A Brief Preview

Let me give you a quick preview of the story we tell in this course. To help you develop an understanding of quantum computing, we will need the key concept of a quantum state. We say more about quantum states below. Like many concepts in science and math, it will be important to think about states using multiple ways of representing quantum states. We will introduce you to the following:

1. Abstract state “vectors”
2. Graphical pictures of quantum state “spaces”
3. Numerical forms of mathematical objects called column vectors and row vectors, which are straightforward generalizations of ordinary vectors

I’m confused already. I don’t know anything about vectors, state spaces, or column vectors.

We don’t expect you to be familiar with the concepts. We will introduce them one by one and give you several ways of thinking about their role in QC and QIS.

We have also found that many introductions to quantum computing and quantum information science use confusing language about quantum measurements, collapse of quantum states, jumps between quantum states, and the results of measurements on entangled states. Keeping in mind how actual quantum measurements are carried out, we will show you a way of talking about those concepts that will help sweep away the fog surrounding many of those discussions.

Before we get too far into the weeds, can you explain how quantum theory is different from quantum mechanics? I’ve heard both terms used in reading about quantum computing.

They are essentially the same. When quantum theory emerged in the first third of the 20th century, its way of thinking about how the world works was significantly different from what physicists had used before. To distinguish the new from the old, which physicists called “Newtonian mechanics” or “classical mechanics,” they invented the term quantum mechanics. In physics, mechanics means theories of motion, forces, and the like.

So, “classical” is like classical music? You know, Joseph Boulogne (Chevalier de Saint George), Wolfgang Amadeus Mozart, Clara Schumann, Richard Wagner, Nadia Boulanger, Marin Alsop, William Grant Still, and the like.

Well, not quite. What “classical” means is physics that does not need quantum mechanics to explain what is going on. Classical physics can be just as mind-bending as quantum physics, particularly when it comes to things like chaos theory.

To give you more context about what we are going to do, let me explain a bit about the differences between the Newtonian worldview and that of quantum mechanics. In Newtonian mechanics—before quantum mechanics came along—we thought of the world as being made of objects whose positions and velocities we can track and predict if we know the interactions among those objects. With the addition of theories of electricity and magnetism (now unified as electromagnetism), physicists thought they had a reasonably comprehensive picture of how the world works. Interactions among material entities (such as atoms, protons, electrons) and fields like the electromagnetic field and the gravitational field could explain everything. Of course, there were many details to be worked out, but the basic conceptual structure was in place, or so they thought. The world was essentially deterministic. What does that mean? Here is an example: If we knew the positions and velocities of all these entities that make up a system at some time, according to the Newtonian worldview, we could predict the future behavior of those entities. If you could apply that method to the bouncing numbered balls in a lottery machine, you could get rich very quickly!

That’s scary. You mean you could predict everything that might happen?

That indeed is the Newtonian worldview. But there is a catch. In real life, trying to do that becomes practically impossible if there are more than a few objects in the system. To make matters worse, even for simple systems with only a few objects, the precision with which we can make those predictions diminishes as the systems evolve in time because we can never really know the current state absolutely perfectly. Furthermore, with chaotic systems (like the weather), the precision of the predictions diminishes surprisingly rapidly because chaotic behavior changes dramatically. Such systems are hyper-sensitive to even small changes in their initial conditions. You may have heard of the butterfly effect: The flapping of a butterfly’s wings in Brazil can cause a tornado in Texas.

All of that said, science has moved beyond the Newtonian worldview. With the discovery of quantum mechanics in the early 20th century, we now know that the world at its most fundamental is not deterministic. As far as we know today, there is randomness and probability at the core of nature.

We will see that the strange aspects of quantum states—combinations of states called superposition states, a property of those states called entanglement, and the role of randomness and probability when we make observations on (“measure”) a quantum system—will turn out to be of crucial importance in quantum information science and quantum computing. In fact, sets of procedures—algorithms—in QIS and QC make use of all these strange aspects.

We will start off with a look at traditional computers, which are often called “classical computers” since they don’t directly involve quantum mechanics. The concepts of classical computers will form a good platform from which to launch ourselves into quantum mechanics and ultimately quantum computing. We hope you will be patient because it will be a while before we get into the meat of quantum computing. We have found that if we don’t spend some time on basic quantum concepts, then quantum computing and quantum information processing more generally will seem even more strange. Well-armed with those concepts, you will find quantum computing, though still strange, much more comprehensible.

Once we have laid the foundation with the crucial quantum concepts, we will show you several algorithms that demonstrate the advantages of quantum computing over classical computing. We will also introduce some key issues in quantum cryptography and error correction. That may sound rather boring but they are of great importance in any kind of computing and their quantum versions raise many intriguing issues about what is information and what is “noise.”

We will end our tour of QC and QIS by visiting several quantum issues that are not directly part of quantum computing but that, given the tools we will have learned about, we will be able to understand. Those topics reinforce the notion that the quantum world is conceptually and experimentally far removed from the Newtonian world. For example, there is some deep quantum weirdness in a famous result called Bell’s theorem (which makes us question the nature of reality) and in the relationship between classical computing and quantum computing.

We’ll conclude our voyage with a look at the future—both your personal future, Cardy, what you ought to do next if you want to learn more about quantum computing, and also the future of quantum computing itself.

That was a big help. All that sounds exciting. I’m ready to get going with the real stuff. But could you give me a quick preview of why we should worry about quantum computing, besides its being cool?

Fair question, but one that is hard to answer because the field of quantum computing in some ways is still in its infancy. But we do know that quantum computing allows us to carry out calculations that would be completely impractical with classical computers. More importantly, quantum computing allows us to think about problems in a completely different way from the way we think about classical computation. As an analogy, I might point to the discovery of microbes (bacteria and viruses) as the carriers of diseases. After that discovery, we had entirely new ways of preventing and treating disease. Similarly, I believe that ultimately quantum computing will open our eyes to whole new ways of thinking and issues that we don’t even recognize today. Quantum computing is not just doing traditional computing more efficiently or faster, though that in itself would be worthwhile, but it is doing computation in entirely new ways. Even though we don’t know exactly how that is going to work out, the possibilities are just mind-blowing. Already we know that QCs can find energies and configurations of molecules more efficiently than classical computers. That opens the door to new methods of drug discovery, for example. We also know that quantum computers can crack many of the encryption methods used to keep data secure. But quantum methods can be used to develop encryption systems that are yet more secure.