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

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 lowerdimensional 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.The fields of study of quantum phases of matter and of quantum computation have been evolving alongside each other for over a decade, such that they are now deeply intertwined. This is on the one hand because many instances of non-trivial quantum order are key to storing or processing quantum informa-Figure 1: Interrelation between symmetry-protected topological (SPT) order, measurement-based quantum computation (MBQC), and quantum cellular automata (QCA). By framing both subsystem SPT order and MBQC in terms of QCA, we develop a new framework for characterizing subsystem SPT order and constructing computationally universal phases of matter.tion, an idea that has stimulated a plethora of theoretical and experimental research in both fields. Perhaps the most familiar example is the idea of topological quantum computation which leverages the anyonic excitations of topologically ordered systems to perform error-resilient quantum computation [1][2][3]. A complementing approach to topological quantum computation uses Majorana fermions located at the edges of one-dimensional (1D) chains with symmetryprotected topological (SPT) order [4][5][6]. On the other hand, quantum states exhibiting SPT order can be used as resources for instances of measurement-based quantum computation [7-22] -an insight most relevant to the present work. Various further examples can be found [23][24][25]; indeed, every time a new type of quantum order is discovered, it is not long before its uses in notions of quantum computation are being investigated.The m...

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

“…This is a particularly illuminating example because it reveals the fundamental importance of cone symmetries in defining the cluster phase in comparison to the emphasis on line symmetries in Ref. [35]). In Sec.…”

“…Besides being of independent geometric interest-they are the only vertex-translative lattices in 2D-they contain lattices more exotic than those studied previously, thus offering an important testbed for our method for constructing cluster phases, which complements the methods of Ref. [35]. Our lattice-first approach yields several new insights, such as a counterexample case to the conjecture of Ref.…”

“…Our lattice-first approach yields several new insights, such as a counterexample case to the conjecture of Ref. [35] that cluster phases with glider QCA should be constructed using line-like symmetries, as well as examples of lattices with one-form symmetries, which represent foliated error correcting codes and have underlying non-unitary QCA.…”

“…[33] were able to describe a 2D cluster phase, extending the 1D results to quasi-1D systems with subsystem symmetries, such as those discussed in Sec. II C, and giving rise to 2D resources phases that are universal for MBQC [33,35]. A self-contained review of MBQC protocols with quasi-1D SPT phases is given in Appendix B.…”

“…[34]. A recent paper [35] has constructed tensor network states with underlying Clifford quantum cellular automaton (QCA) in their virtual space, so that they have subsystem symmetries and support computationally universal subsystem SPTO phases.…”