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arXiv:1812.07115 · Conference on Information Sciences and Systems

A Noncoherent Space-Time Code from Quantum Error Correction

S. Andrew Lanham, Travis C. Cuvelier, Corey Ostrove, Brian R. La Cour, Granville E. Ott, Robert W. Heath

Published Mar 1, 2019• 9 citations
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Abstract

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.

DOI
10.1109/ciss.2019.8692830

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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.
We propose a general framework for noncoherent communication using techniques from the field of quantum error correction (QEC). We first propose an approach for analyzing a classical communication channel as a quantum channel, and develop an extension of the QEC conditions to the classical case. We derive a quantum analogue of the noncoherent multiantenna wireless channel. Restricting to the case in which the number of transmitter and receiver antennas are equal and a power of two, we use the framework to develop a family of space-time block codes for noncoherent multiple-input multiple-output (MIMO) communication. Under a Rayleigh fading assumption, we derive the optimal decoder and bound the probability of symbol detection error. We compare our performance to comparable coherent and noncoherent approaches and achieve competitive performance for various antenna geometries and rates.
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.
This paper presents a novel quantum error correction protocol leveraging space-time encoding to enhance the resilience of quantum information transmission. Traditional quantum error correction protocols predominantly rely on spatial dimensionality encoding, which often proves insufficient against various noise sources. This protocol introduces a dual-dimensional encoding scheme, integrating both spatial and temporal dimensions, to significantly improve error detection and correction capabilities. The core mechanism utilizes the inherent error-correcting properties of quantum codes, amplified by the redundancy introduced through space-time encoding. The proposed protocol demonstrates a marked improvement in the transmission robustness of quantum information compared to conventional spatial-only approaches. The design focuses on optimizing the encoding and decoding processes to minimize the overhead associated with the space-time scheme while maximizing the error correction performance. Mathematical formulations and key parameters are detailed, providing a comprehensive understanding of the protocol's operational principles and potential performance metrics. This research contributes to the advancement of quantum communication technology by offering a more effective strategy for reliable quantum information transfer.
This paper introduces multidimensional generalizations of binary Reed-Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary Reed-Muller codes. The subspaces are associated with projection operators determined by Pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest. The Grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas, where separation with respect to the chordal distance on Grassmannian space guarantees closeness to the channel capacity. It is shown that the capacity of the noncoherent multiple-input-multiple-output (MIMO) channel at both low and moderate signal-to-noise ratio (SNR) (under the constraint that only isotropically distributed unitary matrices are used for information transmission) is closely approximated by these packings.
This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum-mechanical background to be able to understand how quantum error correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.
Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.