# 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

### 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.

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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### Cited by

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

[2] 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).

[3] 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).

[4] 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).

[5] Lorenzo Piroli and J. Ignacio Cirac, "Quantum Cellular Automata, Tensor Networks, and Area Laws", Physical Review Letters 125 19, 190402 (2020).

[6] 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).

[7] 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).

[8] 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).

[9] 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).

[10] Abhinav Prem and Dominic Williamson, "Gauging permutation symmetries as a route to non-Abelian fractons", SciPost Physics 7 5, 068 (2019).

[11] 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).

[12] Nathanan Tantivasadakarn and Sagar Vijay, "Searching for fracton orders via symmetry defect condensation", Physical Review B 101 16, 165143 (2020).

[13] 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).

[14] 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).

[15] Wilbur Shirley, Kevin Slagle, and Xie Chen, "Twisted foliated fracton phases", Physical Review B 102 11, 115103 (2020).

[16] Huan He, Yizhi You, and Abhinav Prem, "Lieb-Schultz-Mattis–type constraints on fractonic matter", Physical Review B 101 16, 165145 (2020).

[17] 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).

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

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

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

[21] 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).

[22] Albert T. Schmitz, "Distilling Fractons from Layered Subsystem-Symmetry Protected Phases", arXiv:1910.04765.

[23] 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 2021-04-21 19:15:20) and SAO/NASA ADS (last updated successfully 2021-04-21 19:15:21). The list may be incomplete as not all publishers provide suitable and complete citation data.