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Classification of quantum wire in tensor network states with a local Pauli symmetry

Paul Herringer

Published Jan 1, 2021• 0 citations
Publisher

Abstract

Certain symmetry-protected topological (SPT) phases also possess the property that every state in the phase is a universal resource for measurement-based quantum computation (MBQC). These phases are called computational phases of quantum matter. Computational phases have been classified in 1D, and several examples are known in 2D. 2D computational phases are of interest because we can simulate any quantum computation using a 2D phase. Therefore, the classification of 2D phases may lead to a better understanding of the relationship between symmetry and quantum computation. Many known 2D phases have a projected entangled pair state (PEPS) representation where the local tensor symmetries define a quantum cellular automaton (QCA) acting on correlation space. However, not every tensor symmetry defines a QCA, which motivates the definition and study of a larger class of PEPS tensors called stabilizer tensors. Stabilizer tensors have symmetries that act as a tensor product of Pauli operators on correlation space but do not necessarily define a QCA. In this thesis, I present a classification of stabilizer tensors by their channel capacity for quantum wire. In particular, I consider the quantum wire capacity of translationally invariant, cylindrical PEPS constructed from stabilizer tensors and find that it falls into one of 13 classes. Furthermore, I prove that no other types of channel capacity exist for stabilizer tensors, regardless of the size of the cylinder. I also discuss the significance of this result for the classification of 2D computational phases along with possible next steps.

DOI
10.14288/1.0402579

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Certain symmetry-protected topological (SPT) phases also possess the property that every state in the phase is a universal resource for measurement-based quantum computation (MBQC). These phases are called computational phases of quantum matter. Computational phases have been classified in 1D, and several examples are known in 2D. 2D computational phases are of interest because we can simulate any quantum computation using a 2D phase. Therefore, the classification of 2D phases may lead to a better understanding of the relationship between symmetry and quantum computation. Many known 2D phases have a projected entangled pair state (PEPS) representation where the local tensor symmetries define a quantum cellular automaton (QCA) acting on correlation space. However, not every tensor symmetry defines a QCA, which motivates the definition and study of a larger class of PEPS tensors called stabilizer tensors. Stabilizer tensors have symmetries that act as a tensor product of Pauli operators on correlation space but do not necessarily define a QCA. In this thesis, I present a classification of stabilizer tensors by their channel capacity for quantum wire. In particular, I consider the quantum wire capacity of translationally invariant, cylindrical PEPS constructed from stabilizer tensors and find that it falls into one of 13 classes. Furthermore, I prove that no other types of channel capacity exist for stabilizer tensors, regardless of the size of the cylinder. I also discuss the significance of this result for the classification of 2D computational phases along with possible next steps.
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.
We consider a class of translation-invariant 2D tensor network states with a stabilizer symmetry, which we call stabilizer PEPS. The cluster state, GHZ state, and states in the toric code belong to this class. We investigate the transmission capacity of stabilizer PEPS for measurement-based quantum wire, and arrive at a complete classification of transmission behaviors. The transmission behaviors fall into 13 classes, one of which corresponds to Clifford quantum cellular automata. In addition, we identify 12 other classes.
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.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of GGG, the so-called GGG-injective PEPS, and on global on-site symmetries coming from a finite group QQQ: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.
We investigate the tensor network representations of fermionic crystalline symmetry-protected topological (SPT) phases on two-dimensional lattices. As a mapping from virtual indices to physical indices, projected entangled-pair state (PEPS) serves as a concrete way to construct the wavefunctions of 2D crystalline fermionic SPT (fSPT) phases protected by 17 wallpaper group symmetries, for both spinless and spin-1/2 fermions. Based on PEPS, the full classification of 2D crystalline fSPT phases with wallpaper groups can be obtained. Tensor network states provide a natural framework for studying 2D crystalline fSPT phases.
Projected entangled pair states (PEPS) provide a natural description of the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We show that on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, can be captured by a virtual symmetry of the tensors expressed as a set of matrix product operators labelled by the different group elements. A classification of SPT phases can hence be obtained by studying the topological obstructions to continuously deforming one set of matrix product operators into another. This leads to the classification of bosonic SPT states in terms of group cohomology, as originally derived by Chen et al. in [1106.4772]. Our formalism accommodates perturbations away from fixed point models, and hence opens up the possibility of studying phase transitions between different SPT phases. We furthermore show how the global symmetries of SPT PEPS can be promoted into a set of local gauge constraints by introducing bosonic degrees of freedom on the links of the PEPS lattice, thereby providing a natural and general mapping between PEPS in SPT phases and topologically ordered phases.