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Local unitary versus local Clifford equivalence of stabilizer and graph states
The equivalence of stabilizer states under local transformations is of
fundamental interest in understanding properties and uses of entanglement. Two
stabilizer states are equivalent under the usual stochastic local operations
and classical communication criterion if and only if they are equivalent under
local unitary (LU) operations. More surprisingly, under certain conditions, two
LU equivalent stabilizer states are also equivalent under local Clifford (LC)
operations, as was shown by Van den Nest et al. [Phys. Rev. \textbf{A71},
062323]. Here, we broaden the class of stabilizer states for which LU
equivalence implies LC equivalence () to include all
stabilizer states represented by graphs with neither cycles of length 3 nor 4.
To compare our result with Van den Nest et al.'s, we show that any stabilizer
state of distance is beyond their criterion. We then further prove
that holds for a more general class of stabilizer states
of . We also explicitly construct graphs representing
stabilizer states which are beyond their criterion: we identify all 58 graphs
with up to 11 vertices and construct graphs with () vertices
using quantum error correcting codes which have non-Clifford transversal gates.Comment: Revised version according to referee's comments. To appear in
Physical Review
Parafermion stabilizer codes
We define and study parafermion stabilizer codes which can be viewed as
generalizations of Kitaev's one dimensional model of unpaired Majorana
fermions. Parafermion stabilizer codes can protect against low-weight errors
acting on a small subset of parafermion modes in analogy to qudit stabilizer
codes. Examples of several smallest parafermion stabilizer codes are given. A
locality preserving embedding of qudit operators into parafermion operators is
established which allows one to map known qudit stabilizer codes to parafermion
codes. We also present a local 2D parafermion construction that combines
topological protection of Kitaev's toric code with additional protection
relying on parity conservation
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