Introduction

The existence of efficient quantum error correction (QEC), combined with the concept of the accuracy threshold [G98, KLZ98a], inspired confidence that reliable quantum computation is achievable in principle. However, a key point to understand is whether there are physical limitations to resilient quantum computation within this framework. In this chapter, we discuss one of the few situations that still poses some doubts [AHH+02, CSG.-B04, KF05, A06, ALZ06] about the effectiveness of QEC codes: critical environments.

The term “critical environment” originates from condensed matter physics, where it refers to physical systems in which quantum correlations decay as power laws. In such an environment, the Born–Markov approximation used to evaluate decoherence rates cannot be formally justified [W99]. For quantum computation, this fact translates into the appearance of errors that can depend on previous events in the computer history. The ultimate nightmare [AHH+02] is that this memory effect may eventually lead to error probabilities above the threshold value and therefore to the breakdown of resilient quantum computation. In this chapter, our goal is to find the minimal conditions for the existence of a finite threshold value; finding a particular value for the error threshold is a more detailed question which we leave for future work.

Introduction and overview

As mentioned in Chapter 4, dynamical decoupling (DD) techniques have developed decades of tradition in the context of NMR spectroscopy, where they continue to play a pervasive role throughout all aspects of imaging and coherent control of multi-spin dynamics in the liquid and solid state. A dedicated experimental characterization and exploitation of DD as a tool for high-fidelity dynamical control and decoherence suppression in different quantum information processing (QIP) platforms has been undertaken only in more recent years, largely enabled by impressive laboratory progress in fast and ultrafast coherent control capabilities over the past decade. Without attempting to provide a comprehensive in-depth account, it is our goal in this chapter to highlight some key experimental advances to illustrate the significance of DD strategies for near-term practical quantum error control in QIP; and justify why, quoting from a recent experimental work [MTA+06], they “will likely form a quintessential element in real quantum computers.”

The large majority of dynamical error-suppression implementations in QIP have aimed thus far to validate DD as a tool for enhanced quantum memory in different qubit devices in the presence of various decoherence mechanisms. Likewise, with the exception of a recent work reporting the suppression of collisional decoherence using a “continuous spin echo” close in spirit to Eulerian DD [SAD10], the majority of experiments have employed (sufficiently) “hard” pulses, such that any evolution other than that due to the control field could be taken as negligible and the bang-bang (BB) limit formalized in Chapter 4 invoked as an adequate approximation.

Introduction

Many of the chapters in this book have developed the theory of quantum error correction (QEC) for the specific purpose of building a fault-tolerant, reliable quantum computer. Specifically, Chapter 5 shows how the theory of QEC is a core component in the theory of fault-tolerant quantum computation. Fault-tolerant quantum computation has received much attention from the theoretical quantum information community because it is a key requirement for a scalable quantum computer. The most celebrated tasks that one could perform with a fault-tolerant quantum computer are factoring a prime number [S94b, S97], searching a database quickly [G96b, G97b], or simulating a quantum system efficiently [F82, L96].

One can exploit quantum phenomena to enhance communication as well. The field of quantum communication encompasses any aspect of quantum theory that is exploited for communicative purposes. In this chapter, we discuss four main topics involving QEC in quantum communication:

1. (i) entanglement distillation,

2. (ii) a security proof of quantum key expansion,

3. (iii) continuous-variable systems, and

4. (iv) implementation of QEC for the purpose of quantum communication.

We first comment briefly on each of the above topics. Entanglement distillation converts a set of noisy ebits to a smaller set of noiseless ones. It is useful to have a procedure for entanglement distillation because noiseless entanglement is the core resource in several quantum communication protocols. We show how the techniques from QEC apply to entanglement distillation. A security proof for quantum key expansion is a mathematical proof that shows that a quantum key expansion protocol is secure against an arbitrary attack by an eavesdropper.

Introduction

This chapter concerns quantum coding schemes based on iterative decoding. So far, most of the focus of this book has been on quantum error-correcting codes (QECCs) that use a relatively small number of qubits, and concatenation thereof. The decoding strategy adopted for these codes consists of identifying the lowest-weight error that is consistent with the syndrome and reversing its effect. The lowest-weight error associated with each syndrome can, in the worst case, be found by exhaustive search and stored in a look-up table for future reference. Each time a syndrome is measured, the appropriate recovery can be rapidly extracted from the table to complete the error-correction cycle.

In contrast, the codes described in this chapter use a very large number of qubits; the larger the better! For a fixed rate r = k/n, the performance of the codes improves as n increases. For such large codes however, more clever decoding schemes are required. For a code with parameters [[n, k, d]], the size of the look-up table – i.e. the number of distinct syndromes – is 2n−k = 2(1−r)n. Thus, look-up tables are only suitable for codes of a few dozen qubits. For larger codes, the decoding process involves a highly nontrivial computation that needs to be executed “on the fly,” so fast and reliable algorithms are needed. Iterative decoding is one such algorithm and it has produced some of the best coding schemes – in particular turbo-codes and sparse codes – widely used in classical wireless communication.

Introduction

What a good code is depends on the particular constraints of the problem at hand. In this chapter we address a constraint that is relevant to many physical settings: locality. In particular, we are interested in situations where geometrical locality is relevant. This typically means that the physical qubits composing the code are placed in some lattice and only interactions between nearby qubits are possible. In this case, it is desirable that syndrome extraction also be local, so that fault tolerance can possibly be achieved. Topological codes offer a natural solution to locality constraints, as they have stabilizer generators with local support.

In topological codes information is stored in global degrees of freedom, so larger lattices provide larger code distances. The nature of these global degrees of freedom is illustrated in Fig. 19.1, where several closed curves in a torus are compared. Consider curves a and b. They look the same if examined locally, as in the region marked with dotted lines. However, curve a is the boundary of a region, whereas curve b is not. In order to decide whether a curve is a boundary or not we need global information about it. This is, as we will see, a core idea in topological codes.

Introduction

In this chapter we continue the introduction to dynamical decoupling techniques for open quantum systems that was started in Chapter 4. Here we focus on the construction of efficient schemes for dynamical decoupling and we highlight some combinatorial constructions. Efficiency of decoupling schemes is measured in terms of the number of control operations to be applied to the system. This number should be small, ideally a polynomial in the number of qubits in the system.

If we assume that the quantum system is governed by a general Hamiltonian HS, having interactions involving any subset of the qubits, then any scheme that achieves decoupling of HS is necessarily inefficient in the above sense. The efficient schemes we consider here arise in physically realistic situations where there are much more stringent restrictions on the types of interactions. The most important example is the case of pair-interaction Hamiltonians. These Hamiltonians can be expressed as sums of interaction terms that involve at most two qubits.

While there is a large body of work on dynamical decoupling schemes for pair-interaction Hamiltonians [S90, EBW94,WHH68, H76], which historically has been used mainly in the context of nuclear magnetic resonance (NMR) theory, the past few years have seen a development of new techniques for decoupling that are based on group theory. We therefore call such techniques, or rather the pulse sequences that an experimenter can apply to a given Hamiltonian in order to selectively switch off unwanted couplings, combinatorial decoupling schemes.

Motivation and overview

The aim of this chapter is to illustrate the basic physical principles and mathematical framework of dynamical decoupling (DD) techniques for open quantum systems, as relevant to quantum information processing (QIP) applications. Historically, the physical origins of DD date back to the idea of coherent averaging of interactions, as pioneered in high-resolution solid-state nuclear magnetic resonance (NMR) by Haeberlen and Waugh using elegantly designed multiple-pulse sequences [HW68, WHH68, H76]. It was in the same landmark work [HW68] that average Hamiltonian theory was developed as a formalism on which the design and analysis of DD sequences has largely relied since then. In the original context of NMR spectroscopy, decoupling serves the purpose of enhancing resolution by simplifying complex spectra. This is achieved by realizing that an otherwise static spin Hamiltonian “can be made to appear time-dependent in a controlled way,” so that “as the characteristic repetition period of the pulses becomes [sufficiently] small, the spin system comes to behave over long times as though under the influence of a time-independent average Hamiltonian” [HW68]. Some basic insight may be gained by revisiting the paradigmatic example offered by the so-called Hahn echo [H50] and Carr–Purcell (CP) sequences [CP54] in the simplest setting of a two spin-1/2 system.

The principles of fault tolerance

In this chapter we will consider some of the practical difficulties in building a large-scale quantum computing device. This discussion relies heavily upon the prior chapters that introduced faulttolerant quantum computation. Assuming that fault tolerance is possible allows us to focus on the physical realization of these ideas with a few specific examples.

Before going into detail, we review the governing ideas behind any fault-tolerant architecture. These are the necessary components that we analyze in this chapter: good quantum memory, high-fidelity quantum operations, long-range quantum gates, and highly parallel operation, such that error correction in different sections of the device can be accomplished at the same time. While a wide variety of potential implementations may be possible, the goal of a fault-tolerant architecture is not only to be scalable, i.e., to be able to run an arbitrarily large computation with at most polynomial overhead [S95, S96e], but also to be as efficient as possible. Efficiency in this context means using the fewest physical resources (quantum bits, time, control operations) necessary to accomplish the desired computation [S03b].

From this perspective, the recipe for fault tolerance is well established. First, we need to identify what quantum operations are available for the various quantum bits at our disposal. Developing error models for these operations forms the bulk of this chapter. Important questions about operations include noise, implementation time, and bandwidth (how many such operations may be performed in parallel).

One of the applications of quantum error correction is protecting quantum computers from noise. Certainly, there is nothing particularly quantum mechanical in the idea of protecting information by encoding it. Even ordinary digital computers use various fault tolerance methods at the software level to correct errors during the storage or the transmission of information; e.g., the integrity of the bits stored in hard disks is verified by using parity checks (checksums). In addition, for critical computing systems such as those inside airplanes or nuclear reactors, software fault tolerance methods are also applied during the processing of information; e.g., airplane control computers compare the results from multiple parallel processors to detect faults. In general, however, the hardware of modern computers is remarkably robust to noise, so that for most applications the processing of information can be executed with high reliability without using software error correction.

In contrast to the ease and robustness with which classical information can be processed, the processing of quantum information appears at present to be much more challenging. Although constructing reliable quantum computing hardware is certainly a daunting task, we have nevertheless strong hopes that large-scale quantum computers, able to implement useful long computations, can in fact be realized. This optimism is founded on methods of quantum fault tolerance, which show that scalable quantum computation is, in principle, possible against a variety of noise processes.

For most of human history we maneuvered our way through the world based on an intuitive understanding of physics, an understanding wired into our brains by millions of years of evolution and constantly bolstered by our everyday experience. This intuition has served us very well, and functions perfectly at the typical scales of human life – so perfectly, in fact, that we rarely even think about it. It took many centuries before anyone even tried to formulate this understanding; centuries more before the slightest evidence suggested that these assumptions might not always hold. When the twin revolutions of relativity and quantum mechanics overturned twentieth-century physics, they also overturned this intuitive notion of the world.

In spite of this, the direct effect at the human scale has been small. Our cars do not run on Szilard engines. Very few freeways, even in Los Angeles, have signs saying “Speed Limit 300,000 km/s.” And human intuition remains rooted in its evolutionary origins. It takes years of training for scientists to learn the habits of thought appropriate to quantum mechanics; and even then, surprises still come along in the areas we think we understand the best.

Technology has transformed how we live our lives. Computers and communications depend on the amazingly rapid developments of electronics, which in turn derive from our understanding of quantum mechanics: we use the microscopic movements of electrons in solids to do work and play games; pulses of coherent light in optical fibers tie the world together.