49,229 research outputs found
Three-dimensional surface codes: Transversal gates and fault-tolerant architectures
One of the leading quantum computing architectures is based on the
two-dimensional (2D) surface code. This code has many advantageous properties
such as a high error threshold and a planar layout of physical qubits where
each physical qubit need only interact with its nearest neighbours. However,
the transversal logical gates available in 2D surface codes are limited. This
means that an additional (resource intensive) procedure known as magic state
distillation is required to do universal quantum computing with 2D surface
codes. Here, we examine three-dimensional (3D) surface codes in the context of
quantum computation. We introduce a picture for visualizing 3D surface codes
which is useful for analysing stacks of three 3D surface codes. We use this
picture to prove that the and gates are transversal in 3D surface
codes. We also generalize the techniques of 2D surface code lattice surgery to
3D surface codes. We combine these results and propose two quantum computing
architectures based on 3D surface codes. Magic state distillation is not
required in either of our architectures. Finally, we show that a stack of three
3D surface codes can be transformed into a single 3D color code (another type
of quantum error-correcting code) using code concatenation.Comment: 23 pages, 24 figures, v2: published versio
Subspace subcodes of Reed-Solomon codes
We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed ν-dimensional vector subspace S of GF (2m). SSRS codes are constructed using properties of the Galois field GF(2m). They are not linear over the field GF(2ν), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2ν) is a subfield of GF(2m), it may not be the best ν-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems
Almost-Euclidean subspaces of via tensor products: a simple approach to randomness reduction
It has been known since 1970's that the N-dimensional -space contains
nearly Euclidean subspaces whose dimension is . However, proofs of
existence of such subspaces were probabilistic, hence non-constructive, which
made the results not-quite-suitable for subsequently discovered applications to
high-dimensional nearest neighbor search, error-correcting codes over the
reals, compressive sensing and other computational problems. In this paper we
present a "low-tech" scheme which, for any , allows to exhibit nearly
Euclidean -dimensional subspaces of while using only
random bits. Our results extend and complement (particularly) recent work
by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1)
simplicity (we use only tensor products) and (2) yielding "almost Euclidean"
subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor
change
An Upper-Bound on the Decoding Failure Probability of the LRPC Decoder
Low Rank Parity Check (LRPC) codes form a class of rank-metric
error-correcting codes that was purposely introduced to design public-key
encryption schemes. An LRPC code is defined from a parity check matrix whose
entries belong to a relatively low dimensional vector subspace of a large
finite field. This particular algebraic feature can then be exploited to
correct with high probability rank errors when the parameters are appropriately
chosen. In this paper, we present theoretical upper-bounds on the probability
that the LRPC decoding algorithm fails
Fault-tolerance techniques for hybrid CMOS/nanoarchitecture
The authors propose two fault-tolerance techniques for hybrid CMOS/nanoarchitecture implementing logic functions as look-up tables. The authors compare the efficiency of the proposed techniques with recently reported methods that use single coding schemes in tolerating high fault rates in nanoscale fabrics. Both proposed techniques are based on error correcting codes to tackle different fault rates. In the first technique, the authors implement a combined two-dimensional coding scheme using Hamming and Bose-Chaudhuri-Hocquenghem (BCH) codes to address fault rates greater than 5. In the second technique, Hamming coding is complemented with bad line exclusion technique to tolerate fault rates higher than the first proposed technique (up to 20). The authors have also estimated the improvement that can be achieved in the circuit reliability in the presence of Don-t Care Conditions. The area, latency and energy costs of the proposed techniques were also estimated in the CMOS domain
A Rice method proof of the Null-Space Property over the Grassmannian
The Null-Space Property (NSP) is a necessary and sufficient condition for the
recovery of the largest coefficients of solutions to an under-determined system
of linear equations. Interestingly, this property governs also the success and
the failure of recent developments in high-dimensional statistics, signal
processing, error-correcting codes and the theory of polytopes. Although this
property is the keystone of -minimization techniques, it is an open
problem to derive a closed form for the phase transition on NSP. In this
article, we provide the first proof of NSP using random processes theory and
the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds
for NSP with respect to unitarily invariant distributions. Furthermore, we
derive a simple sufficient condition for NSP.Comment: 18 Pages, some Figure
Stability of Homomorphisms, Coverings and Cocycles I: Equivalence
This paper is motivated by recent developments in group stability, high
dimensional expansion, local testability of error correcting codes and
topological property testing. In Part I, we formulate and motivate three
stability problems: 1. Homomorphism stability: Are almost homomorphisms close
to homomorphisms? 2. Covering stability: Are almost coverings of a cell complex
close to genuine coverings of it? 3. Cocycle stability: Are 1-cochains whose
coboundary is small close to 1-cocycles? We then prove that these three
problems are equivalent.Comment: 32 page
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