We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [26]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing a renormalization group flow. Furthermore it is shown that those states of the hierarchy associated with Kitaev's original quantum double models are related to each other by the condensation of topological charges. We conjecture that charge condensation is the physical mechanism underlying the hierarchy in general.
The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular nonabelian ones. Here we analyze a class of nonabelian fractional quantum Hall model states which are generalizations of the abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey nonabelian fusion rules of type su(q)_k. For a subset of these states we are able to derive the conformal field theory description that makes…
Detecting and identifying 2D symmetry-protected topological, symmetry-breaking and intrinsic topological phases with modular matrices via tensor-network methods
We construct a tensor network representation of the 3d toric code ground state that is stable to a generating set of uniform local tensor perturbations, including those that do not map to local operators on the physical Hilbert space. The stability is established by mapping the phase diagram of the perturbed tensor network to that of the 3d Ising gauge theory, which has a non-zero finite temperature transition. More generally, we find that the stability of a topological tensor network state is determined by the form of its virtual symmetries and the topological excitations created by virtual o…
Tensor networks are a central tool in condensed matter physics. In this paper, we study the task of tensor network non-zero testing (TNZ): Given a tensor network T, does T represent a non-zero vector? We show that TNZ is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of TNZ on non-negative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MA-complete stoquastic k-SAT problem on D-dimensional qudits is in NP for logarithmic k and constant D. This reveals …
Hierarchy of mixed symmetry protected topological states in extended cluster states under subsystem decoherence
Graduate School of Engineering Science, Akita University（秋田大学工学研究科）；Department of Physics, College of Humanities and Sciences, Nihon University（日本大学文理学部物理学科）
Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic
Institute for Advanced Study（高等研究院）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）；Kavli Institute of Theoretical Physics（卡弗里理论物理研究所）；King’s College London（伦敦国王学院）
Despite the success in describing a range of quantum many-body states using tensor networks, there is a no-go theorem that rules out strictly local tensor networks as topologically nontrivial groundstates of gapped parent Hamiltonians with short-range (including exponentially decaying) couplings. In this work, we show that for free fermions, strictly local tensor networks may describe nonzero temperature averages with respect to gapped Hamiltonians with exponentially decaying couplings. Parent Hamiltonians in this sense may be constructed for any dimensionality and without any obstructions due…
The non-abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wavefunction with two tuning parameters dual with each other, and the norm can be exactly mapped into a partition function of the two-coupled $ϕ^{2}$-state Potts models, where $ϕ=(\sqrt{5}+1)/2$ is the golden ratio. By developing the tensor network representation of this wavefunction on a square lattice, we can accurately calculate the full phase diagram with the …
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 2…
The zipper condition for $4$-tensors in two-dimensional topological order and the higher relative commutants of a subfactor arising from a commuting square
iTHEMS Research Group, RIKEN（理化学研究所 iTHEMS 研究组）；Kavli IPMU (WPI), the University of Tokyo（东京大学 Kavli 宇宙物理数学统一中心（世界顶尖研究中心））
