The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit

Dynamic Iterative Pursuit

For compressive sensing of dynamic sparse signals, we develop an iterative pursuit algorithm. A dynamic sparse signal process is characterized by varying sparsity patterns over time/space. For such signals, the developed algorithm is able to incorporate sequential predictions, thereby providing better compressive sensing recovery performance, but not at the cost of high complexity. Through experimental evaluations, we observe that the new algorithm exhibits a graceful degradation at deteriorating signal conditions while capable of yielding substantial performance gains as conditions improve.Comment: 6 pages, 7 figures. Accepted for publication in IEEE Transactions on Signal Processin

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute. Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist? We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of $[[N,k,d,\varepsilon]]$ approximate QLDPC codes that encode $k = \widetilde{\Omega}(N)$ logical qubits into $N$ physical qubits with distance $d = \widetilde{\Omega}(N)$ and approximation infidelity $\varepsilon = \mathcal{O}(1/\textrm{polylog}(N))$. The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in $\mathcal{O}(\textrm{polylog} N)$ projectors. We prove the existence of an efficient encoding map, and we show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth. Finally, we show that the spectral gap of the code Hamiltonian is $\widetilde{\Omega}(N^{-3.09})$ by analyzing a spacetime circuit-to-Hamiltonian construction for a bitonic sorting network architecture that is spatially local in $\textrm{polylog}(N)$ dimensions.Comment: 51 pages, 13 figure

O(N) methods in electronic structure calculations

Linear scaling methods, or O(N) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high performance computers. The linear scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas is then discussed. The applications of linear scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear scaling methods are discussed.Comment: 85 pages, 15 figures, 488 references. Resubmitted to Rep. Prog. Phys (small changes

Network-coded NOMA with antenna selection for the support of two heterogeneous groups of users

The combination of Non-Orthogonal Multiple Access (NOMA) and Transmit Antenna Selection (TAS) techniques has recently attracted significant attention due to the low cost, low complexity and high diversity gains. Meanwhile, Random Linear Coding (RLC) is considered to be a promising technique for achieving high reliability and low latency in multicast communications. In this paper, we consider a downlink system with a multi-antenna base station and two multicast groups of single-antenna users, where one group can afford to be served opportunistically, while the other group consists of comparatively low power devices with limited processing capabilities that have strict Quality of Service (QoS) requirements. In order to boost reliability and satisfy the QoS requirements of the multicast groups, we propose a cross-layer framework including NOMAbased TAS at the physical layer and RLC at the application layer. In particular, two low complexity TAS protocols for NOMA are studied in order to exploit the diversity gain and meet the QoS requirements. In addition, RLC analysis aims to facilitate heterogeneous users, such that, sliding window based sparse RLC is employed for computational restricted users, and conventional RLC is considered for others. Theoretical expressions that characterize the performance of the proposed framework are derived and verified through simulation results