subsystem symmetry-protected topological phases from subsystem SymTFT of 2-foliated exotic tensor gauge theory", Journal of High Energy Physics 2025 9, 170 (2025).
[46] Ho Tat Lam, Jung Hoon Han, and Yizhi You, "Topological dipole insulator", SciPost Physics 17 5, 137 (2024).
[47] Tom Holden-Dye, Lluis Masanes, and Arijeet Pal, "Fundamental charges for dual-unitary circuits", Quantum 9, 1615 (2025).
[48] Hosho Katsura and Yu Nakayama, "Spontaneously broken supersymmetric fracton phases with fermionic subsystem symmetries", Journal of High Energy Physics 2022 8, 72 (2022).
[49] P Harata, K Wongsaprom, and P Srivilai, "Electron number fluctuations and charge stability in a triple-island system via path integral Monte Carlo simulation ", Journal of Physics: Condensed Matter 38 7, 075302 (2026).
[50] Zheng Qin, Xiufan Li, Yang Zhou, Shikun Zhang, Rui Li, Chunxiao Du, and Zhisong Xiao, "Emerging applications of measurement-based quantum computing", Quantum Information Processing 23 12, 389 (2024).
[51] David T. Stephen and Oliver Hart, "Preparing matrix product states via fusion: Constraints and extensions", Physical Review B 112 23, 235127 (2025).
[52] Grace M. Sommers, David A. Huse, and Michael J. Gullans, "Crystalline Quantum Circuits", PRX Quantum 4 3, 030313 (2023).
[53] Melanie Swan, Renato P. Dos Santos, and Frank Witte, "Quantum Matter Overview", J 5 2, 232 (2022).
[54] Robert Raussendorf, Cihan Okay, Michael Zurel, and Polina Feldmann, "The role of cohomology in quantum computation with magic states", Quantum 7, 979 (2023).
[55] Jie-Yu Zhang, Meng-Yuan Li, and Peng Ye, "Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi Point Strange Correlators", PRX Quantum 5 3, 030342 (2024).
[56] Isaac D. Smith, Maxime Cautrès, David T. Stephen, and Hendrik Poulsen Nautrup, "Optimally Generating su(2N) Using Pauli Strings", Physical Review Letters 134 20, 200601 (2025).
[57] Chengkang Zhou, Meng-Yuan Li, Zheng Yan, Peng Ye, and Zi Yang Meng, "Detecting subsystem symmetry protected topological order through strange correlators", Physical Review B 106 21, 214428 (2022).
[58] Lorenzo Siro Trezzini, Alessandro Bisio, and Paolo Perinotti, "Renormalisation of Quantum Cellular Automata", Quantum 9, 1756 (2025).
[59] Austin K. Daniel, Rafael N. Alexander, and Akimasa Miyake, "Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices", Quantum 4, 228 (2020).
[60] Cole Kelson-Packer and Akimasa Miyake, "Phase transitions in (2+1)D subsystem-symmetric monitored quantum circuits", Physical Review B 111 7, 075109 (2025).
[61] Han-Xie Wang, Shuai A. Chen, and Peng Ye, "Fractonic superfluids. III. Hybridizing higher moments", Physical Review Research 7 3, 033118 (2025).
[62] Andrea Pizzamiglio, Alessandro Bisio, and Paolo Perinotti, "Classification of Qubit Cellular Automata on Hypercubic Lattices", Physical Review Letters 134 24, 240601 (2025).
[63] Shi-Hao Zhang, Xiang-Dong Zhang, and Lü-Zhou Li, "Research progress of measurement-based quantum computation", Acta Physica Sinica 70 21, 210301 (2021).
[64] Po-Shen Hsin, David Stephen, Arpit Dua, and Dominic Williamson, "Subsystem symmetry fractionalization and foliated field theory", SciPost Physics 18 5, 147 (2025).
[65] Paul Herringer and Robert Raussendorf, "Classification of measurement-based quantum wire in stabilizer PEPS", Quantum 7, 1041 (2023).
[66] Robert Raussendorf, Wang Yang, and Arnab Adhikary, "Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power", Quantum 7, 1215 (2023).
[67] David T. Stephen, José Garre-Rubio, Arpit Dua, and Dominic J. Williamson, "Subsystem symmetry enriched topological order in three dimensions", Physical Review Research 2 3, 033331 (2020).
[68] Zoltán Zimborás, Terry Farrelly, Szilárd Farkas, and Lluis Masanes, "Does causal dynamics imply local interactions?", Quantum 6, 748 (2022).
[69] Jian-Hao Zhang, Ke Ding, Shuo Yang, and Zhen Bi, "Fractonic higher-order topological phases in open quantum systems", Physical Review B 108 15, 155123 (2023).
[70] Zhangjie Qin and V. W. Scarola, "Scaling of computational order parameters in Rydberg-atom graph states", Physical Review A 111 4, 042617 (2025).
[71] Sounak Biswas, Yves H. Kwan, and S. A. Parameswaran, "Beyond the freshman's dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models", Physical Review B 105 22, 224410 (2022).
[72] Caroline de Groot, David T Stephen, Andras Molnar, and Norbert Schuch, "Inaccessible entanglement in symmetry protected topological phases", Journal of Physics A: Mathematical and Theoretical 53 33, 335302 (2020).
[73] Bo-Xi Li and Peng Ye, "Infinite-component BF topological field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification", Physical Review B 113 12, 125141 (2026).
[74] 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", Physical Review Letters 125 12, 120502 (2020).
[75] Dong‐Sheng Wang, "A comparative study of universal quantum computing models: Toward a physical unification", Quantum Engineering 3 4(2021).
[76] Arunava Majumder, Marius Krumm, Tina Radkohl, Lukas J. Fiderer, Hendrik Poulsen Nautrup, Sofiene Jerbi, and Hans J. Briegel, "Variational measurement-based quantum computation for generative modeling", Physical Review A 110 6, 062616 (2024).
[77] Terry Farrelly, "A review of Quantum Cellular Automata", Quantum 4, 368 (2020).
[78] David T. Stephen, Wen Wei Ho, Tzu-Chieh Wei, Robert Raussendorf, and Ruben Verresen, "Universal Measurement-Based Quantum Computation in a One-Dimensional Architecture Enabled by Dual-Unitary Circuits", Physical Review Letters 132 25, 250601 (2024).
[79] Alberto D. Verga, "Entanglement dynamics and phase transitions of the Floquet cluster spin chain", Physical Review B 107 8, 085116 (2023).
[80] Kevissen Sellapillay, Alberto D. Verga, and Giuseppe Di Molfetta, "Entanglement dynamics and ergodicity breaking in a quantum cellular automaton", Physical Review B 106 10, 104309 (2022).
[81] Jung Hoon Han, Ethan Lake, Ho Tat Lam, Ruben Verresen, and Yizhi You, "Topological quantum chains protected by dipolar and other modulated symmetries", Physical Review B 109 12, 125121 (2024).
[82] David Gunn, Georgios Styliaris, Tristan Kraft, and Barbara Kraus, "Phases of matrix product states with symmetric quantum circuits and symmetric measurements with feedforward", Physical Review B 111 11, 115110 (2025).
[83] Lorenzo S. Trezzini, Matteo Lugli, Paolo Meda, Alessandro Bisio, Paolo Perinotti, and Alessandro Tosini, "Fermionic cellular automata in one dimension", Journal of Mathematical Physics 67 5, 052204 (2026).
[84] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).
[85] Trithep Devakul, Dominic J. Williamson, and Yizhi You, "Classification of subsystem symmetry-protected topological phases", Physical Review B 98 23, 235121 (2018).
[86] 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).
[87] Dominic J. Williamson, Arpit Dua, and Meng Cheng, "Spurious Topological Entanglement Entropy from Subsystem Symmetries", Physical Review Letters 122 14, 140506 (2019).
[88] Trithep Devakul and Dominic J. Williamson, "Universal quantum computation using fractal symmetry-protected cluster phases", Physical Review A 98 2, 022332 (2018).
[89] Salvatore D. Pace, Guilherme Delfino, Ho Tat Lam, and Ömer M. Aksoy, "Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections", arXiv:2406.12962, (2024).
[90] Shang-Qiang Ning, Hiromi Ebisu, and Ho Tat Lam, "Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints", arXiv:2603.19381, (2026).
[91] 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).
[92] Shuai A. Chen and Peng Ye, "Many-body physics of spontaneously broken higher-rank symmetry: from fractonic superfluids to dipolar Hubbard model", arXiv:2305.00941, (2023).
[93] Thierry N. Kaldenbach, Isaac D. Smith, Hendrik Poulsen Nautrup, Matthias Heller, and Hans J. Briegel, "Efficient Preparation of Resource States for Hamiltonian Simulation and Universal Quantum Computation", arXiv:2509.05404, (2025).
[94] Salvatore D. Pace and Daniel Bulmash, "Lieb-Schultz-Mattis constraints from stratified anomalies of modulated symmetries", arXiv:2602.11266, (2026).
[95] Albert T. Schmitz, "Distilling Fractons from Layered Subsystem-Symmetry Protected Phases", arXiv:1910.04765, (2019).
[96] Avijit Maity, Vikram Tripathi, and Andriy H. Nevidomskyy, "Web of Non-invertible Dualities for (2+1) Dimensional Models with Subsystem Symmetries", arXiv:2511.18969, (2025).
[97] Robert Raussendorf, Cihan Okay, Michael Zurel, and Polina Feldmann, "The role of cohomology in quantum computation with magic states", arXiv:2110.11631, (2021).
[98] Tomohiro Yamazaki and Yuki Takeuchi, "Measurement-based quantum computation on weighted graph states with arbitrarily small weight", arXiv:2512.01327, (2025).
[99] Dong-Sheng Wang, "Quantum computation by teleportation and symmetry", International Journal of Modern Physics B 33 15, 1930004-526 (2019).
[100] Ryohei Weil, Dmytro Bondarenko, Arnab Adhikary, and Robert Raussendorf, "Testing measurement-based computational phases of quantum matter on a quantum processor", arXiv:2601.03426, (2026).
[101] Frederik Hahn, Rose McCarty, Hendrik Poulsen Nautrup, and Nathan Claudet, "The Structure of Circle Graph States", arXiv:2603.08847, (2026).
[102] Aleksander Głódkowski, "Effective theories for many-body systems with nonuniform symmetries", arXiv:2607.09427, (2026).
The above citations are from Crossref's cited-by service (last updated successfully 2026-07-17 06:58:10) and SAO/NASA ADS (last updated successfully 2026-07-16 17:19:16). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch ADS cited-by data during last attempt 2026-07-17 06:58:10: Cannot retrieve data from ADS due to rate limitations.
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.