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Register a new userEfficient color code decoders in $dgeq 2$ dimensions from toric code decoders
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We introduce an efficient decoder of the color code in $dgeq 2$ dimensions, the Restriction Decoder, which uses any $d$-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-filp and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold $p_{textrm{2D}} approx 10.2%$ on the square-octagon lattice is on a par with the toric code threshold on the square lattice.
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The performance of short polar codes under successive cancellation (SC) and SC list (SCL) decoding is analyzed for the case where the decoder messages are coarsely quantized. This setting is of particular interest for applications requiring low-complexity energy-efficient transceivers (e.g., internet-of-things or wireless sensor networks). We focus on the extreme case where the decoder messages are quantized with 3 levels. We show how under SCL decoding quantized log-likelihood ratios lead to a large inaccuracy in the calculation of path metrics, resulting in considerable performance losses with respect to an unquantized SCL decoder. We then introduce two novel techniques which improve the performance of SCL decoding with coarse quantization. The first technique consists of a modification of the final decision step of SCL decoding, where the selected codeword is the one maximizing the maximum-likelihood decoding metric within the final list. The second technique relies on statistical knowledge about the reliability of the bit estimates, obtained through a suitably modified density evolution analysis, to improve the list construction phase, yielding a higher probability of having the transmitted codeword in the list. The effectiveness of the two techniques is demonstrated through simulations.
The design of decoding algorithms is a significant technological component in the development of fault-tolerant quantum computers. Often design of quantum decoders is inspired by classical decoding algorithms, but there are no general principles for building quantum decoders from classical decoders. Given any pair of classical codes, we can build a quantum code using the hypergraph product, yielding a hypergraph product code. Here we show we can also lift the decoders for these classical codes. That is, given oracle access to a minimum weight decoder for the relevant classical codes, the corresponding $[[n,k,d]]$ quantum code can be efficiently decoded for any error of weight smaller than $(d-1)/2$. The quantum decoder requires only $O(k)$ oracle calls to the classical decoder and $O(n^2)$ classical resources. The lift and the correctness proof of the decoder have a purely algebraic nature that draws on the discovery of some novel homological invariants of the hypergraph product codespace. While the decoder works perfectly for adversarial errors, it is not suitable for more realistic stochastic noise models and therefore can not be used to establish an error correcting threshold.
Polar codes are a class of capacity achieving error correcting codes that has been recently selected for the next generation of wireless communication standards (5G). Polar code decoding algorithms have evolved in various directions, striking different balances between error-correction performance, speed and complexity. Successive-cancellation list (SCL) and its incarnations constitute a powerful, well-studied set of algorithms, in constant improvement. At the same time, different implementation approaches provide a wide range of area occupations and latency results. 5G puts a focus on improved error-correction performance, high throughput and low power consumption: a comprehensive study considering all these metrics is currently lacking in literature. In this work, we evaluate SCL-based decoding algorithms in terms of error-correction performance and compare them to low-density parity-check (LDPC) codes. Moreover, we consider various decoder implementations, for both polar and LDPC codes, and compare their area occupation and power and energy consumption when targeting short code lengths and rates. Our work shows that among SCL-based decoders, the partitioned SCL (PSCL) provides the lowest area occupation and power consumption, whereas fast simplified SCL (Fast-SSCL) yields the lowest energy consumption. Compared to LDPC decoder architectures, different SCL implementations occupy up to 17.1x less area, dissipate up to 7.35x less power, and up to 26x less energy.
Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a $[[4,2,2]]$ code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.
Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations and prospects for protecting stored quantum information against errors. Here we show that the Hamiltonian of the central model of this class of systems, the Toric Code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via Superconducting Quantum Interference Devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along non-contractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single qubit operations allow to create and propagate elementary excitations of the Toric Code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.
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