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.
 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).
 A. Y. Kitaev ``Unpaired Majorana fermions in quantum wires'' Physics-Uspekhi 44, 131-136 (2001).
 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).
 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).
 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).
 A. Miyake ``Quantum Computation on the Edge of a Symmetry-Protected Topological Order'' Phys. Rev. Lett. 105, 040501 (2010).
 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).
 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).
 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).
 J. Miller and A. Miyake ``Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation'' Phys. Rev. Lett. 114, 120506 (2015).
 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).
 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).
 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).
 H. Poulsen Nautrup and T.-C. Wei ``Symmetry-protected topologically ordered states for universal quantum computation'' Phys. Rev. A 92, 052309 (2015).
 J. Miller and A. Miyake ``Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry'' Phys. Rev. Lett. 120, 170503 (2018).
 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).
 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).
 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).
 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).
 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).
 S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, ``Symmetry-protected topological order at nonzero temperature'' Phys. Rev. A 96, 022306 (2017).
 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).
 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).
 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).
 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).
 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).
 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).
 Z. Nussinov and G. Ortiz ``Sufficient symmetry conditions for Topological Quantum Order'' Proceedings of the National Academy of Sciences 106, 16944-16949 (2009).
 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).
 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).
 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).
 S. Jiang and Y. Ran ``Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases'' Phys. Rev. B 95, 125107 (2017).
 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).
 D. J. Williamson, N. Bultinck, and F. Verstraete, ``Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation'' (2017).
 N. Bultinck, D. J. Williamson, J. Haegeman, and F. Verstraete, ``Fermionic projected entangled-pair states and topological phases'' J. Phys. A 51, 025202 (2017).
 J. F. Fitzsimons ``Private quantum computation: an introduction to blind quantum computing and related protocols'' npj Quantum Information 3, 23 (2017).
 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).
 J. Fitzsimons and J. Twamley ``Globally Controlled Quantum Wires for Perfect Qubit Transport, Mirroring, and Computing'' Phys. Rev. Lett. 97, 090502 (2006).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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. Cirac, ``Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems'' Adv. Phys. 57, 143-224 (2008).
 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).
 D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, ``Matrix Product State Representations'' Quantum Info. Comput. 7, 401-430 (2007).
 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).
 D. Sauerwein, A. Molnar, J. I. Cirac, and B. 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. G. Berkovich and E. Zhmud ``Characters of finite groups'' American Mathematical Soc. (1998).
 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).
 A. S. Darmawan and S. D. Bartlett ``Graph states as ground states of two-body frustration-free Hamiltonians'' New J. Phys. 16, 073013 (2014).
 R. Verresen, R. Moessner, and F. Pollmann, ``One-dimensional symmetry protected topological phases and their transitions'' Phys. Rev. B 96, 165124 (2017).
 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).
 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).
 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).
 R Raussendorf, J Harrington, and K Goyal, ``Topological fault-tolerance in cluster state quantum computation'' New Journal of Physics 9, 199-199 (2007).
 B. 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).
 Abhinav Prem and Dominic Williamson, "Gauging permutation symmetries as a route to non-Abelian fractons", SciPost Physics 7 5, 068 (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).
 Nathanan Tantivasadakarn and Sagar Vijay, "Searching for fracton orders via symmetry defect condensation", Physical Review B 101 16, 165143 (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).
 Huan He, Yizhi You, and Abhinav Prem, "Lieb-Schultz-Mattis–type constraints on fractonic matter", Physical Review B 101 16, 165145 (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).
 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).
 Terry Farrelly, "A review of Quantum Cellular Automata", arXiv:1904.13318.
 Arpit Dua, Pratyush Sarkar, Dominic J. Williamson, and Meng Cheng, "Bifurcating entanglement-renormalization group flows of fracton stabilizer models", arXiv:1909.12304.
 J. P. Ibieta-Jimenez, L. N. Queiroz Xavier, M. Petrucci, and P. Teotonio-Sobrinho, "Fracton-like phases from subsystem symmetries", arXiv:1908.07601.
 Albert T. Schmitz, "Distilling Fractons from Layered Subsystem-Symmetry Protected Phases", arXiv:1910.04765.
The above citations are from Crossref's cited-by service (last updated successfully 2020-06-02 07:20:14) and SAO/NASA ADS (last updated successfully 2020-06-02 07:20:15). 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.