Quantum Computation: a Quantum Primer

[draft]

Quantum computers have been in the news a lot lately, but from the comments I've been reading around the net it's clear that a lot of people don't understand what quantum computers are and aren't or what they can and can't do. That's no surprise -- quantum mechanics is a difficult concept to get your head around, let around quantum computation. That's why I've decided to write a few short articles on the subject of Quantum Computation -- a sort of math-less primer for the masses.

I'm starting with a brief introduction to Quantum Mechanics, because it's critical to understanding Quantum Computation and because it's one of my favorite subjects to ramble on about. IMHO, one of the worst injustices wrought upon our educated youth is a failure to introduce them to the wonders of the quantum world when they are young and able to accept mystery. There is much we do not understand about why quantum mechanics is the way it is, but it's not very hard to understand quantum mechanics itself. Unforunately, well-intentioned educators poison children's minds with the crazy idea that there are actually hard and fast rules by which one could calculate the future of the universe using Newton's Approximations, more poorly known as "Newton's Laws." Teaching the truth would be more interesting, more amazing, and engage more imaginations that Newton's work ever could.

So to start out with, if you've never been educated about Quantum Mechanics or Relativity, forget just about everything you know about matter and energy because most of it is probably wrong. I have to warn you: the truth is more crazy and amazing than you probably realize.

You can't watch the SciFi channel or read a SciFi book these days without a good chance of running across the words, "wave-particle duality," and the 'particle' half of that phrase is our first stop on this strange journey. Mater, as the grade school interpretation goes, is made up of atoms, each of which is further composed of electrons, protons, and neutrons. In this view, an atom consists of a hard nucleus with electrons spinning around it. You might already know that the proton and neutron are further composed of other more fundamental particles, but that is not really important to our discussion.

What is important is the idea of "size." When you think of a particle, it's completely normal to think of how large it is -- so many inches or microns in diameter. But for a single particle, reality is quite different.

Zen Shocker #1: "Take the red pill..."

You might have heard that atoms are mostly empty space because their constituent particles are very tiny, but reality is stranger yet: Individual particles -- the stuff of which matter is made -- are infinitely small, no matter how much mass they have. Infinitely small points have no volume, so everyday objects are composed of "empty" space that still contains mass. Take a moment to let the confusion that this causes in most people to wash over your mind.

Let all the thoughts of, "no, that can't be right!" drain slowly away.

Now let's quickly reconcile this strange idea with everyday reality by means of a somewhat inaccurate, but temporarily useful, analogy. Think of a star constellation: each star is but a tiny spot in the sky, but taken together they describe the shape of a recognizable object. Even though a point in space has no size, the distance between two points does. A baseball is like a constellation of infinitely small particles. The constellation consists of small groups of particles grouped together as "atoms", and clusters of atoms grouped together as "molecules", but in the end, it's still just a giant constellation of "particles". When taken together these particles describe the shape and mass of baseball even though what you are holding is mostly "empty" space.

Don't get too attached to this analogy though...

Zen Shocker #2: "Beam me up, Scotty..."

Particles don't generally 'move'. Instead, they "blink out of existence" and then "blink back into existence" somewhere else.

Again, breath deeply.

This is a very real form of short-range teleportation. You can think of it almost like an animated flip-book: An animator takes a picture of your baseball constellation, moves one of the particles to a new location, and then snaps another frame. In general the particles don't move far -- but they jump very rapidly. When these frames are shown in rapid succession they give the appearance of motion, even though none is actually taking place.

The analogy only holds just so far though... Not all the particles teleport at the same time; rather, they teleport randomly one at a time at random intervals.

Zen Shocker #4: "Get tea and no tea."

We've been using the word 'particle' rather loosely, and it is now time to more carefully define the word. It should be quite clear now that the "particles" we've been discussing don't behave anything like the particles you're probably used to.

The word 'particle' unfortunately has three meanings which are easily confused, and we need to eliminate that confusion.

In general when you hear the word 'particle(1)' you think a small bit of something, such as a grain of sand. In this sense, a 'particle(1)' has definite dimensions, mass, and is not at all what we are concerned with in quantum mechanics. A particle(1) is an idealization of a particle(2) -- no such beast really exists. Instead grains of sand are made of many particles(2).

When a physicist says 'particle(2)', as in, "an electron particle(2)," he or she is referring to a single quantum mechanical entity. I say 'entity' because the 'particle(2)' may or may not have mass, has no size, and generally behaves very "strangely" compared to a "normal particle(1)." These are the particles(2) we've been discussing.

Finally, 'particles(2)' "behave like 'particles(3)'," which is physicist-speak for "particles(2) have particle(1)-like properties". This means particles(2) can have mass, that, at times, but not always, particles(2) exists or are detected at a very specific and well-defined point in space, can collide with other particles(2), and impart energy to them just like idealized particles(1), but lack some idealized particle(1) properties like size.

Keeping straight what someone means when they say the word particle can be a bear, but it's really important if you want to understand Quantum Mechanics. A particle(2) might not be in a state in which it is currently behaving like a particle(3), and is never a particle(1) at any time.

Ok, go have a stiff drink.

Zen Shocker #4: "Read the directions and directly you will be directed in the right direction."

Now that you have a clear idea that particles aren't what you thought they were, we're going to really start turning the screws.

The 'Wave' part of "Wave-Particle Duality" refers to the fact that particles are also probability waves. When a particle "winks out of existence" as we talked about earlier, it takes on a new life as a probability wave. Probability waves are funny things and have a lot of weird properties. Most importantly a particle behaving like a wave still has mass.

A "probability wave" isn't something that can be directly measured, unlike an electric or magnetic field, but it's useful to think of it as being measurable anyway. The stronger the probability wave is in any location, the more likely the particle is to teleport to that location.

The shape of the probability wave is what gives rise to the "electron shell" shapes in atoms that you may have heard about in a Chemistry class. It's also what is meant by the very poorly worded term, "electron cloud". An s-shell, or spherical shell, is a probability wave that is spherical in shape. A p-shell is a probability wave that is dumbbell shaped. The funny shapes of higher-order shells are a result of quantizing rules which force the probability clouds to take on shapes defined by 'spherical harmonics', and is something we're definitely not going to cover here.

The important bit to take away from all this is that particles "wink out of existence", become a probability wave which describes the likelihood the particle will re-appear somewhere else, and then eventually "winks back into existence".

Zen Shocker #5: "Excuse me!"

What causes a particle to "wink back into existence" is the interaction with another particle. This is a complicated subject that is not something we're going to talk about much. Basically, when two particles' waves start interacting with each other, at some point both particles wink back into existence, interact, and then wink back out of existence. When this happens the probability wave is said to have 'collapsed'.

Crazy enough yet?

Zen Shocker #6: "There is no spoon."

This is really an extension of the previous idea, but think about it: if particles "wink into existence" and then immediately become a probability wave, they don't actually spend any time as particles. Actually, they generally spend an infinitesimally small time as a particle, and so really particles are waves. In fact, you would be hard pressed to prove that a particle is nothing more than a probability wave which has collapsed to a single point in space.

So particles are actually waves that occasionally collapses down to a single point and are thus localized "like a particle" for a brief moment in time.

Now think about that baseball. You are holding a constellation of separate probability waves which, at a very rapid rate, individually collapse into particles and then become probability waves again. What you are holding isn't a bunch of hard little particles: it's almost always "empty" space that contains mass and energy. At any given instant in time you're probably not holding any 'particles', but even when you are you're probably holding just one of them. There really is no baseball. Rather, there's just something that causes you to think, "baseball." Why do you see the baseball? Why does it have weight?

Because what you experience isn't the particles themselves; rather, it's the net interaction of those particles with the particles in your hand and the photons in the air all around you.

Zen Shocker #7: "This is not a test."

Quantum mechanics is one of the most carefully tested theories ever. It has so far stood up to every test applied to it.

That's it for part one.

In part two we'll look more closely at probability waves and find out that they're stranger than they already sound.

-- ThoughtKeeper - 21 Feb 2007