Quantum computing is one of the first great technologies of the 21st century, but the details are still shrouded in mystery. I can explain conventional digital computing down to the electron in a MOSFET, and with this newsletter, I have made it my mission to do the same for quantum computing.

Welcome to the Quantum Edge newsletter. Here you will read about the physics, chemistry, and all sciences that create the foundation for quantum computing. Join me in my quest to translate the mysteries of the quantum world to the language of the dinner table and the coffee shop.

In today’s newsletter: A look at vector addition through the eyes of a pool player and carnival goer

Qubits hold information (or numbers) as muti dimensional vectors. Conventional bits hold individual numbers (that can be classified as scalers, since they only hold one piece of information).

What else can hold multi-dimensional information? Written and spoken languages, like English or French can. A sentence is a common way that humans use descriptive language to hold and deliver multiple pieces of information. If you haven’t figured out where this is going yet - it is going to lead to math story problems.

Back in my school days, many people lamented at their dislike of story problems. Myself included. But I think that’s because all school-delivered story problems involve trains simultaneously leaving their respective stations in Pittsburgh and Chicago. My story problems involve bumper cars, which are much more fun, and unlike the trains, are expected to collide.

Qubits, as I’ve said, store information as vectors. Bumper cars, when operating, can also be described by vectors. They have speed and direction. When one bumper car hits another, speed and momentum is transferred between the two.

In my pool playing example in the last issue, the moving cue ball hit a stationary eight ball. The cueball was a vector that equaled its speed and direction. If we use a compass to describe the direction, we could say that the cue ball vector was (6, 180). Six inches per second at 180 degrees by the compass.

The eight ball was also a vector, but it started as (0, 0). Zero speed and zero direction. After the collision both pool balls were vectors with speed and direction. If the two balls were lined up perfectly in the direction 180 degrees and the player hit in a straight line with no curve or spin, post collision the vectors might be (0, 0) for the cue ball and (5.995, 180) for the eight ball.

With a perfect hit, virtually all of the energy is transferred from the cue ball to the eight ball. A small amount of the motion energy is converted to heat energy so the cue ball vector plus the eight ball vector (0, 0) + (5.995, 180) ends up being less than the starting cue ball vector of (6, 180). Basically, the cue ball vector jumps to the eight ball and a small amount of the motion energy is converted to heat energy.

And, yes, playing pool heats up the pool balls as they hit each other. My example exaggerates the amount of heat. In reality, it might be very small, like (0.0005, 180) converted to heat, but it is a real amount. if you hit two pool balls together enough times, fast enough, they will get hot.

The direction stayed at 180, because my story problem exists in a perfect world where everything is lined up perfectly. In the real world, the cue ball vector would have been something along the lines of (6, 175) and the hit would not be perfectly center to center. The resulting energy distribution would be less easy and convenient.

In the real world where I didn’t line my shot up perfectly, the vector would not be perfectly transferred to the eight ball, so the cue ball would keep some motion energy - maybe (1.5, 150) - and the eight ball would get less of the motion and would be bounced in a little different direction, maybe (4.495, 195)

I’m tired of writing out “cue ball” and “eight ball” so I’m going to abbreviate and say the cue ball vector is cbV and the eight ball vector is ebV. I’ll put an S for start and an E for end so we can know if our vectors are before or after the collision.

Starting cue ball vector + starting eight ball vector = ending cue ball vector + ending eight ball vector + ending heat vector

ScbV + SebV = EcbV + EebV + EhV

(6, 175) + (0, 0) = (1.5, 150) + (4.495, 195) + (0.005, 0).

Cue ball starting at 6 inches per second at 175 degrees, hits eight ball, which sits at 0 inches per second and 0 degrees. The result is cue ball going 1.5 inches per second at 150 degrees and the eight ball going 5.495 inches per second at 195 degrees and 0.005 inches per second in 0 direction converted from movement to heat.

The formula (f) for playing pool is ScbV + SebV = EcbV + EebV + EhV

I didn’t actually add the vectors nor calculate exactly how much energy is converted to heat, so don’t double check my math. You will be disappointed. It’s close enough, though, to be used as a good example.

By the way, you just did a story problem with vector math.

If the cue ball has spin on it, as all fancy pool players seem to manage, the spin becomes an additional dimension (or direction) in the cue ball force vector. No spin and the cue ball in motion has a two-dimensional vector. Add spin and it has a three-dimensional vector: cue ball starting at 6 inches per second at 175 degrees, with left spin (-180 degrees direction) of 10 rotations per minute.

Whoops. That’s actually four dimensions because the spin has both direction (-180 degrees) and velocity (10 RPM). The vector could be written out as: (6, 175, -180, 10). Each of those numbers in the vector is called either a component or a coordinate.

Each of the components is the value for an additional dimension of the vector. The pool ball itself is a 3-dimensional object. It has width, depth, and height. Some people would put time in as a fourth dimension as well.

In our example, we put a movement vector into the ball when we hit it: speed, direction, spin direction, and spin speed. If you want to describe the object, you can actually combine the vectors.

The cue ball is:

2.25” wide by 2.25” tall 2.25” deep.

It weighs 6 ounces.

The cue ball is 2.25” x 2.25” x 2.25”, 6 ounces, moving at 6 inches per second at 175 degrees, with left spin (-180 degrees direction) of 10 rotations per minute.

You can represent it in math as an eight-dimensional vector:

(2.25, 2.25, 2.25, 6, 6, 175, -180, 10)

The order of the components doesn’t really matter as long as everyone who is performing math with similar vectors puts the components in the same places. Like on a map, convention is to put the X dimension (X is horizontal) first and the Y dimension (X is vertical) second: (x, y). For a 3D graph, convention is to put the Z dimension (Z is depth) third: (x, y, z). It works no matter how you order the components as long as everyone does it the same.

You may have noted that I slipped weight (6 ounces) in the vector as a component:

Weight is the first “6” in bold: (2.25, 2.25, 2.25, 6, 6, 175, -180, 10)

Weight is a vector representing gravitational force on an object. Gravitational force, according to Einstein, is a bend in space, so weight is also a vector component. (For those saying: “but you should use mass not weight”, mass is the amount of something and weight is the gravitational force, so weight it is).

When playing pool, the initial shot tends to start with the cue ball in motion and the other balls stationary. Most or virtually all of the movement energy is transferred out of the cue ball to the target and then from it to any other balls it subsequently hits. The more directly centered the connection between the ball in motion and the stationary ball, the greater the percentage of energy that is transferred.

Bumper cars, like a pool ball, have height, width, depth, and weight. When in motion, they have speed, direction, and can spin, so they have spin direction and spin velocity. Unlike pool balls, bumper cars have a motor so both cars have a motion vector from the outset. That makes the motion transfer (and the vector math) far more complicated. Not only that, having a motor means they can accelerate. That adds acceleration as another component to the vector. We’ve just added another dimension!

That’s about all I have to say about bumper cars. The important message here is a way to think of more than three dimensions at the same time. We live in a 3D world (or 4 if you are a time-as-dimension advocate). But it is not uncommon to hear physicists talk of more than three or four dimensions in the science world. String theory, one of the attempts to explain the fundamental workings of the universe, can operate in 11-dimensions. I don’t know what those 11 dimensions are, but by taking a look at our pool balls and bumper cars helps me to better understand how we might end up with 11 dimensions, or even more.

Bits are simple pieces of electronic hardware that have one property (or behavior) that we use: electric charge. They are either holding an electric charge or are not holding an electric charge. When holding an electric charge, we say they have a value of 1. When not holding an electric charge, we say they have a value of 0. Each bit only holds that one piece of information: is it 0 or is it 1? We combine lots of these bits together to be able to hold and manipulate a lot of information, but each bit, as we use it, only holds a simple one-dimensional, scaler, number: (0) or (1).

Qubits are based on subatomic particles which have a number of properties and behaviors that we can use. As we have covered before, we call the main property “spin.” Spin is a three-dimensional vector, so it holds more information than the one-dimensional bit. We still have some time and words to go before we understand how we use the three-dimensional qubit vectors, but the small view into vector math that we did above is a start.

We did a bit of 2D vector math here with our cue ball and eight ball:

(6, 175) + (0, 0) = (1.5, 150) + (4.495, 195) + (0.005, 0)

But that was we before we looked at all eight dimensions. Not to worry. We didn’t do anything wrong. We just calculated with a part of the vector. That’s okay depending on what we are trying to understand. Qubit math will be a bit like that. There are a lot more properties to a subatomic particle than the few we will be using for qubit math, but we only use the ones we need.

Until next time…

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Positive Edge is the consulting arm of Duane Benson, Tech journalist, Futurist, Entrepreneur. Positive Edge is your conduit to decades of leading-edge technology development, management and communications expertise.