A Noncoherent Space-Time Code from Quantum Error Correction

In this work, we develop a space-time block code for noncoherent communication using techniques from the field of quantum error correction. We decompose the multiple-input multiple-output (MIMO) channel into operators from quantum mechanics, and design a non-coherent space time code using the quantum stabilizer formalism. We derive an optimal decoder, and analyze the former through a quantum mechanical lens. We compare our approach to a comparable coherent approach and a noncoherent differential approach, achieving comparable or better performance.Comment: 6 pages, one figure, accepted at the 53rd annual Conference on Information Sciences and System

Union bound minimization approach for designing grassmannian constellations

In this paper, we propose an algorithm for designing unstructured Grassmannian constellations for noncoherent multiple-input multiple-output (MIMO) communications over Rayleigh block-fading channels. Unlike the majority of existing unitary space-time or Grassmannian constellations, which are typically designed to maximize the minimum distance between codewords, in this work we employ the asymptotic pairwise error probability (PEP) union bound (UB) of the constellation as the design criterion. In addition, the proposed criterion allows the design of MIMO Grassmannian constellations specifically optimized for a given number of receiving antennas. A rigorous derivation of the gradient of the asymptotic UB on a Cartesian product of Grassmann manifolds, is the main technical ingredient of the proposed gradient descent algorithm. A simple modification of the proposed cost function, which weighs each pairwise error term in the UB according to the Hamming distance between the binary labels assigned to the respective codewords, allows us to jointly solve the constellation design and the bit labeling problem. Our simulation results show that the constellations designed with the proposed method outperform other structured and unstructured Grassmannian designs in terms of symbol error rate (SER) and bit error rate (BER), for a wide range of scenarios.This work was supported by Huawei Technologies, Sweden under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033. The work of I. Santamaria was also partly supported under grant PID2019-104958RB-C43 (ADELE) funded by MCIN/ AEI /10.13039/501100011033

Boolean Functions, Projection Operators and Quantum Error Correcting Codes

This paper describes a fundamental correspondence between Boolean functions and projection operators in Hilbert space. The correspondence is widely applicable, and it is used in this paper to provide a common mathematical framework for the design of both additive and non-additive quantum error correcting codes. The new framework leads to the construction of a variety of codes including an infinite class of codes that extend the original ((5,6,2)) code found by Rains [21]. It also extends to operator quantum error correcting codes.Comment: Submitted to IEEE Transactions on Information Theory, October 2006, to appear in IEEE Transactions on Information Theory, 200

Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to 2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is included in the arXiv packag

Quantum information processing approaches in classical systems

The engineering problem of building scalable quantum computers has prompted the development of a rich theory modeling the evolution of quantum systems as well as techniques to preserve quantum information in the presence of noise. Such techniques offer systems-level approaches to the problem of robustly encoding and preserving information and, as a result, see applicability in a wide variety of architectures for computing systems. In this thesis, we visit the mathematical underpinnings of quantum information and apply strategies inspired by quantum information processing to two non-quantum systems to demonstrate advantage. We first describe the construction of a quantum emulation device, an analog electronic system with the same mathematical structure as a gate-based quantum computer, and introduce novel time-domain information encoding methods to increase the computational capacity of the device. We confirm the sustained performance of the improved system by successfully transforming emulated states by randomly selected quantum gates. We then visit similarities between quantum information processing and signal processing in the noncoherent wireless communication setting, the latter being an environment characterized by a lack of instantaneous channel knowledge. We describe the theoretical underpinnings of the noncoherent communication environment from both an information theoretic and signal processing perspective. This leads us to propose a multi-antenna space-time code construction based on a family of quantum error correcting codes known as stabilizer codes. For this code, we derive the optimal decoder in Rayleigh and Ricean fading and benchmark the its performance against coherent and differential coding at comparable rates.Electrical and Computer Engineerin

Two are better than one: Fundamental parameters of frame coherence

This paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first use worst-case and average coherence to derive near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. Next, we provide a catalog of nearly tight frames with small worst-case and average coherence. Later, we find a new lower bound on worst-case coherence; we compare it to the Welch bound and use it to interpret recently reported signal reconstruction results. Finally, we give an algorithm that transforms frames in a way that decreases average coherence without changing the spectral norm or worst-case coherence

Kerdock Codes Determine Unitary 2-Designs

The binary non-linear Kerdock codes are Gray images of ℤ_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=−√-1 by these ℤ_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits