Neural Network Decoders for Large-Distance 2D Toric Codes

We still do not have perfect decoders for topological codes that can satisfy all needs of different experimental setups. Recently, a few neural network based decoders have been studied, with the motivation that they can adapt to a wide range of noise models, and can easily run on dedicated chips without a full-fledged computer. The later feature might lead to fast speed and the ability to operate at low temperatures. However, a question which has not been addressed in previous works is whether neural network decoders can handle 2D topological codes with large distances. In this work, we provide a positive answer for the toric code \cite{Kitaev2003Faulttolerantanyon}. The structure of our neural network decoder is inspired by the renormalization group decoder \cite{duclos2010fast, duclos2013fault}. With a fairly strict policy on training time, when the bit-flip error rate is lower than 9% and syndrome extraction is perfect, the neural network decoder performs better when code distance increases. With a less strict policy, we find it is not hard for the neural decoder to achieve a performance close to the minimum-weight perfect matching algorithm. The numerical simulation is done up to code distance d=64. Last but not least, we describe and analyze a few failed approaches. They guide us to the final design of our neural decoder, but also serve as a caution when we gauge the versatility of stock deep neural networks. The source code of our neural decoder can be found at \cite{sourcecodegithub}.

Magic State Distillation: Not as Costly as You Think

Despite significant overhead reductions since its first proposal, magic state distillation is often considered to be a very costly procedure that dominates the resource cost of fault-tolerant quantum computers. The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.

The implementation of L-codes in the system of residual classes

The possibilities of R-codes for error correction in the system SRC are being intensively investigated. This is due to the simplicity of the structure of R-codes and good corrective capabilities, as well as the comparative simplicity of their construction for any given minimum code distance. It is important and interesting to consider the so-called linear codes (L-codes) in the SRC. In the literature, L-codes are described qualitatively rather than quantitatively. Until now no one has researched in depth the properties of systems of residual classes, whose bases are not mutually prime numbers. Such a system also has certain corrective properties, which makes it necessary to assess the possibility and feasibility of using such systems to improve the reliability of computer systems and components. Therefore, this important and promising issue is considered in this article.

Efficient magic state factories with a catalyzed|CCZ⟩to2|T⟩transformation

We present magic state factory constructions for producing|CCZ⟩states and|T⟩states. For the|CCZ⟩factory we apply the surface code lattice surgery construction techniques described in \cite{fowler2018} to the fault-tolerant Toffoli \cite{jones2013, eastin2013distilling}. The resulting factory has a footprint of12d×6d(wheredis the code distance) and produces one|CCZ⟩every5.5dsurface code cycles. Our|T⟩state factory uses the|CCZ⟩factory's output and a catalyst|T⟩state to exactly transform one|CCZ⟩state into two|T⟩states. It has a footprint25%smaller than the factory in \cite{fowler2018} but outputs|T⟩states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the caseθ=22.5∘produces a particularly efficient circuit for producing|T⟩states. Compared to using the12d×8d×6.5d|T⟩factory of \cite{fowler2018}, our|CCZ⟩factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm \cite{shor1994} and the chemistry algorithm of Babbush et al. \cite{babbush2018}. Assuming a physical gate error rate of10−3, our CCZ factory can produce∼1010states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to∼1012by increasing the factory footprint∼20%.

INCREASE IN EFFICIENCY OF DECODING OF CODES OF READ-SOLOMON ON THE GENERALIZED MINIMUM DISTANCE

In the modern systems of transfer and storage of information for correction of the arising mistakes noiseproof codes of Read-Solomon widely are used. With use of soft decisions apply decoding of these codes on the generalized minimum distance which advantage is simplicity of realization to correction of mistakes. In work the algorithm of decoding of codes of Read-Solomon on the generalized minimum distance which feature is use of the algebraic decoder correcting errors abroad a half of the minimum code distance with use of soft decisions is offered. The algebraic decoder realizes syndromic decoding and is based on application of analytical continuation of an algorithm of Berlekempa-Messi for 2τ iterations (τ-number of in addition corrected wrong symbols). He provides search of positions of tC+τ of wrong symbols in a code word (tC - number of the wrong symbols which are guaranteed corrected by a code) which locators would be the return to roots of a possible polynom of locators of errors of degree tC + τ. Search of positions of mistakes is carried out in ascending order of nadezhnost of symbols of the accepted code word. The efficiency of correction of mistakes was investigated by the offered algorithm in the channel with additive white Gaussian noise by imitating modeling on the COMPUTER. Researches were conducted for Read-Solomon's codes defined over the field of GF(28). The additional code prize provided with an algorithm at correction on iteration of three additional mistakes in relation to Read-Solomon (255,239,17) code reaches 0,26 dB. The additional code prize for Read-Solomon (255,127,129) code at correction on iteration of two additional mistakes has made about 0,1 dB. The additional code prize for Read-Solomon (255,41,215) code at correction on iteration of three additional mistakes has made about 0,17 dB.

Biorthogonal wavelet codes with prescribed code distance

We propose a scheme of construction of 2-circulant codes with given code distance on the basis of biorthogonal filters with the property of perfect reconstruction over a finite filed of odd characteristic. The corresponding algorithm for constructing biorthogonal filters utilizes the Euclidean algorithm for finding the gcd of polynomials.