We provide the first example of a symmetry protected quantum phase that has universal computational power. This two-dimensional phase is protected by one-dimensional line-like symmetries that can be understood in terms of local symmetries of a tensor network. These local symmetries imply that every ground state in the phase is a universal resource for measurement based quantum computation.arXiv:1803.00095v2 [quant-ph]

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...

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). We show that, for a wide range of SPT phases, the computational power of ground states is uniform throughout each phase. This computational power, defined as the Lie group of executable gates in MBQC, is determined by the same algebraic information that labels the SPT phase itself. We prove that these Lie groups always contain a full set of single-qubit gates, thereby affirming the long-standing conjecture that general SPT phases can serve as computationally useful phases of matter. Introduction. In many-body physics, the essential properties of a quantum state are determined by the phase of matter in which it resides. Recent years have witnessed tremendous progress in the discovery and classification of quantum phases [1][2][3][4][5][6][7][8][9][10], and it is thus pertinent to ask: what can a phase of matter be used for? A traditional example is the ubiquitous superconductor, while newly discovered phases such as topological insulators [11] and quantum spin liquids [12] have promising future applications. Quantum phases are useful in quantum information processing as well: certain topological phases allow for error-resilient topological quantum computation via the braiding and fusion of their anyonic excitations [13,14]. These applications all operate due to properties of a phase rather than a particular quantum state, hence they enjoy passive protection against certain sources of noise and error.

We investigate the usefulness of ground states of quantum spin chains with symmetry-protected topological order (SPTO) for measurement-based quantum computation. We show that, in spatial dimension one, if an SPTO phase supports quantum wire, then, subject to an additional symmetry condition that is satisfied in all cases so far investigated, it can also be used for quantum computation.

Resource states that contain nontrivial symmetry-protected topological order are identified for universal single-qudit measurement-based quantum computation. Our resource states fall into two classes: one as the qudit generalizations of the 1D qubit cluster state, and the other as the higher-symmetry generalizations of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state, namely, with unitary, orthogonal, or symplectic symmetry. The symmetry in cluster states protects information propagation (identity gate), while the higher symmetry in AKLT-type states enables nontrivial gate computation. This work demonstrates a close connection between measurement-based quantum computation and symmetry-protected topological order.

We introduce a model of three-dimensional (3D) topological order enriched by planar subsystem symmetries. The model is constructed starting from the 3D toric code, whose ground state can be viewed as an equal-weight superposition of two-dimensional (2D) membrane coverings. We then decorate those membranes with 2D cluster states possessing symmetry-protected topological order under linelike subsystem symmetries. This endows the decorated model with planar subsystem symmetries under which the looplike excitations of the toric code fractionalize, resulting in an extensive degeneracy per unit length of the excitation. We also show that the value of the topological entanglement entropy is larger than that of the toric code for certain bipartitions due to the subsystem symmetry enrichment. Our model can be obtained by gauging the global symmetry of a short-range entangled model which has symmetry-protected topological order coming from an interplay of global and subsystem symmetries. We study the nontrivial action of the symmetries on boundary of this model, uncovering a mixed boundary anomaly between global and subsystem symmetries. To further study this interplay, we consider gauging several different subgroups of the total symmetry. The resulting network of models, which includes models with fracton topological order, showcases more of the possible types of subsystem symmetry enrichment that can occur in 3D.

Subsystem symmetry-protected topological (SSPT) order is a type of quantum order that is protected by symmetries acting on lower-dimensional subsystems of the entire system. In this paper, we show how SSPT order can be characterized and detected by a constant correction to the entanglement area law, similar to the topological entanglement entropy. Focusing on the paradigmatic two-dimensional cluster phase as an example, we use tensor network methods to give an analytic argument that almost all states in the phase exhibit the same correction to the area law, such that this correction may be used to reliably detect the SSPT order of the cluster phase. Based on this idea, we formulate a numerical method that uses tensor networks to extract this correction from ground state wave functions. We use this method to study the fate of the SSPT order of the cluster state under various external fields and interactions, and find that the correction persists unless a phase transition is crossed, or the subsystem symmetry is explicitly broken. Surprisingly, these results uncover that the SSPT order of the cluster state persists beyond the cluster phase, thanks to a new type of subsystem time-reversal symmetry. Finally, we discuss the correction to the area law found in 3D cluster states on different lattices, indicating rich behaviour for general subsystem symmetries.The modern perspective of quantum phases of matter is based on global patterns of entanglement [1][2][3]. Two quantum states are said to lie in distinct (symmetryprotected) topological phases when they cannot be connected by (symmetric) local unitary evolution. As such, topological phases of matter can be characterized and detected by entanglement-based quantities. A well-known example is the topological entanglement entropy (TEE): For states with non-trivial topological order, the scaling of entanglement entropy exhibits a correction to the area law [4][5][6][7]. That is, for a subregion A of a lattice, the entropy for ground states of gapped, local Hamiltonians takes the general form,for some constants a, γ, where ∂A is the boundary of region A and the dots indicate terms that decay exponentially with |A|. The correction γ takes a uniform non-zero value within non-trivial topological phases as a consequence of the global entanglement patterns. It can therefore be used to detect and characterize topological order and topological phase transitions analytically, numerically, and potentially even experimentally [8][9][10][11][12][13][14][15][16][17].Recently, however, it has been observed that γ may deviate from the expected value due to the presence of symmetry-protected topological (SPT) order localized around ∂A [18-23]. One setting in which this occurs is for states with subsystem SPT (SSPT) order [21,[24][25][26]. Such order is non-trivial only in the presence of subsystem symmetries, which are defined as symmetries that act on rigid lower-dimensional subsystems of the entire system. In cases where these symmetries act on 1D lines spanning a 2D lattice, one may ...

We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group G, which we call G-LOCC, and we define the inaccessible entanglement E inacc as the entanglement that cannot be used for distillation under G-LOCC. We derive a tight bound on E inacc which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely log ( D ω 2 ) ⩽ E i n a c c ⩽ log ( | G | ) , where D ω is the topologically protected edge mode degeneracy of the SPT phase ω with symmetry G. For particular phases such as the Haldane phase, D ω = | G | so the bound becomes an equality. We numerically investigate the distribution of states throughout the bound, and show that typically the region near the upper bound is highly populated, and also determine the nature of those states lying on the upper and lower bounds. We then discuss the relation of E inacc to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.