61,247 research outputs found
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
Minimal instances for toric code ground states
A decade ago Kitaev's toric code model established the new paradigm of
topological quantum computation. Due to remarkable theoretical and experimental
progress, the quantum simulation of such complex many-body systems is now
within the realms of possibility. Here we consider the question, to which
extent the ground states of small toric code systems differ from LU-equivalent
graph states. We argue that simplistic (though experimentally attractive)
setups obliterate the differences between the toric code and equivalent graph
states; hence we search for the smallest setups on the square- and triangular
lattice, such that the quasi-locality of the toric code hamiltonian becomes a
distinctive feature. To this end, a purely geometric procedure to transform a
given toric code setup into an LC-equivalent graph state is derived. In
combination with an algorithmic computation of LC-equivalent graph states, we
find the smallest non-trivial setup on the square lattice to contain 5
plaquettes and 16 qubits; on the triangular lattice the number of plaquettes
and qubits is reduced to 4 and 9, respectively.Comment: 14 pages, 11 figure
Protected gates for topological quantum field theories
We study restrictions on locality-preserving unitary logical gates for
topological quantum codes in two spatial dimensions. A locality-preserving
operation is one which maps local operators to local operators --- for example,
a constant-depth quantum circuit of geometrically local gates, or evolution for
a constant time governed by a geometrically-local bounded-strength Hamiltonian.
Locality-preserving logical gates of topological codes are intrinsically fault
tolerant because spatially localized errors remain localized, and hence
sufficiently dilute errors remain correctable. By invoking general properties
of two-dimensional topological field theories, we find that the
locality-preserving logical gates are severely limited for codes which admit
non-abelian anyons; in particular, there are no locality-preserving logical
gates on the torus or the sphere with M punctures if the braiding of anyons is
computationally universal. Furthermore, for Ising anyons on the M-punctured
sphere, locality-preserving gates must be elements of the logical Pauli group.
We derive these results by relating logical gates of a topological code to
automorphisms of the Verlinde algebra of the corresponding anyon model, and by
requiring the logical gates to be compatible with basis changes in the logical
Hilbert space arising from local F-moves and the mapping class group.Comment: 50 pages, many figures, v3: updated to match published versio
Topological fault-tolerance in cluster state quantum computation
We describe a fault-tolerant version of the one-way quantum computer using a
cluster state in three spatial dimensions. Topologically protected quantum
gates are realized by choosing appropriate boundary conditions on the cluster.
We provide equivalence transformations for these boundary conditions that can
be used to simplify fault-tolerant circuits and to derive circuit identities in
a topological manner. The spatial dimensionality of the scheme can be reduced
to two by converting one spatial axis of the cluster into time. The error
threshold is 0.75% for each source in an error model with preparation, gate,
storage and measurement errors. The operational overhead is poly-logarithmic in
the circuit size.Comment: 20 pages, 12 figure
Quantum computation with Turaev-Viro codes
The Turaev-Viro invariant for a closed 3-manifold is defined as the
contraction of a certain tensor network. The tensors correspond to tetrahedra
in a triangulation of the manifold, with values determined by a fixed spherical
category. For a manifold with boundary, the tensor network has free indices
that can be associated to qudits, and its contraction gives the coefficients of
a quantum error-correcting code. The code has local stabilizers determined by
Levin and Wen. For example, applied to the genus-one handlebody using the Z_2
category, this construction yields the well-known toric code.
For other categories, such as the Fibonacci category, the construction
realizes a non-abelian anyon model over a discrete lattice. By studying braid
group representations acting on equivalence classes of colored ribbon graphs
embedded in a punctured sphere, we identify the anyons, and give a simple
recipe for mapping fusion basis states of the doubled category to ribbon
graphs. We explain how suitable initial states can be prepared efficiently, how
to implement braids, by successively changing the triangulation using a fixed
five-qudit local unitary gate, and how to measure the topological charge.
Combined with known universality results for anyonic systems, this provides a
large family of schemes for quantum computation based on local deformations of
stabilizer codes. These schemes may serve as a starting point for developing
fault-tolerance schemes using continuous stabilizer measurements and active
error-correction.Comment: 53 pages, LaTeX + 199 eps figure
A fault-tolerant one-way quantum computer
We describe a fault-tolerant one-way quantum computer on cluster states in
three dimensions. The presented scheme uses methods of topological error
correction resulting from a link between cluster states and surface codes. The
error threshold is 1.4% for local depolarizing error and 0.11% for each source
in an error model with preparation-, gate-, storage- and measurement errors.Comment: 26 page
Virtual Knot Cobordism
This paper defines a theory of cobordism for virtual knots and studies this
theory for standard and rotational virtual knots and links. Non-trivial
examples of virtual slice knots are given. Determinations of the four-ball
genus of positive virtual knots are given using the results of a companion
paper by the author and Heather Dye and Aaron Kaestner. Problems related to
band-passing are explained, and a theory of isotopy of virtual surfaces is
formulated in terms of a generalization of the Yoshikawa moves.Comment: 32 pages, 43 figures, LaTeX documen
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