Superconductivity was discovered by Onnes (1911), who observed that the electrical resistance of various metals dropped to zero when the temperature of the sample was lowered below a certain critical value Tc, the transition temperature, which depends on the specific material being dealt with.

Another equally important feature of this new phase was its perfect diamagnetism which was discovered by Meissner and Ochsenfeld (1933), the so-called Meissner effect. A metal in its superconducting phase completely expels the magnetic field from its interior (see Fig. 3.1). The very fact that many molecules and atoms are repelled by the presence of an external magnetic field is quite well known, as we have already seen in the preceding chapter. The difference here lies in the perfect diamagnetism, which means that it is the whole superconducting sample that behaves as a giant atom!

This effect persists (for certain kinds of metal) until we reach a critical value of the external magnetic field, H = Hc(T), above which the superconducting sample returns to its normal metallic state. Moreover, at fixed temperature, this effect is completely reversible, suggesting that the superconducting phase is an equilibrium state of the electronic system. The temperature dependence of the critical field is such that Hc(Tc) = O and Hc(0) attains its maximum value as shown in Fig. 3.2.

In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables.

In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems.

The immediate problem we have to face concerns the choice of dissipative system to be studied.

On deciding to write this book, I had two main worries: firstly, what audience it would reach and secondly, to avoid as far as possible overlaps with other excellent texts already existing in the literature.

Regarding the first issue I have noticed, when discussing with colleagues, super-vising students, or teaching courses on the subject, that there is a gap between the standard knowledge on the conventional areas of physics and the way macroscopic quantum phenomena and quantum dissipation are presented to the reader. Usually, they are introduced through phenomenological equations of motion for the appropriate dynamical variables involved in the problem which, if we neglect dissipative effects, are quantized by canonical methods. The resulting physics is then interpreted by borrowing concepts of the basic areas involved in the problem – which are not necessarily familiar to a general readership – and adapted to the particular situation being dealt with. The so-called macroscopic quantum effects arise when the dynamical variable of interest, which is to be treated as a genuine quantum variable, refers to the collective behavior of an enormous number of microscopic (atomic or molecular) constituents. Therefore, if we want it to be appreciated even by more experienced researchers, some general background on the basic physics involved in the problem must be provided.

In order to ill this gap, I decided to start the presentation of the book by introducing some very general background on subjects which are emblematic of macroscopic quantum phenomena: magnetism and superconductivity.

Introduction

In the standard model of quantum computation [D95], some simple quantum state is prepared and evolved by a partially ordered set of unitary gates, and finally is measured in the computation basis. Up until the late 1990s, transformations that deviate from unitarity were commonly associated with irreversibility, and thus incoherence and harm induced in the quantum data of interest. However, this viewpoint was challenged by three notable results: (1) teleportation [BBC+93], (2) syndrome measurements [S95, S96e], and (3) the fault-tolerant Toffoli and π/8-gate constructions [S96b, BMP+99]. In these schemes, well-designed measurements project the quantum data of interest onto a subspace of the original ambient space for each outcome, up to some known unitary transformation that is either reversible (and harmless) or intended. Subsequently, researchers found many useful error-correction procedures and unitary gates that rely heavily on measurements.

Two “measurement models” of quantum computation were proposed in the early 2000s that make extreme use of measurements. In the cluster model [RB01], any quantum computation can be performed by applying a partially time-ordered set of single-qubit measurements on a certain entangled state. The state must be of appropriate size, but is otherwise independent of the computation. The related graph state model [RBB03] uses an initial state that depends on the computation itself. In the seemingly different teleportation-based model [N03, L01a, L04], any quantum computation can be implemented by applying one- or two-qubit nondemolition measurements on an arbitrary initial state, so as to “teleport” each gate in the simulated circuit.

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.

Quantum error correction (QEC) is all for naught if it cannot be implemented experimentally in the laboratory. Further, even if it can be implemented in small laboratory experiments, these experiments need not lead to a technology that is scalable to large quantum computers (where large is defined as “big enough to contemporaneously outperform the best classical computer on some problem”). In this chapter we will survey some of the experiments that have been performed to implement QEC. These experiments are all a long way from demonstrating viable QEC, but demonstrate the proof-of-principle methods that will need to be implemented in the future if quantum computation is to be made viable. This chapter deals with experimental implementations whose goal is to implement QEC, and not experiments using passive or open-loop methods, which are covered in Chapter 22.

Experiments in liquid-state NMR

The first experiments that attempted to perform QEC were performed using room-temperature liquid-state NMR. Liquid-state NMR [CFH97, GC97] is a testbed for quantum computing ideas in which one uses the internal states of coupled nuclear spins from a molecule as the qubits in a quantum computer. In its original and experimentally implemented form, liquid-state NMR is not generally thought to be scalable: the signal used to read out the result of the quantum computation decays exponentially [W97] in the number of qubits in the system and the mixed states produced in these experiments can be shown to possess no quantum entanglement.

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:

1. (i) Random driving forces from the environment (e.g., Brownian motion).

2. (ii) Interactions that produce entanglement between the system and the environment.

3. (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.

In Chapter 1 we saw that open quantum systems could interact with an environment and that this coupling could turn pure states into mixed states. This process will detrimentally impact any quantum computation, because it can lessen or destroy the interference effects that are vital to distinguishing a quantum computer from a classical computer. The problem of overcoming this effect is called the decoherence problem. Historically, the problem of overcoming decoherence was thought to be a major obstacle towards building a quantum computer. However, it was discovered that, under suitable conditions, the decoherence problem could be overcome. The main idea behind how this can be achieved is through the theory of quantum error correction (QEC). In this chapter we give an introduction into the way in which the decoherence problem can be overcome via the method of QEC. It is important to note that the scope of the introduction is not comprehensive, and focuses only on the basics of QEC without reference to the notion of fault-tolerant quantum computation, which is covered in Chapter 5. Quantum error correction should be thought of as a (major) tool in this larger theory of fault-tolerant quantum computing.

Introduction

The idea of fault-tolerant quantum computation presents a bold challenge to the rather well established principles called the strong Church–Turing thesis and the Bohr correspondence principle. One of the variations of the strong Church–Turing thesis (SCTT), which is not due to Church or Turing, but rather gradually developed in the field of computational complexity theory, is the following [BV97]:

Any “reasonable” model of computation can be efficiently simulated on a probabilistic Turing machine.

This can be expressed even less formally but more practically as:

No computer can be more efficient than a digital one equipped with a random number generator.

Here, a computer A is “more efficient” than a computer B if A can solve, in polynomial time, a problem that cannot be solved in a polynomial time by computer B.

On the other hand the Bohr correspondence principle (BCP) demands that:

Classical physics and quantum physics give the same answer when the systems become large.

A more rigorous form of the BCP is the following:

For large systems the experimental data are consistent with classical probabilistic models. Here, by a classical probabilistic model we mean a model in which all observables are described by real functions, and all states by probability distributions on a certain set (“phase-space”).

Introduction

In the previous chapter we saw how quantum information can be protected by active means. A combination of encoding and syndrome measurements, followed by corrective operations conditioned on the syndrome measurement outcomes, led to a method by which quantum information could be error corrected. In this chapter we will see how quantum information can be protected passively. Rather than attempting to fix errors, we will hide the quantum information from the environment, so that it never becomes corrupted. This “quiet corner” of the system's Hilbert space is called a decoherence-free subspace (DFS) or noiseless subsystem (NS). The main mechanism that will allow us to play this trick is symmetry. Every student of physics knows that symmetries lead to conservation laws. In our case the conserved quantity will turn out to be quantum information. Of course such symmetries must be exceedingly rare, or else protected quantum information would be all around us. Thus we will be forced to make some very strong assumptions, and ask the environment to “cooperate.” While we will definitely allow the environment to couple to our system, we will only allow it to do so in a certain symmetric manner. We will later discuss how relaxing this “cooperation requirement” still allows quantum information to be passively protected for a short time. In Chapter 4 we will see how, by using dynamical decoupling techniques, it becomes possible to force the environment to cooperate, i.e., to dynamically generate symmetries leading to a DFS or NS.