3,322 research outputs found
Topological order in 1D Cluster state protected by symmetry
We demonstrate how to construct the Z2*Z2 global symmetry which protects the
ground state degeneracy of cluster states for open boundary conditions. Such a
degeneracy ultimately arises because the set of stabilizers do not span a
complete set of integrals of motion of the cluster state Hamiltonian for open
boundary conditions. By applying control phase transformations, our
construction makes the stabilizers into the Pauli operators spanning the qubit
Hilbert space from which the degeneracy comes.Comment: 1 figure, To be published in Quantum Information Processin
Complete characterization of spin chains with two Ising symmetries
Spin chains with two Ising symmetries are the Jordan-Wigner duals of
one-dimensional interacting fermions with particle-hole and time-reversal
symmetry. From earlier works on Majorana chains, it is known that this class of
models has 10 distinct topological phases. In this paper, we analyze the
physical properties of the correspondent 10 phases of the spin model. In
particular, thanks to a set of two non-commuting dualities, we determine the
local and non-local order parameters of the phases. We find that 4 phases are
topologically protected by the Ising symmetries, while the other 6 break at
least one symmetry. Our study highlights the non-trivial relation between the
topological classifications of interacting bosons and fermions.Comment: 7 page
Detecting subsystem symmetry protected topological order via entanglement entropy
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 three-dimensional cluster states on
different lattices, indicating rich behavior for general subsystem symmetriesComment: 17 pages. v2: Published version, minor changes throughou
Symmetry protected topological order at nonzero temperature
We address the question of whether symmetry-protected topological (SPT) order
can persist at nonzero temperature, with a focus on understanding the thermal
stability of several models studied in the theory of quantum computation. We
present three results in this direction. First, we prove that nontrivial SPT
order protected by a global on-site symmetry cannot persist at nonzero
temperature, demonstrating that several quantum computational structures
protected by such on-site symmetries are not thermally stable. Second, we prove
that the 3D cluster state model used in the formulation of topological
measurement-based quantum computation possesses a nontrivial SPT-ordered
thermal phase when protected by a global generalized (1-form) symmetry. The SPT
order in this model is detected by long-range localizable entanglement in the
thermal state, which compares with related results characterizing SPT order at
zero temperature in spin chains using localizable entanglement as an order
parameter. Our third result is to demonstrate that the high error tolerance of
this 3D cluster state model for quantum computation, even without a protecting
symmetry, can be understood as an application of quantum error correction to
effectively enforce a 1-form symmetry.Comment: 42 pages, 10 figures, comments welcome; v2 published versio
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