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arXiv:2311.13040

The Penrose Tiling is a Quantum Error-Correcting Code

Zhi Li, Latham A. Boyle

Published Nov 21, 2023• 0 citations
arXivHTMLPDF

Abstract

The Penrose tiling (PT) is an intrinsically non-periodic way of tiling the plane, with many remarkable properties. A quantum error-correcting code (QECC) is a clever way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Although PTs and QECCs might seem completely unrelated, in this paper we point out that PTs give rise to (or, in a sense, are) a remarkable new type of QECC. In this code, quantum information is encoded through quantum geometry, and any local errors or erasures in any finite region, no matter how large, may be diagnosed and corrected. We also construct variants of this code (based on the Ammann-Beenker and Fibonacci tilings) that can live on finite spatial tori, in discrete spin systems, or in an arbitrary number of spatial dimensions. We discuss connections to quantum computing, condensed matter physics, and quantum gravity.

DOI
10.48550/arxiv.2311.13040

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The Penrose tiling (PT) is an intrinsically non-periodic way of tiling the plane, with many remarkable properties. A quantum error-correcting code (QECC) is a clever way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Although PTs and QECCs might seem completely unrelated, in this paper we point out that PTs give rise to (or, in a sense, are) a remarkable new type of QECC. In this code, quantum information is encoded through quantum geometry, and any local errors or erasures in any finite region, no matter how large, may be diagnosed and corrected. We also construct variants of this code (based on the Ammann-Beenker and Fibonacci tilings) that can live on finite spatial tori, in discrete spin systems, or in an arbitrary number of spatial dimensions. We discuss connections to quantum computing, condensed matter physics, and quantum gravity.
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In a gauge theory, a collection of kinematical degrees of freedom is used to redundantly describe a smaller amount of gauge-invariant information. In a quantum error correcting code (QECC), a collection of computational degrees of freedom that make up a device's physical layer is used to redundantly encode a smaller amount of logical information. We elaborate this parallel in terms of quantum reference frames (QRFs), which are a universal toolkit for dealing with symmetries in quantum systems and which define the gauge theory analog of encodings. The result is a precise dictionary between QECCs and QRF setups within the perspective-neutral framework for gauge systems. Concepts from QECCs like error sets and correctability translate to novel insights into the informational architecture of gauge theories. Conversely, the dictionary provides a systematic procedure for constructing symmetry-based QECCs and characterizing their error correcting properties. In this initial work, we scrutinize the dictionary between Pauli stabilizer codes and their corresponding QRF setups. We show that there is a one-to-one correspondence between maximal correctable error sets and tensor factorizations splitting system from error-generated QRF degrees of freedom. Relative to this split, errors corrupt only redundant frame data, leading to a novel characterization of correctability. When passed through the dictionary, standard Pauli errors behave as electric excitations that are dual, via Pontryagin duality, to magnetic excitations related to gauge-fixing. This gives rise to a new class of correctable errors and a systematic error duality. We illustrate our findings in surface codes, which themselves connect quantum error correction with gauge systems. Our exploratory investigations pave the way for foundational applications to gauge theories and for eventual practical applications to quantum simulation.
The aim of this project is to bring together quantum error-correcting codes theory and the study of finite geometries. A quantum code is used to protect quantum information from errors that may occur due to quantum decoherence. We give a geometric interpretation of the codes as sets of lines in certain finite projective spaces. We exploit the geometric aspect of codes to rewrite proofs in a more intuitive way and explore their properties through visualization. Some examples of stabiliser codes and their associated quantum sets of lines are presented. We also discuss how to build nonadditive codes as the union of stabiliser codes. Finite geometry has proved to be a powerful tool to work on quantum error-correcting codes. Some of its applications include finding new codes or proving the non-existence of codes with certain parameters.
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Computer architectures which exploit quantum mechanical effects can solve computing tasks that are otherwise impossible to perform. A quantum computer operates on a number of small quantum mechanical systems, known as quantum bits, or qubits. Since these systems are realized on the scale of atoms, they are very prone to errors. Errors occur when the environment interacts with the qubits, a process called decoherence. It is widely accepted that it will not be possible to shield qubits completely from the outside world. If one were to perform a quantum computation on the qubits directly, then after a short period of time the information present in the qubits would be lost. To counter decoherence the state of a qubit can be encoded into multiple physical ones. This is called a quantum error correcting code. Performing quantum error correction allows one to extend the life time of the encoded qubit arbitrarily, assuming that the rate of errors remains below a certain threshold value. The use of quantum codes creates an overhead in resources, as for every logical qubit many more physical qubits are needed. The resource overhead for fault-tolerance is problematic, since realizing qubits will be costly, and in the early stages of building quantum computers the number of physical qubits will be limited. The currently favored coding architecture is the toric code and its variant the surface code in which the physical qubits are put on a square grid in which interactions are only between nearest neighbors. In this thesis we will explore quantum codes in which qubits interact as if they were nearest neighbors in more exotic spaces. In the first part we will consider closed surfaces with constant negative curvature. We show how such surfaces can be constructed and enumerate all quantum codes derived from them which have less than 10.000 physical qubits. For codes that are extremal in a certain sense we perform numerical simulations to determine the value of their threshold. Furthermore, we give evidence that these codes can be used for more overhead efficient storage as compared to the surface code by orders of magnitude. We also show how to read and write the encoded qubits while keeping their connectivity low. In the second part we consider codes in which qubits are layed-out according to a four- dimensional geometry. Such codes allow for much simpler decoding schemes compared to codes which are two-dimensional. In particular, measurements do not necessarily have to be repeated to obtain reliable information about the error and the classical hardware performing the error correction is greatly simplified. We perform numerical simulations to analyze the performance of these codes using decoders based on local updates. We also introduce a novel decoder based on techniques from machine learning and image recognition to decode four-dimensional codes.
We discuss the connection between quantum error-correcting codes (QECCS) and algebraic coding theory. We start with an introduction to the relevant concepts of quantum mechanics, including the general error model. A quantum error-correcting code is a subspace of a complex Hilbert space, and its error-correcting properties are characterized by the Knill-Laflamme conditions. Using the stabilizer formalism, we illustrate how QECCs for can be constructed using techniques from algebraic coding theory. We also sketch how the information obtained via a quantum measurement can be interpreted as syndrome of the related classical code. Additionally, we present secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs. This includes the puncture code by Rains and construction X for quantum codes.