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 and what those terms actually mean.

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: How a quantum dot qubit may be constructed and bringing in imaginary numbers

Before diving into imaginary numbers, I’m taking a brief side trip into a specific type of qubit. Part 1 of this newsletter covers the construction of one type of qubit that shows promise for more stable quantum computing. Part 2 jumps into imaginary numbers which are important for doing math with qubits. Two different but related subjects today.

One of the many forms of qubits is called a quantum dot spin qubit. It uses a small stack of atoms (the dot) and electric fields to control the spin of a single electron within the stack of atoms in the dot.

In a few years, quantum dot spin qubits may be the dominant form of qubit. The jury is still far from rendering a verdict, but they have been shown to possibly have significant advantages in fabrication and coherence. They can be built using mostly standard conventional computer fabrication equipment and techniques while most other qubit forms require exotic processes and materials.

Almost all integrated circuit (ICs) powering our technological world today are built using silicon and a process call MOS, or CMOS. MOS stands for metal oxide semiconductor, and CMOS stands for complementary metal oxide semiconductor. A metal or material with the conducting properties of metal is layered on an insulating layer of silicon-dioxide (SiO₂). Silicon, as a semiconductor, is also layered on to form the active switching portion of the circuit. Silicon metal oxide semiconductor (SiMOS) quantum dot qubits can be built by using the same processing equipment and materials.

The “C” for “complementary” indicates the use of two different types of transistors in a pair. It is related to the circuit design architecture, not the construction architecture. Our quantum dot qubits don’t use the “C” circuit architecture. I just included the explanation because you might see “CMOS” in many different technical articles, but not often see just “MOS.”

Just as the quantum dot spin qubit is one of many possible types of qubits, the illustration, figure 1, depicts one of several ways to make SiMOS quantum dot qubits. The light grey area is a silicon substrate with a darker layer of SiO₂ (the oxide) insulator on top. A series of gates* are fabricated on top of the SiO₂ using a conductor (the metal) to create the qubits, sense the condition, and conduct operations.

The three arrows in figure 1 indicated by “A” point to the locations of quantum dot qubits. One is between the SET gate and the SLB/SRB layers, one between P1 and LCB and a third between P2 and LCB. Signals are sent from the supervising computer to the SET, P1 and P2 gates to control, write and read the three qubits. The signals come in the form of small microwave packets (photons) of specific frequency and amplitude.

Now, back to the original subject at hand - imaginary numbers.

The science of Quantum mechanics, at the foundation of quantum computing, is weird. That statement should come as no surprise if you’ve been reading this newsletter or studying the quantum universe on your own. One of the ways we make quantum mechanics less weird is through the use of imaginary numbers. That last sentence alone is pretty weird.

An imaginary number is the square root of a negative number. Before being taught about imaginary numbers, I was told that the square root of a negative number is impossible. In some respects it is, but as a concept in the language of math, it is possible and very useful in calculating quantum things.

A fair number of math things are sort of impossible. Like a simple negative number. You can’t have negative apples. How would that work? I have one apple and I give it to you. Then I have no apples. I can’t give you another apple to have a negative amount of them. It is nonsensical. But in life, we deal with negative numbers frequently.

As a descriptive construct negative numbers can and do have logical meaning. We can use them to describe a physical transaction. It can mean that I promised you one more apple. You could then say that I have negative one (-1) apple. If I get another apple, I immediately give it to you, my negative one apple canceled out my positive one apple and I have no apples.

-1 + 1 = 0

We couldn’t really do that apple math without negative numbers.

Imaginary numbers work something like that. In many respects, imaginary numbers seem like nonsense. But as a descriptive construct, they can make math easier. As with negative numbers, there are some things we can do or describe easily only because of imaginary numbers.

A square root of a number is a second number that, when multiplied by itself, equals the first number. For example, the square root of nine is three. Three times three equals nine.

When you multiply a number by itself, it is called squaring the number. When you say “number times number”, an equivalent but shorter phrase is “number squared”, so saying “three times three” is the same as saying “three squared.”

The square root of 1,764 is 42, thus 42 squared = 1,764. Square roots don’t need to be whole numbers. The square root of two, as we discussed in the prior newsletter issue equals an irrational number that is approximately 1.4142135. 1.4142135 squared equals 1.9999998235. It works with fractions also. ¾ squared equals 9/16, thus the square root of 9/16 equals ¾.

Starting to get weird.

If you square a negative number, like minus five (-5), you multiply minus five times minus five. The minus signs cancel each other out and you get positive 25. The square root of 25 equals five. So, now we’ve run into a limitation in the language of math. The square of five equals 25. The square of minus five equals 25. The square root of 25 just equals five, never minus five. But minus five does not equal five.

Using some basic algebra and

5 × 5 = 25

-5 x -5 = 25

Therefore

5 × 5 = -5 x -5

A little more algebra gets you:

5 × 5 = 5 x -1 × 5 x -1

Since -1 x -1 = 1, and one times anything is just the same number, the negative ones can be taken out, and the equation ends up as:

5 × 5 = 5 × 5

or

5 × 5 = 25

In the language of math, you can illustrate 5i (the square root of negative 25). Figure 2 below shows the math steps to get from the square root of negative 25 to 5i.

Qubits live in a strange world that defies clear explanation. Imaginary numbers help add some clarity in the math of quantum mechanics.

Remember the Bloch sphere from last newsletter issue? It represents a qubit’s value in the form of vectors. If you have a qubit with a value, and you multiply it by i, the vector is rotated by 90 degrees in the sphere. There is no real number that can directly rotate a qubit vector. In a manner similar to the way having negative apples can explain a fruit transaction, imaginary numbers can help explain a quantum transaction.

Conventional computer bits can be inverted - zero gets inverted to one or one gets inverted to zero. They only have one dimension, so the only fundamental operation that is needed to change a bit is to invert it.

Qubits, on the other hand, are multidimensional so they need more than one fundamental operation. Inverting is just one of several fundamental operations than can affect a qubit value. Multiplying the qubit by i is another fundamental qubit operation.

That will become quite important as we dive deeper into quantum gates and how math gets done in a quantum computer.

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.