72 research outputs found
Topological phases with generalized global symmetries
We present simple lattice realizations of symmetry-protected topological
(SPT) phases with -form global symmetries where charged excitations have
spatial dimensions. Specifically, we construct space-dimensional models
supported on a -colorable graph by using a family of unitary phase
gates, known as multi-qubit control- gates in quantum information community.
In our construction, charged excitations of different dimensionality may
coexist and form a short-range entangled state which is protected by symmetry
operators of different dimensionality. Non-triviality of proposed models, in a
sense of quantum circuit complexity, is confirmed by studying protected
boundary modes, gauged models and corresponding gapped domain walls. We also
comment on applications of our construction to quantum error-correcting codes,
and discuss corresponding fault-tolerant logical gates.Comment: 32 pages, 17 figures, single column (v2, corrected minor mistakes and
typos, to appear in PRB
Exotic topological order in fractal spin liquids
We present a large class of three-dimensional spin models that possess
topological order with stability against local perturbations, but are beyond
description of topological quantum field theory. Conventional topological spin
liquids, on a formal level, may be viewed as condensation of string-like
extended objects with discrete gauge symmetries, being at fixed points with
continuous scale symmetries. In contrast, ground states of fractal spin liquids
are condensation of highly-fluctuating fractal objects with certain algebraic
symmetries, corresponding to limit cycles under real-space renormalization
group transformations which naturally arise from discrete scale symmetries of
underlying fractal geometries. A particular class of three-dimensional models
proposed in this paper may potentially saturate quantum information storage
capacity for local spin systems.Comment: 18 pages, 10 figure
Fault-tolerant logical gates in quantum error-correcting codes
Recently, Bravyi and K\"onig have shown that there is a tradeoff between
fault-tolerantly implementable logical gates and geometric locality of
stabilizer codes. They consider locality-preserving operations which are
implemented by a constant depth geometrically local circuit and are thus
fault-tolerant by construction. In particular, they shown that, for local
stabilizer codes in D spatial dimensions, locality preserving gates are
restricted to a set of unitary gates known as the D-th level of the Clifford
hierarchy. In this paper, we elaborate this idea and provide several extensions
and applications of their characterization in various directions. First, we
present a new no-go theorem for self-correcting quantum memory. Namely, we
prove that a three-dimensional stabilizer Hamiltonian with a
locality-preserving implementation of a non-Clifford gate cannot have a
macroscopic energy barrier. Second, we prove that the code distance of a
D-dimensional local stabilizer code with non-trivial locality-preserving m-th
level Clifford logical gate is upper bounded by . For codes with
non-Clifford gates (m>2), this improves the previous best bound by Bravyi and
Terhal. Third we prove that a qubit loss threshold of codes with non-trivial
transversal m-th level Clifford logical gate is upper bounded by 1/m. As such,
no family of fault-tolerant codes with transversal gates in increasing level of
the Clifford hierarchy may exist. This result applies to arbitrary stabilizer
and subsystem codes, and is not restricted to geometrically-local codes. Fourth
we extend the result of Bravyi and K\"onig to subsystem codes. A technical
difficulty is that, unlike stabilizer codes, the so-called union lemma does not
apply to subsystem codes. This problem is avoided by assuming the presence of
error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig
is recovered.Comment: 13 pages, 4 figure
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