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Science

Jump-Starting Quantum Error Correction with Entanglement

Daniel Gottesman

Published Oct 19, 2006• 0 citations
Publisher

Abstract

Today's communications networks are filled with Web pages, e-mails, telephone conversations, video clips, financial transactions, and more, all encoded digitally as bursts of light flashing through fiber-optic cables. All of these messages can be boiled down to sequences of bits, 0's and 1's—usually long sequences, measured in kilobits, megabits, or sometimes even gigabits. But light, like everything else, behaves according to the laws of quantum mechanics, and a sequence of bits is not a good way to describe the properties of a quantum object. Instead, we should turn to quantum bits, or “qubits,” and someday, perhaps, today's network traffic will be supplemented by streams of such quantum information. Quantum transmissions would enable cryptographic systems to detect attempted eavesdropping, and quantum computers capable of manipulating quantum information could solve some problems far more rapidly than any conceivable classical computer. Just as classical information is encoded today in error-correcting codes to protect it against errors in transmission, quantum information will need to be encoded in quantum error-correcting codes in order to reliably reach its destination. Indeed, quantum information is more sensitive to errors than is classical information, so error correction will be even more important. On page 436 of this issue, Brun et al. (1) show how it is possible to improve the performance of quantum error-correcting codes, using a quantum state previously shared between the sender and receiver.

DOI
10.1126/science.1133797

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Today's communications networks are filled with Web pages, e-mails, telephone conversations, video clips, financial transactions, and more, all encoded digitally as bursts of light flashing through fiber-optic cables. All of these messages can be boiled down to sequences of bits, 0's and 1's—usually long sequences, measured in kilobits, megabits, or sometimes even gigabits. But light, like everything else, behaves according to the laws of quantum mechanics, and a sequence of bits is not a good way to describe the properties of a quantum object. Instead, we should turn to quantum bits, or “qubits,” and someday, perhaps, today's network traffic will be supplemented by streams of such quantum information. Quantum transmissions would enable cryptographic systems to detect attempted eavesdropping, and quantum computers capable of manipulating quantum information could solve some problems far more rapidly than any conceivable classical computer. Just as classical information is encoded today in error-correcting codes to protect it against errors in transmission, quantum information will need to be encoded in quantum error-correcting codes in order to reliably reach its destination. Indeed, quantum information is more sensitive to errors than is classical information, so error correction will be even more important. On page 436 of this issue, Brun et al. (1) show how it is possible to improve the performance of quantum error-correcting codes, using a quantum state previously shared between the sender and receiver.
Introduction Quantum error-correcting codes (QECCs) have turned out to have many applications; but their primary purpose, of course, is to protect quantum information from noise. This need manifests itself in two rather different situations. The first is in quantum computing. The qubits in a quantum computer are subject to noise due both to imprecision in the operations (or gates) and to interactions with the external environment. They undergo errors when they are acted upon, and when they are just being stored (though hopefully not at the same rate). All the qubits in the computer must be kept error-free long enough for the computation to be completed. This is the principle of fault tolerance. A rather different situation occurs in quantum communication . Here, the sender and receiver (Alice and Bob) are assumed to be physically separated, and the qubits travel from the sender to the receiver over a (presumably noisy) channel. This channel is assumed to be the dominant source of errors. While the qubits may undergo processing before and after transmission, errors during this processing are considered negligible compared to the channel errors. This picture of transmission through a channel is similar in spirit to the paradigm underlying much of classical information theory.
Bits as well as qubits are affected by noise in transmission, the process of computation or even storage. We will refer to the error-prone area as a channel. The error is introduced by unwanted but unavoidable interaction with the rest of the world and needs to be corrected before the data can be reliable. Classically this is important in communication systems and there is a whole thriving field of study of classical error correction. In quantum systems, states are intrinsically so fragile that unless they are impossibly isolated from the rest of the world, errors or noise would make computation or communication using them infeasible. Fortunately, it was discovered early in the history of this subject that it is possible to encode qubits in special ways so as to make the information resilient to errors. The qubits are decoded at the end of the computation/communication (schematic of Figure 10.1).
In the preceding chapter we mentioned the inevitable errors that occur when one tries to send quantum signals over, say, an optical fiber, even when there is no eavesdropper. But errors in transmission are not a problem just for quantum cryptography. For this entire chapter we forget about sending quantum information and instead focus on simply transmitting ordinary data faithfully over some kind of channel. Moreover, we assume that the data either is not sensitive or has already been encrypted. Unfortunately, many methods for transmitting data are susceptible to outside influences that can cause errors. How do we protect information from these errors? Error-correcting codes provide a mathematical method of not only detecting these errors, but also correcting them. Nowadays error-correcting codes are ubiquitous; they are used, for example, in cell-phone transmissions and satellite links, in the representation of music on a compact disk, and even in the bar codes in grocery stores. The story of modern error-correcting codes began with Claude Shannon's famous paper “A Mathematical Theory of Communication,” which was published in 1948. Shannon worked for Bell Labs where he specialized in finding solutions to problems that arose in telephone communication. Quite naturally, he started considering ways to correct errors that occurred when information was transmitted over phone lines. Richard Hamming, who also worked at Bell Labs on this problem, published a groundbreaking paper in 1950 on the subject.
This paper is a tutorial-style introduction to methods and protocols for reliable transmission of quantum information over noisy quantum links. The exposition is based on a design philosophy that parallels classical link-layer mechanisms such as forward error correction and Automatic Repeat reQuest (ARQ). It starts by drawing analogies and highlights differences between classical and quantum links. Such analogies are used as guiding principles, with explicit attention to where the classical correspondence no longer applies and where non-classical constraints and resources govern the protocol design. The emphasis is on the fact that classical reliability is achieved through fungible and cloneable bits, whereas unknown quantum states cannot be copied or freely retransmitted. In addition, quantum information is subject to stringent physical constraints due to decoherence as well as the need for transduction between stationary and flying qubits. Quantum error correction is presented as the analogue of classical forward error correction, where logical qubits are protected by encoding them into larger entangled code spaces and using syndrome measurements to infer and correct errors without collapsing the encoded state. It then introduces teleportation-based link abstractions, in which transmission of an unknown qubit is realized by consuming shared Bell pairs together with classical messages. Entanglement purification protocols play a role similar to ARQ-style reliability enhancement by converting many noisy entangled pairs into fewer, higher-fidelity pairs at the cost of additional resources. The paper is written from the perspective of classical communication engineering and is intended as a guide for translating the relevant ideas toward reliable quantum links and networks.
Information processing based on the laws of quantum mechanics promises to be a revolutionary new avenue in information technology. This emerging field of quantum information processing (QIP) is however challenged by the fragile nature of the quantum bits (qubits) in which quantum information is stored and processed. An error in even a single qubit makes the quantum processor go off-track, corrupting the calculation as a whole. Therefore, the chance for an erroneous outcome increases with the number of qubits in the processor. Large-scale QIP thus hinges on the ability to correct for these errors. Classical information processing often uses error correction algorithms to identify errors by checking whether information is consistent in multiple copies. This strategy is unfortunately not applicable to QIP as quantum states cannot be copied. Moreover, direct measurements on qubits collapse their quantum states, reducing them to classical information. Fortunately, the theory of quantum error correction (QEC) overcomes these complications by encoding quantum information in entangled states of many qubits and performing parity measurements to identify errors in the system without destroying the encoded information. Implementing these codes is challenging as it requires many qubits and quick interleaving of operations and measurements. Moreover, to not introduce more errors in the system than QEC can solve for, these operations and measurements need to be of sufficient fidelity and speed...
This paper investigates a novel approach to information encoding and error correction leveraging the unique properties of quantum entanglement. The core claim is to construct an information encoding scheme based on entangled states, complemented by quantum error correction codes, to achieve high-efficiency information transmission and storage. The proposed mechanism utilizes multi-body quantum entanglement for encoding, exploiting the correlated nature of entangled states, alongside the development of a specialized quantum error correction code designed to specifically address errors within entangled states. The research demonstrates a potential pathway exceeding the limitations of classical encoding, offering the possibility of superluminal information transfer by directly utilizing the correlations inherent in entanglement and implementing targeted error correction. The theoretical framework presented provides a foundation for future research into scalable quantum communication and data storage systems. The key mathematical components explored include the density matrix formalism, Pauli operators, and the concepts of quantum error correction codes, specifically focusing on their application to entangled states. The primary objective is to establish a robust and efficient method for encoding and correcting information using quantum entanglement, a crucial step towards realizing practical quantum communication networks.