87 research outputs found
Approximating local observables on projected entangled pair states
Tensor network states are for good reasons believed to capture ground states
of gapped local Hamiltonians arising in the condensed matter context, states
which are in turn expected to satisfy an entanglement area law. However, the
computational hardness of contracting projected entangled pair states in two
and higher dimensional systems is often seen as a significant obstacle when
devising higher-dimensional variants of the density-matrix renormalisation
group method. In this work, we show that for those projected entangled pair
states that are expected to provide good approximations of such ground states
of local Hamiltonians, one can compute local expectation values in
quasi-polynomial time. We therefore provide a complexity-theoretic
justification of why state-of-the-art numerical tools work so well in practice.
We comment on how the transfer operators of such projected entangled pair
states have a gap and discuss notions of local topological quantum order. We
finally turn to the computation of local expectation values on quantum
computers, providing a meaningful application for a small-scale quantum
computer.Comment: 7 pages, 1 figure, minor changes in v
A Non-Commuting Stabilizer Formalism
We propose a non-commutative extension of the Pauli stabilizer formalism. The
aim is to describe a class of many-body quantum states which is richer than the
standard Pauli stabilizer states. In our framework, stabilizer operators are
tensor products of single-qubit operators drawn from the group , where and . We
provide techniques to efficiently compute various properties related to
bipartite entanglement, expectation values of local observables, preparation by
means of quantum circuits, parent Hamiltonians etc. We also highlight
significant differences compared to the Pauli stabilizer formalism. In
particular, we give examples of states in our formalism which cannot arise in
the Pauli stabilizer formalism, such as topological models that support
non-Abelian anyons.Comment: 52 page
Double Semion Phase in an Exactly Solvable Quantum Dimer Model on the Kagome Lattice
Quantum dimer models typically arise in various low energy theories like
those of frustrated antiferromagnets. We introduce a quantum dimer model on the
kagome lattice which stabilizes an alternative topological
order, namely the so-called "double semion" order. For a particular set of
parameters, the model is exactly solvable, allowing us to access the ground
state as well as the excited states. We show that the double semion phase is
stable over a wide range of parameters using numerical exact diagonalization.
Furthermore, we propose a simple microscopic spin Hamiltonian for which the
low-energy physics is described by the derived quantum dimer model.Comment: 7 pages, 5 figure
Explicit tensor network representation for the ground states of string-net models
The structure of string-net lattice models, relevant as examples of
topological phases, leads to a remarkably simple way of expressing their ground
states as a tensor network constructed from the basic data of the underlying
tensor categories. The construction highlights the importance of the fat
lattice to understand these models.Comment: 5 pages, pdf figure
From entanglement renormalisation to the disentanglement of quantum double models
We describe how the entanglement renormalisation approach to topological
lattice systems leads to a general procedure for treating the whole spectrum of
these models, in which the Hamiltonian is gradually simplified along a parallel
simplification of the connectivity of the lattice. We consider the case of
Kitaev's quantum double models, both Abelian and non-Abelian, and we obtain a
rederivation of the known map of the toric code to two Ising chains; we pay
particular attention to the non-Abelian models and discuss their space of
states on the torus. Ultimately, the construction is universal for such models
and its essential feature, the lattice simplification, may point towards a
renormalisation of the metric in continuum theories.Comment: 46 pages, 25 eps figure
A hierarchy of topological tensor network states
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.Comment: 61 page
Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
We exhibit a mapping identifying Kitaev's quantum double lattice models
explicitly as a subclass of Levin and Wen's string net models via a completion
of the local Hilbert spaces with auxiliary degrees of freedom. This
identification allows to carry over to these string net models the
representation-theoretic classification of the excitations in quantum double
models, as well as define them in arbitrary lattices, and provides an
illustration of the abstract notion of Morita equivalence. The possibility of
generalising the map to broader classes of string nets is considered.Comment: 8 pages, 6 eps figures; v2: matches published versio
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