Quantum circuits

If the state of a quantum system is a kind of information, then the dynamics of that system is a kind of information processing. This is the basic idea behind quantum computing, which seeks to exploit the physics of quantum systems to do useful information processing. In this chapter we will acquaint ourselves with a few of the ideas of quantum computing, using the idealized quantum circuit model. Then we will turn our attention to an actual quantum process, nuclear magnetic resonance, that can be understood as a realization of quantum information processing.

In a quantum circuit, we have a set of n qubit systems whose dynamical evolution is completely under our control. We represent the evolution as a sequence of unitary operators, each of which acts on one or more of the qubits. Graphically, we represent the qubits as a set of horizontal lines, and the various stages of their evolution as a series of boxes or other symbols showing the structure of the sequence of operations. In such a diagram, time runs from left to right (see Fig. 18.1.). The n qubits form a kind of “computer,” and its overall evolution amounts to a “computation.” The key idea is that very complicated unitary operations on the n qubits can be built up step-by-step from many simple operations on one, two or a few qubits.

The delta function

In this appendix, we review some techniques of mathematical physics. We present the mathematics in “physics style” – that is, with apparent disregard for the mathematical niceties. We will use “functions” whose properties cannot be matched by any actual function. We will exchange the order of limit operations by commuting integrals, derivatives, and infinite sums, all without any apparent consideration of the deep analytical issues involved. If the math police gave out tickets for reckless deriving, we would probably get one.

Why risk it? Often, the “reckless” derivation is a useful shorthand for a more sophisticated (and rigorous) chain of mathematical reasoning. An ironclad proof of a result may have to deal with many technical issues that, though necessary to close all of the logical loopholes, act to obscure the central ideas. The less formal approach is therefore both briefer and more revealing.

Spin-s systems

Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum, which a system of particles possesses due to particle motion through space; and spin angular momentum, which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description.

We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states |z±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {|x±〉} and {|y±〉} in terms of the |z±〉 states. From basis states and measurement values we constructed operators for the spin components Sx, Sy, and Sz. With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56).

Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.

Information and bits

On the evening of 18 April 1775, British troops garrisoned in Boston prepared to move west to the towns of Lexington and Concord to seize the weapons and capture the leaders of the rebellious American colonists. The colonists had anticipated such a move and prepared for it. However, there were two possible routes by which the British might leave the city: by land via Boston Neck, or directly across the water of Boston Harbor. The colonists had established a system of spies and couriers to carry the word ahead of the advancing troops, informing the colonial militias exactly when, and by what road, the British were coming.

The vital message was delivered first by signal lamps hung in the steeple of Christ Church in Boston and observed by watchers over the harbor in Charlestown. As Henry Wadsworth Longfellow later wrote,

Two lamps: the British were crossing the harbor. A silversmith named Paul Revere, who had helped to organize the communication network, was dispatched on horseback to carry the news to Lexington. He stopped at houses all along the way and called out the local militia. By dawn on 19 April, the militiamen were facing the British on Lexington Common.

The last two decades have seen the development of the new field of quantum information science, which analyzes how quantum systems may be used to store, transmit, and process information. This field encompasses a growing body of new insights into the basic properties of quantum systems and processes and sheds new light on the conceptual foundations of quantum theory. It has also inspired a great deal of contemporary research in optical, atomic, molecular, and solid state physics. Yet quantum information has so far had little impact on the way that quantum mechanics is taught.

Quantum Processes, Systems, and Information is designed to be both an undergraduate textbook on quantum mechanics and an exploration of the physical meaning and significance of information. We do not regard these two aims as incompatible. In fact, we believe that attention to both subjects can lead to a deeper understanding of each. Therefore, the essential “story” of this book is very different from that found in most existing undergraduate textbooks.

Roughly speaking, the book is organized into five parts:

• Part I (Chapters 1–5) presents the basic outline of quantum theory, including a development of the essential ideas for simple “qubit” systems, a more general mathematical treatment, basic theorems about information and uncertainty, and an introduction to quantum dynamics.

• Part II (Chapters 6–9) extends the theory in several ways, discussing quantum entanglement, ideas of quantum information, density operators for mixed states, and dynamics and measurement on open systems.

• […]

Continuous degrees of freedom

The quantum systems we have discussed so far have been described by finite-dimensional Hilbert spaces. Basic measurements on such systems have a finite number of possible outcomes, and a quantum state predicts a discrete probability distribution over these. Now we wish to extend our theory to handle systems with one or more continuous degrees of freedom, such as the position of a particle that can move in one dimension. This will require an extension of our theory to Hilbert spaces of infinite dimension, and to systems with continuous observables.

There is a philosophical issue here. How do we know that there really are infinitely many distinct locations for a particle? The short answer is, we don't. It might be that space itself is both discrete (at the tiniest scales) and bounded (at the largest), so that the number of possible locations of a particle is some very large but finite number. If this is the case, then the continuum model for space is nothing more than a convenient approximation. Infinity is just a simplified way of describing a quantity that is immense, but still finite.

In this section, we will adopt this view of infinity. We will imagine that any continuous variable is really an approximation of a “true” discrete variable. This idea will motivate the continuous quantities and operations that we need.

The photon in the interferometer

This chapter introduces many of the ideas of quantum theory by exploring three specific “case studies” of quantum systems. Each is an example of a qubit, a generic name for the simplest type of quantum system. The concepts we develop will be incorporated into a rigorous mathematical framework in the next chapter. Our business here is to provide some intuition about why that mathematical framework is reasonable and appropriate for dealing with the quantum facts of life.

Interferometers

In Section 1.2 we discussed the two-slit interference experiment with a single photon. In that experiment, the partial waves of probability amplitude were spread throughout the entire region of space beyond the two slits. It is much easier to analyze the situation in an interferometer, an optical apparatus in which the light is restricted to a finite number of discrete beams. The beams may be guided from one point to another, split apart or recombined as needed, and when two beams are recombined into one, the result may show interference effects. At the end of the interferometer, one or more sensors can measure the intensity of various beams. (A beam is just a possible path for the light, so there is nothing paradoxical in talking about a beam of zero intensity.) Figure 2.1 shows the layout of a Mach–Zehnder interferometer, which is an example of this kind of apparatus.