5,427 research outputs found
The LU-LC conjecture is false
The LU-LC conjecture is an important open problem concerning the structure of
entanglement of states described in the stabilizer formalism. It states that
two local unitary equivalent stabilizer states are also local Clifford
equivalent. If this conjecture were true, the local equivalence of stabilizer
states would be extremely easy to characterize. Unfortunately, however, based
on the recent progress made by Gross and Van den Nest, we find that the
conjecture is false.Comment: Added a new part explaining how the counterexamples are foun
The LU-LC conjecture is false
The LU-LC conjecture is an important open problem concerning the structure of entanglement of states described in the stabilizer formalism. It states that two local unitary equivalent stabilizer states are also local Clifford equivalent. If this conjecture were true, the local equivalence of stabilizer states would be extremely easy to characterize. Unfortunately, however, based on the recent progress made by Gross and Van den Nest, we find that the conjecture is false. © Rinton Press
On Local Equivalence, Surface Code States and Matroids
Recently, Ji et al disproved the LU-LC conjecture and showed that the local
unitary and local Clifford equivalence classes of the stabilizer states are not
always the same. Despite the fact this settles the LU-LC conjecture, a
sufficient condition for stabilizer states that violate the LU-LC conjecture is
missing. In this paper, we investigate further the properties of stabilizer
states with respect to local equivalence. Our first result shows that there
exist infinitely many stabilizer states which violate the LU-LC conjecture. In
particular, we show that for all numbers of qubits , there exist
distance two stabilizer states which are counterexamples to the LU-LC
conjecture. We prove that for all odd , there exist stabilizer
states with distance greater than two which are LU equivalent but not LC
equivalent. Two important classes of stabilizer states that are of great
interest in quantum computation are the cluster states and stabilizer states of
the surface codes. To date, the status of these states with respect to the
LU-LC conjecture was not studied. We show that, under some minimal
restrictions, both these classes of states preclude any counterexamples. In
this context, we also show that the associated surface codes do not have any
encoded non-Clifford transversal gates. We characterize the CSS surface code
states in terms of a class of minor closed binary matroids. In addition to
making connection with an important open problem in binary matroid theory, this
characterization does in some cases provide an efficient test for CSS states
that are not counterexamples.Comment: LaTeX, 13 pages; Revised introduction, minor changes and corrections
mainly in section V
Cops and Robbers is EXPTIME-complete
We investigate the computational complexity of deciding whether k cops can
capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that
the problem is EXPTIME-complete when both G and k are part of the input; we
prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point
Compact set of invariants characterizing graph states of up to eight qubits
The set of entanglement measures proposed by Hein, Eisert, and Briegel for
n-qubit graph states [Phys. Rev. A 69, 062311 (2004)] fails to distinguish
between inequivalent classes under local Clifford operations if n > 6. On the
other hand, the set of invariants proposed by van den Nest, Dehaene, and De
Moor (VDD) [Phys. Rev. A 72, 014307 (2005)] distinguishes between inequivalent
classes, but contains too many invariants (more than 2 10^{36} for n=7) to be
practical. Here we solve the problem of deciding which entanglement class a
graph state of n < 9 qubits belongs to by calculating some of the state's
intrinsic properties. We show that four invariants related to those proposed by
VDD are enough for distinguishing between all inequivalent classes with n < 9
qubits.Comment: REVTeX4, 9 pages, 1 figur
Edge local complementation for logical cluster states
A method is presented for the implementation of edge local complementation in
graph states, based on the application of two Hadamard operations and a single
controlled-phase (CZ) gate. As an application, we demonstrate an efficient
scheme to construct a one-dimensional logical cluster state based on the
five-qubit quantum error-correcting code, using a sequence of edge local
complementations. A single physical CZ operation, together with local
operations, is sufficient to create a logical CZ operation between two logical
qubits. The same construction can be used to generate any encoded graph state.
This approach in concatenation may allow one to create a hierarchical quantum
network for quantum information tasks.Comment: 15 pages, two figures, IOP styl
Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms
The local minimum degree of a graph is the minimum degree that can be reached
by means of local complementation. For any n, there exist graphs of order n
which have a local minimum degree at least 0.189n, or at least 0.110n when
restricted to bipartite graphs. Regarding the upper bound, we show that for any
graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n)
for bipartite graphs, improving the known n/2 upper bound. We also prove that
the local minimum degree is smaller than half of the vertex cover number (up to
a logarithmic term). The local minimum degree problem is NP-Complete and hard
to approximate. We show that this problem, even when restricted to bipartite
graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which
W[1]-hardness is a long standing open question. Finally, we show that the local
minimum degree is computed by a O*(1.938^n)-algorithm, and a
O*(1.466^n)-algorithm for the bipartite graphs
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