1,437 research outputs found
Layer-by-layer disentangling two-dimensional topological quantum codes
While local unitary transformations are used for identifying quantum states
which are in the same topological class, non-local unitary transformations are
also important for studying the transition between different topological
classes. In particular, it is an important task to find suitable non-local
transformations that systematically sweep different topological classes. Here,
regarding the role of dimension in the topological classes, we introduce
partially local unitary transformations namely Greenberger-Horne-Zeilinger
(GHZ) disentanglers which reduce the dimension of the initial topological model
by a layer-by-layer disentangling mechanism. We apply such disentanglers to
two-dimensional (2D) topological quantum codes and show that they are converted
to many copies of Kitaev's ladders. It implies that the GHZ disentangler causes
a transition from an intrinsic topological phase to a symmetry-protected
topological phase. Then, we show that while Kitaev's ladders are building
blocks of both color code and toric code, there are different patterns of
entangling ladders in 2D color code and toric code. It shows that different
topological features of these topological codes are reflected in different
patterns of entangling ladders. In this regard, we propose that the
layer-by-layer disentangling mechanism can be used as a systematic method for
classification of topological orders based on finding different patterns of the
long-range entanglement in topological lattice models.Comment: 9 pages, 9 figures, submitted to PR
Minimising surface-code failures using a color-code decoder
The development of practical, high-performance decoding algorithms reduces
the resource cost of fault-tolerant quantum computing. Here we propose a
decoder for the surface code that finds low-weight correction operators for
errors produced by the depolarising noise model. The decoder is obtained by
mapping the syndrome of the surface code onto that of the color code, thereby
allowing us to adopt more sophisticated color-code decoding algorithms.
Analytical arguments and exhaustive testing show that the resulting decoder can
find a least-weight correction for all weight depolarising errors for
even code distance . This improves the logical error rate by an exponential
factor compared with decoders that treat bit-flip and dephasing
errors separately. We demonstrate this improvement with analytical arguments
and supporting numerical simulations at low error rates. Of independent
interest, we also demonstrate an exponential improvement in logical error rate
for our decoder used to correct independent and identically distributed
bit-flip errors affecting the color code compared with more conventional
color-code decoding algorithms
Structure of 2D Topological Stabilizer Codes
We provide a detailed study of the general structure of two-dimensional
topological stabilizer quantum error correcting codes, including subsystem
codes. Under the sole assumption of translational invariance, we show that all
such codes can be understood in terms of the homology of string operators that
carry a certain topological charge. In the case of subspace codes, we prove
that two codes are equivalent under a suitable set of local transformations if
and only they have equivalent topological charges. Our approach emphasizes
local properties of the codes over global ones.Comment: 54 pages, 11 figures, version accepted in journal, improved
presentation and result
Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries
We investigate topological order on fractal geometries embedded in
dimensions. In particular, we diagnose the existence of the topological order
through the lens of quantum information and geometry, i.e., via its equivalence
to a quantum error-correcting code with a macroscopic code distance or the
presence of macroscopic systoles in systolic geometry. We first prove a no-go
theorem that topological order cannot survive on any fractal
embedded in 2D. For fractal lattice models embedded in 3D or higher spatial
dimensions, topological order survives if the boundaries of the
interior holes condense only loop or membrane excitations. Moreover, for a
class of models containing only loop or membrane excitations, and are hence
self-correcting on an -dimensional manifold, we prove that topological order
survives on a large class of fractal geometries independent of the type of hole
boundaries. We further construct fault-tolerant logical gates using their
connection to global and higher-form topological symmetries. In particular, we
have discovered a logical CCZ gate corresponding to a global symmetry in a
class of fractal codes embedded in 3D with Hausdorff dimension asymptotically
approaching for arbitrarily small , which hence only
requires a space-overhead with being the code
distance. This in turn leads to the surprising discovery of certain exotic
gapped boundaries that only condense the combination of loop excitations and
gapped domain walls. We further obtain logical gates
with on fractal codes embedded in D. In particular, for the
logical in the level of Clifford
hierarchy, we can reduce the space overhead to .
Mathematically, our findings correspond to macroscopic relative systoles in
fractals.Comment: 46+10 pages, fixed typos and the table content, updated funding
informatio
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