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Chapman & Hall/CRC eBooks

Quantum Noise

Binayak Samadder Choudhury, Soumen Samanta

Published Dec 10, 2025• 0 citations
Publisher

Abstract

This chapter is on quantum noise. The origin of noise as a part of interaction with the environment is discussed. The Kraus operator formulation of noise along with various special types of noises are presented. References [ 14 , 32 , 44 , 77 , 109 , 113 , 115 , 131 ] are helpful to understand different aspects of quantum noise.
DOI
10.1201/9781003561439-6

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The simplest quantum system is the physical quantum bit or qubit . The qubit is a two-level quantum system—example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. We do not worry too much about physical implementations in this chapter, but instead focus on the mathematical postulates of the quantum theory and operations that we can perform on qubits. We progress from qubits to a study of physical qudits . Qudits are quantum systems that have d levels and are an important generalization of qubits. Again, we do not discuss physical realizations of qudits. Noise can affect quantum systems, and we must understand methods of modeling noise in the quantum theory because our ultimate aim is to construct schemes for protecting quantum systems against the detrimental effects of noise. In Chapter 1, we remarked on the different types of noise that occur in nature. The first, and perhaps more easily comprehensible type of noise, is that which is due to our lack of information about a given scenario. We observe this type of noise in a casino, with every shuffle of cards or toss of dice. These events are random, and the random variables of probability theory model them because the outcomes are unpredictable. This noise is the same as that in all classical information-processing systems.
The paradigm of quantum computing is a new frontier of computation based on phenomena of quantum physics. Quantum computing offers a gateway to the realm of scientific exploration that is inaccessible to conventional computing. But to fully realize this capability, we often have to deal with roadblocks due to inherent problems such as the fragile nature of qubits, which makes them highly susceptible to noise from the external environment and within. Therefore, an in-depth understanding of various types of noise in qubits and the possible ways to tackle them is vital and indispensable for devising a solution. In this work, we explore the different kinds of noise that plague a quantum computing system. In addition, it contains a discussion on the techniques to suppress them, which are open for further investigation.
The simplest quantum system is the physical quantum bit or qubit . The qubit is a two-level quantum system—example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. We do not worry too much about physical implementations in this chapter, but instead focus on the mathematical postulates of the quantum theory and operations that we can perform on qubits. From qubits we progress to a study of physical qudits . Qudits are quantum systems that have d levels and are an important generalization of qubits. Again, we do not discuss physical realizations of qudits. Noise can affect quantum systems, and we must understand methods of modeling noise in the quantum theory because our ultimate aim is to construct schemes for protecting quantum systems against the detrimental effects of noise. In Chapter 1, we remarked on the different types of noise that occur in nature. The first, and perhaps more easily comprehensible type of noise, is that which is due to our lack of information about a given scenario. We observe this type of noise in a casino, with every shuffle of cards or toss of dice. These events are random, and the random variables of probability theory model them because the outcomes are unpredictable. This noise is the same as that in all classical information-processing systems. On the other hand, the quantum theory features a fundamentally different type of noise. Quantum noise is inherent in nature and is not due to our lack of information, but is due rather to nature itself. An example of this type of noise is the “Heisenberg noise” that results from the uncertainty principle. If we know the momentum of a given particle from performing a precise measurement of it, then we know absolutely nothing about its position—a measurement of its position gives a random result. Similarly, if we know the rectilinear polarization of a photon by precisely measuring it, then a future measurement of its diagonal polarization will give a random result. It is important to keep the distinction clear between these two types of noise.
This paper is an introduction to the eleven works of the special issue on Quantum and Classical Frontiers of Noise. The weather, and its butterfly effect, is the typical example that explain why many natural phenomena are, in fact, not predictable with certainty. Noise in classical or quantum phenomena, understood as the difference between the empirical output of an experiment and its statistical prediction, is a measure of such uncertainty. One of the great contributions of noise appeared in 1905 when Einstein showed how the noisy (Brownian) motion of a dust particle seen in a microscope provided uncontroversial evidences on the existence of the (unseen) atoms. The aim of this special issue is to provoke discussions on how noise can help us to better understand (the reality behind) classical and quantum theories, and the frontier between them.
Introduction This chapter gives the physical and mathematical basis of decoherence in quantum systems, with particular emphasis on its importance in quantum information theory and quantum computation. We assume that the reader is already familiar with the basics of quantum mechanics, such as wave functions, the Schrödinger equation, and Hilbert spaces, which are covered in standard textbooks. Nevertheless, all basic concepts will be at least briefly defined and described. A working knowledge of graduate-level linear algebra and calculus is also assumed. In quantum information processing, the term decoherence is often used loosely to describe any kind of noise that can affect a quantum system. This can include sources of noise that seem, qualitatively, to be quite different from each other. These noise sources include: (i) Random driving forces from the environment (e.g., Brownian motion). (ii) Interactions that produce entanglement between the system and the environment. (iii) Statistical imprecision in the experimental controls on the system (e.g., timing errors, frequency fluctuations, etc.). Remarkably, all of these sources of noise can be described by an almost identical mathematical framework, in which pure quantum states can evolve into mixed quantum states, and the quantum effects necessary for information processing – interference, entanglement, and reversible evolution – are disrupted. The chapter begins by examining a highly schematic model of decoherence, involving systems of qubits, but uses this model to present the mathematical formalism used in describing more realistic systems as well.