Quantum error correction

Summary

Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum preparation, and faulty measurements. This would allow algorithms of greater circuit depth. Classical error correction employs redundancy. The simplest albeit inefficient approach is the repetition code. The idea is to store the information multiple times, and—if these copies are later found to disagree—take a majority vote; e.g. suppose we copy a bit in the one state three times. Suppose further that a noisy error corrupts the three-bit state so that one of the copied bits is equal to zero but the other two are equal to one. Assuming that noisy errors are independent and occur with some sufficiently low probability p, it is most likely that th

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The robustness of the delivery of digital compressed video streams has always been one of the main topic of the standardization bodies to assure the Quality of the Service (QoS). The MPEG standard which is nowadays the reference for digital television defines how to format the various component parts of a multimedia program and how they are combined into a single transmission bit stream. Although the MPEG stream may be directly used over a wide variety of media, it may also be used over a communication network and it is already packetized into short data packets. The multiplexed stream can be transmitted over a variety of communication networks such as Radio Frequency links (UHF/VHF), Digital Satellite links, Cable TV Networks, Standard Terrestrial Communication Links (PDH, SDH) or Packet/Cell Links (ATM, IP, IPv6, Ethernet). In critical and error prone channels the robustness of the stream has to be granted by employing strong error correction schemes. Depending on the used media the transmission has its own coding scheme and modulation methods designed for the particular environment. For example, in the case of IP based links the transfer protocol accepts loss of data and the problem can be easily overcome by retransmission. Wireless simplex transmission cannot exploit the same approach. Wireless transmission always means variable channel conditions especially when one or both terminals are moving on the outfield. The channel fading evolution and the signal impairments increase the error rate and corrupt the data. In the recent past the standardization of the DVB-T shows the possibility to provide a robust service for digital video delivery. This standard provides superior picture quality with the opportunity to view pictures in standard format or wide screen (4/3 or 16/9) formats, services including subtitling, multiple audio tracks, interactive content, multimedia content where, for instance, programs may be linked to world wide web material. Beside the provided services and features the standard offers also a modulation and transmission scheme capable to cope with the impairments of the terrestrial channel and to provide a BER after forward-error correction below 10-12. Unfortunately, these performances are guaranteed only for fixed reception, since the standard was originally aimed to replace classical analog television broadcasting in the UHF band. This thesis addresses the study of enhanced error-correction schemes in the scope of a compact, flexible and robust high-throughput real time video transmission system able to cope with high mobility channels. The study of a transmission system and error correction scheme tailored for this kind of application is here presented. The system exploits the OFDM modulation technique. This modulation scheme has already proven to be very effective in hostile environments. A flexible modulation scheme is employed to obtain variable bandwidth transmission and enhanced performances especially for mobile applications. The main topic of this research is the study of an effective error correction scheme adapted to this application. While nowadays the digital television standards use a concatenated error correction scheme based on convolutional inner coder, in this work the proposed solution is based on a turbo inner coder. Turbo codes has already shown good performances compared to convolutional codes. These results are here applied to a concatenated scheme to obtain very low BER. A comparison of the performances of various turbo coding techniques under certain implementation constraints is done in this work.The second element of the error correction scheme is the channel interleaver used to average the channel fading in time and frequency. This mechanism is conceived to be adapted to the modulated bandwidth and is optimized for high throughput real time video. This study shows that the proposed system proves to be very effective in high mobile channels because of the deep averaging strategies in conjunction with the high efficiency turbo based concatenated error correction scheme.

EPFL2005

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Motivated by the numerous examples of 1/3 magnetization plateaux in the triangular-lattice Heisenberg antiferromagnet with spins ranging from 1/2 to 5/2, we revisit the semiclassical calculation of the magnetization curve of that model, with the aim of coming up with a simple method that allows one to calculate the full magnetization curve and not just the critical fields of the 1/3 plateau. We show that it is actually possible to calculate the magnetization curve including the first quantum corrections and the appearance of the 1/3 plateau entirely within linear spin-wave theory, with predictions for the critical fields that agree to order 1/S with those derived a long time ago on the basis of arguments that required going beyond linear spin-wave theory. This calculation relies on the central observation that there is a kink in the semiclassical energy at the field where the classical ground state is the collinear up-up-down structure and that this kink gives rise to a locally linear behavior of the energy with the field when all semiclassical ground states are compared to each other for all fields. The magnetization curves calculated in this way for spin 1/2, 1, and 5/2 are shown to be in good agreement with available experimental data.

2016

Novel materials and algorithms for quantum technologies

Francesco Libbi

The enormous advancements in the ability to detect and manipulate single quantum states have lead to the emerging field of quantum technologies. Among these, quantum computation is the most far-reaching and challenging, aiming to solve problems that the classic computers could never address because of the exponential scaling, while quantum sensing exploits the ability to address single quantum states to realize ultra-sensitive and precise detectors. Defect centers in semiconductors play a primary role in these fields. The possibility to store information in the spin of their ground state, manipulate it through microwaves, and read it optically allows to use them as qubits. In addition, the very sharp dependence of their properties on temperature, strain and magnetic fields makes them very promising quantum sensors. In this Thesis we aim at contributing to the progress of quantum technologies both at the hardware and software level. From a hardware point of view, we study a key property of defect centers in semiconductors, the phonon-assisted luminescence, which can be measured to perform the readout of the information stored in a quantum bit, or to detect temperature variations. We predict the luminescence and study the exciton-phonon couplings within a rigorous many-body perturbation theory framework,an analysis that has never been performed for defect centers.In particular, we study the optical emission of the negatively-charged boron vacancy in 2D hexagonal boron nitride, which currently stands out among defect centers in 2D materials thanks to its promise for applications in quantum information and quantum sensing. We show that phonons are responsible for the observed luminescence, which otherwise would be dark due to symmetry. We also show that the symmetry breaking induced by the static Jahn-Teller effect is not able to describe the presence of the experimentally observed peak at 1.5 eV.The knowledge of the coupling between electrons and phonons is fundamental for the accurate prediction of all the features of the photoluminescence spectrum. However, the large number of atoms in a defect supercell hinders the possibility use density functional perturbation theory to study this coupling. In this work we present a finite-differences technique to calculate the electron-phonon matrix elements, which exploits the symmetries of the defect in such a way to use the very same set of displacement needed for the calculation of phonons. The computation of electron-phonon coupling thus becomes a simple post-processing of the finite-differences phonons calculation. On the quantum software side, we propose an improved quantum algorithm to calculate the Green's function through real-time propagation, and use it to compute the retarded Green's function for the 2-, 3- and 4-site Hubbard models. This novel protocol significantly reduces the number of controlled operations when compared to those previously suggested in literature. Such reduction is quite remarkable when considering the 2-site Hubbard model, for which we show that it is possible to obtain the exact time propagation of the

\ket{N\pm 1}

states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.

EPFL2022

Novel materials and algorithms for quantum technologies

Francesco Libbi

\ket{N\pm 1}

states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.

EPFL2022