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.
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.
 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).
 A Yu Kitaev ``Unpaired Majorana fermions in quantum wires'' Physics-Uspekhi 44, 131–136 (2001).
 Jason Alicea, Yuval Oreg, Gil Refael, Felix Oppen, and Matthew P. A. Fisher, ``Non-Abelian statistics and topological quantum information processing in 1D wire networks'' Nature Physics 7, 412–417 (2011).
 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).
 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).
 Akimasa Miyake ``Quantum Computation on the Edge of a Symmetry-Protected Topological Order'' Phys. Rev. Lett. 105, 040501 (2010).
 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).
 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).
 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).
 Jacob Miller and Akimasa Miyake ``Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation'' Phys. Rev. Lett. 114, 120506 (2015).
 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).
 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).
 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).
 Hendrik Poulsen Nautrup and Tzu-Chieh Wei ``Symmetry-protected topologically ordered states for universal quantum computation'' Phys. Rev. A 92, 052309 (2015).
 Jacob Miller and Akimasa Miyake ``Hierarchy of universal entanglement in 2D measurement-based quantum computation'' npj Quantum Information 2, 16036 (2016).
 Jacob Miller and Akimasa Miyake ``Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry'' Phys. Rev. Lett. 120, 170503 (2018).
 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).
 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).
 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).
 Trithep Devakul and Dominic J. Williamson ``Universal quantum computation using fractal symmetry-protected cluster phases'' Phys. Rev. A 98, 022332 (2018).
 Arpit Dua, Boris Malomed, Meng Cheng, and Liang Jiang, ``Universal quantum computing with parafermions assisted by a half fluxon'' (2018).
 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).
 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).
 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).
 Sam Roberts, Beni Yoshida, Aleksander Kubica, and Stephen D. Bartlett, ``Symmetry-protected topological order at nonzero temperature'' Phys. Rev. A 96, 022306 (2017).
 Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, ``Measurement-based quantum computation on cluster states'' Phys. Rev. A 68, 022312 (2003).
 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).
 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).
 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).
 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).
 T. Senthil ``Symmetry-Protected Topological Phases of Quantum Matter'' Ann. Rev. Cond. Mat. Phys. 6, 299–324 (2015).
 A. Nietner, C. Krumnow, E. J. Bergholtz, and J. Eisert, ``Composite symmetry-protected topological order and effective models'' Phys. Rev. B 96, 235138 (2017).
 Ryan Thorngren and Dominic V. Else ``Gauging Spatial Symmetries and the Classification of Topological Crystalline Phases'' Phys. Rev. X 8, 011040 (2018).
 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).
 Trithep Devakul, Dominic J. Williamson, and Yizhi You, ``Classification of subsystem symmetry-protected topological phases'' Phys. Rev. B 98, 235121 (2018).
 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).
 Sagar Vijay, Jeongwan Haah, and Liang Fu, ``Fracton topological order, generalized lattice gauge theory, and duality'' Phys. Rev. B 94, 235157 (2016).
 Zohar Nussinov and Gerardo Ortiz ``Sufficient symmetry conditions for Topological Quantum Order'' Proceedings of the National Academy of Sciences 106, 16944–16949 (2009).
 Jacob C Bridgemanand Christopher T Chubb ``Hand-waving and interpretive dance: an introductory course on tensor networks'' J. Phys. A 50, 223001 (2017).
 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).
 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).
 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).
 Shenghan Jiang and Ying Ran ``Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases'' Phys. Rev. B 95, 125107 (2017).
 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).
 Didier Poilblanc, J. Ignacio Cirac, and Norbert Schuch, ``Chiral topological spin liquids with projected entangled pair states'' Phys. Rev. B 91, 224431 (2015).
 Dominic J. Williamson, Nick Bultinck, and Frank Verstraete, ``Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation'' (2017).
 Nick Bultinck, Dominic J Williamson, Jutho Haegeman, and Frank Verstraete, ``Fermionic projected entangled-pair states and topological phases'' J. Phys. A 51, 025202 (2017).
 Henrik Dreyer, J. Ignacio Cirac, and Norbert Schuch, ``Projected entangled pair states with continuous virtual symmetries'' Phys. Rev. B 98, 115120 (2018).
 Joseph F. Fitzsimons ``Private quantum computation: an introduction to blind quantum computing and related protocols'' npj Quantum Information 3, 23 (2017).
 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).
 Joseph Fitzsimons and Jason Twamley ``Globally Controlled Quantum Wires for Perfect Qubit Transport, Mirroring, and Computing'' Phys. Rev. Lett. 97, 090502 (2006).
 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).
 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).
 Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert, ``Anticoncentration theorems for schemes showing a quantum speedup'' Quantum 2, 65 (2018).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, ``Matrix Product State Representations'' Quantum Info. Comput. 7, 401–430 (2007).
 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).
 David Sauerwein, Andras Molnar, J. Ignacio Cirac, and Barbara Kraus, ``Matrix Product States: Entanglement, symmetries, and state transformations'' (2019).
 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).
 I_A G Berkovich and EM Zhmud ``Characters of finite groups'' American Mathematical Soc. (1998).
 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).
 A. S. Darmawan and S. D. Bartlett ``Graph states as ground states of two-body frustration-free Hamiltonians'' New J. Phys. 16, 073013 (2014).
 Ruben Verresen, Roderich Moessner, and Frank Pollmann, ``One-dimensional symmetry protected topological phases and their transitions'' Phys. Rev. B 96, 165124 (2017).
 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).
 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).
 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).
 R Raussendorf, J Harrington, and K Goyal, ``Topological fault-tolerance in cluster state quantum computation'' New Journal of Physics 9, 199–199 (2007).
 Burton Voorhees ``A note on injectivity of additive cellular automata'' Complex Systems 8, 151–160 (1994).
 Trithep Devakul, "Classifying local fractal subsystem symmetry-protected topological phases", Physical Review B 99 23, 235131 (2019).
 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).
 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).
 Lorenzo Piroli and J. Ignacio Cirac, "Quantum Cellular Automata, Tensor Networks, and Area Laws", Physical Review Letters 125 19, 190402 (2020).
 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).
 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).
 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).
 Abhinav Prem and Dominic Williamson, "Gauging permutation symmetries as a route to non-Abelian fractons", SciPost Physics 7 5, 068 (2019).
 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).
 Nathanan Tantivasadakarn and Sagar Vijay, "Searching for fracton orders via symmetry defect condensation", Physical Review B 101 16, 165143 (2020).
 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).
 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).
 Wilbur Shirley, Kevin Slagle, and Xie Chen, "Twisted foliated fracton phases", Physical Review B 102 11, 115103 (2020).
 Huan He, Yizhi You, and Abhinav Prem, "Lieb-Schultz-Mattis–type constraints on fractonic matter", Physical Review B 101 16, 165145 (2020).
 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).
 Trithep Devakul, Dominic J. Williamson, and Yizhi You, "Classification of subsystem symmetry-protected topological phases", Physical Review B 98 23, 235121 (2018).
 Dominic J. Williamson, Arpit Dua, and Meng Cheng, "Spurious Topological Entanglement Entropy from Subsystem Symmetries", Physical Review Letters 122 14, 140506 (2019).
 Trithep Devakul and Dominic J. Williamson, "Universal quantum computation using fractal symmetry-protected cluster phases", Physical Review A 98 2, 022332 (2018).
 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).
 Albert T. Schmitz, "Distilling Fractons from Layered Subsystem-Symmetry Protected Phases", arXiv:1910.04765.
 Jonathan San Miguel, Arpit Dua, and Dominic Williamson, "Bifurcating subsystem symmetric entanglement renormalization in two dimensions", arXiv:2010.15124.
 Sam Roberts and Dominic J. Williamson, "3-Fermion topological quantum computation", arXiv:2011.04693.
The above citations are from Crossref's cited-by service (last updated successfully 2021-01-22 19:15:57) and SAO/NASA ADS (last updated successfully 2021-01-22 19:15:58). The list may be incomplete as not all publishers provide suitable and complete citation data.
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.