Quantum computing is one of the first great technologies of the 21st century, and it will change everything someday.

Welcome to the Quantum Edge newsletter. Here you will learn more than just: “quantum computing works because of superposition and entanglement.” The Quantum Edge newsletter will tell you what goes with superposition and entanglement, what those terms actually mean, and how they lead to quantum computing.

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: Single qubit gates and a special* number - the square root of two

The language of math has a lot of special numbers. Probably the best-known special number is pi (π). It equals about 3.141592653589793. It doesn’t end there with 16 digits. It keeps going in a never ending, never repeating list. We call that an irrational number. Any number that continues infinitely without repeating is categorized as an irrational number.

Pi is easy to calculate but not easy to calculate accurately. The formula is π = circumference of a circle divided by the diameter of a circle. The more accurately you can measure the circumference and diameter, the more accurately you can calculate it. There are also algorithms that math and computer folks use to calculate pi.

Most of us never overtly use pi (or any other irrational** numbers, for that matter) in our daily lives. However, if you have ever driven a car, baked a cake, taken an airline flight, used a cell phone, or taken advantage of just about any other modern convenience, you have made use of pi and other irrational numbers. But that’s not what I really want to talk about.

What I want to talk about, specifically, is the square root of 2, also called Pythagoras’ constant. It equals about 1.414213562373095, and like pi, it keeps going and never repeats. It equals the diagonal between corners in a square where each side is 1. It is the first number ever proven to be an irrational number (and the number someone was allegedly killed over).

The Pythagorean theorem can be used to calculate that long side (called the hypotenuse) of a right triangle (a right triangle has a square, or 90-degree corner). If side (a) (figure 1, right) were equal to 3 inches and side (b) equal to 4 inches, side c can be calculated with the theorem: 3² + 4² = 9 + 16 = 25 = 5². Side (c) = 5. In the case of a 1-inch by 1-inch right triangle, 1² + 1² = 1 + 1 = 2 = 1.414235…². The long side is equal to 1.414235…

The Bloch sphere, our graphic representation of a cubit in superposition holds one and zero at the same time. It holds the probability of both states. When it is examined and the wave function collapses, it ends up just being one or zero, but while in superposition in a quantum computer, it holds both values.

The square root of 2 is used to calculate the states and probabilities within a qubit. Figure 2 left, below, shows a Bloch sphere with the axes values labeled. Figure 2 right shows how the square root of two comes about within the sphere.

We can call the radius of the qubit equal to one. It’s not one inch. Not one millimeter. Not even a really small unit. One is more of an analogy than an accurate value, but it works to call it equal to one, so that’s what we do. The vertical axis, (Z) is equal to one from the center to the top and the horizontal axis, (Y), is equal to one going from the center to the side. The second horizontal axis (X) is also equal to one going from the center to the edge.

That means the diagonal cut from the outside point of one axis to the outside point of another axis is equal to the square root of 2. In a later issue, I’ll explain in more detail why that is important, but for now, just understand that this is where the weird (irrational) number comes from.

Quantum gates, as we have mentioned, are used to manipulate quantum states and ultimately perform calculations with qubits. If you need to go back and get a quick refresher on logic gates, look at newsletter issue 25. I’ll wait.

I mentioned in that issue that a Pauli-X gate is the closest to a conventional NOT gate. It takes a qubit in one of the states |0⟩ and |1⟩ and flips it to the other. It rotates the phase of the qubit around the X axis. The qubit, as represented in the Bloch sphere in figure 2, has three axes: X, Y and Z. Therefore, there are three Pauli gates: Pauli-X, Pauli-Y and Pauli-Z. Each rotates the qubit around its respective axis.

A conventional NOT gate switches the value of a bit from one to zero or from zero to one. It’s pretty simple because the bit holds a simple scalar number one or nothing (equaling zero). Qubits, though, hold vectors. A simple qubit inversion is not so simple. As a vector, a qubit can be manipulated in three dimensions.

The conventional NOT gate is shown in a truth table in figure 4, left. (A) is the input to the NOT gate and (Q) is the output. Logic operations flow from the input to the output in conventional computer logic gates. A zero in becomes a one at the output and a one in becomes a zero at the output.

The quantum Pauli-X gate representation looks similar but there are a few key differences. It shows the same numbers but in the form of a matrix instead of a truth table. Basically, it does the same thing as a NOT gate. It swaps the values. But it swaps multi-dimensional vector values while a conventional NOT gate swaps one-dimensional scalar values.

The quantum gates can also operate in reverse. The gate can be commanded to operate on the qubit, swapping the values. The gate can also be commanded to undo the operation: put back the way it was. You can’t do that with a conventional computer gate.

The ability of quantum gates to operate in both directions is called “unitary operation.” Quantum gates are unitary devices that can perform an operation and then un-perform an operation. It’s kind of like having a built-in undo capability within the quantum logic. Your conventional computer word processor has an undo function, but that’s different. The computer stores in memory what you do as you do it so it can go back through the list if you ask for an undo. Quantum computer logic doesn’t need that saving-a-list capability because it inherently knows what the original value was.

Qubits, when in superposition, hold both one and zero as values. Quantum gates, when in operation, hold both the question and the answer even after the question has been answered. The caveat is that once the answer is read by conventional digital systems, superposition collapses and neither the question nor the answer are in the qubits anymore. Up until the result is read and recorded, a quantum computer can calculate backward and forward repeatedly.

That’s all for today.

New to the Quantum Edge newsletter?

Thinking about re-reading it but want a more transportable format?

I’ve wrapped the first ten issues of The Quantum Edge newsletter into book form. The collection, called “The Quantum Computing Anthology, Volume 1”, is now available in Kindle and paperback on Amazon. The book collects newsletter issues 1 through 10 and has some additional material and edits for continuity and clarity.

Coming soon: Volume 2, collecting newsletter issues 11 though 20 is in the works. Look for it on Amazon soon.

In the meantime, you can order the Volume 1 Kindle or paperback editions on Amazon today: The Quantum Computing Anthology, Volume 1

Check your email box Thursday - probably. (Okay, some of these weekly issues have come out on Friday, or not at all. But, in a quantum world, how can you tell?)

If you received this newsletter as a forward and wish to subscribe yourself, you can do so at quantumedge.today/subscribe.

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.