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 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. 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: Logic gates in conventional computers and the equivalent in quantum computers.

We’ve talked about the smallest parts of computers (conventional and quantum): the transistor and qubit. Here, we’re going to look at how those two fundamental parts are turned into circuitry that can compute.

The MOSFET transistor is the fundamental building block of a conventional digital computer, and the Qubit holds the same function for a quantum computer. But transistors and qubits are just simple parts. They must be formed into electronic structures to perform work. We call those structures logic gates. Gates are then connected to form memory, adders, multipliers and everything needed to make a computer.

What do we mean by the term “gate”

You will see the term “gate” many times from here on. It refers to the set of basic building blocks of computer circuitry. We call them gates because, like a physical gate in the real world, gates separate the inside from the outside. On one side of the gate is the input and on the other side is the output. In a computer, we call them logic gates, or just gates for short, because each gate performs a logic function on the input and the result is delivered as output.

Regular every day math uses four base expressions: plus (+), minus (-), multiply (x or *), and divide (÷ or /). Regular math has more complex expressions, but they can all be boiled down to combinations of those four basic expressions.

Boolean logic math has three fundamental expressions called: NOT, AND, and OR. They are capitalized to differentiate them from human language words “not”, “and”, and “or.” Boolean logic also has more complex expressions that can be made up of combinations of the three basic expressions.

The NOT gate

“NOT” is a logic function that flips a bit value from 1 to 0 or 0 to 1. It’s called NOT because the output is NOT equal to the input. The NOT gate has one input and one output and the only choices for input or output are 0 and 1. Another word for flipping a bit value is “invert”, so we sometimes call the electronic circuit that makes up a NOT gate an inverter. It is the simplest of all digital logic circuits.

The basic inverter is the first computer circuit I ever built from scratch. It was an exercise back in my high school electronics class. The circuit consists of a transistor and two resistors. You put a 1 into it and a 0 comes out. If you put a 0 into it, a 1 comes out.

Why do we need a NOT gate, you might ask? Well, for starters, when counting in binary (000, 001, 010, 011, 100…) the digits frequently go from 0 to 1 and from 1 to 0. They are inverted. An inverter is a part of the circuit used for counting. Without the inverter, computers cannot work.

Quantum Inverter

One of the features of quantum entanglement is similar to an inverter (a NOT gate). When one entangled qubit collapses out of superposition to a distinct value, the entangled qubit ends up with the inverse value. This means that entanglement can be used in a role similar to that of the digital electronic NOT gate.

Other Gates

Other logic gates take more than one input and combine the inputs to create an output. An AND gate is another type of logic gate. It takes two inputs, referred to as A and B, and produces one output, referred to a Q (why Q? I don’t know). The Boolean algebra formula is A x B. If A is 1 and B is 1, the output is 1. If either A or B is 0, the output, Q, is zero

If A = 0 AND B = 0, Q = 0 (0 × 0 = 0)

If A = 0 AND B = 1, Q = 0 (0 × 1 = 0)

If A = 1 AND B = 0, Q = 0 (1 × 0 = 0)

If A = 1 AND B = 1, Q = 1 (1 × 1 = 1)

The output is 1 only if both A and B are 1. Hence the label “AND” gate.

The OR Gate

The OR gate has the Boolean formula A + B. The OR gate:

If A = 0 OR B = 0, Q = 0 (0 + 0 = 0)

If A = 0 OR B = 1, Q = 1 (0 + 1 = 1)

If A = 1 OR B = 0, Q = 1 (1 + 0 = 1)

If A = 1 OR B = 1, Q = 1 (1 + 1 = 1)

Computer texts use small tables, called “truth tables” to show what inputs to the gate give as outputs. The three truth tables below give the same information as I have written above in paragraph form.

The NOT gate only has one input and one output, so there is only A and Q. The AND and OR have two inputs and one output, so there is A, B, and Q. More complex logic functions built up out of these three can also be shown as truth tables. Something with three inputs and one output would have A, B, C, and Q. The small image above the name of the gate is the schematic diagram symbol, which is used when drawing out a more complex circuit using these gates.

The most common gates that are compounds of these three basics are the NAND, NOR, and XOR, as shown in the truth tables below.

A NAND gate is an AND gate followed by a NOT and a NOR is an OR followed by a NOT. You can see that each output is the inverse of the AND and OR respectively. The XOR, or exclusive OR is a variation on the OR such that you only get a Q = 1 when only one or the other input is 1, not both. The original OR is kind of an OR/AND. That is needed for a lot of digital circuits, but there are times when the “either/or, but not both” functionality is needed. There is also an exclusive NOR, or XNOR, which inverts the Q values to 1, 0, 0, 1.

Figure 4 shows an example of an adder circuit built up out of basic digital logic gates. The circuit adds two 4-bit binary numbers together, delivering the results on the outputs (Q₀ - Q₃) on the right. This adder circuit includes a carry bit (Q₄) which would connect to additional adders to allow adding larger numbers. Notice that while most of the gates are two input gates, there are also three, four and five-input gates.

Quantum computers also use a set of basic gates. The quantum gates do not exactly match up to the logic gates used in a conventional computer. As a qubit differs from a bit, quantum logic gates differ from conventional logic gates.

Pauli-X Gate (Closest to a NOT gate)

The rough equivalent to a NOT gate in the quantum world is called a Pauli-X gate and is comprised of a single qubit. The Pauli-X reverses the spin of the qubit. It is a single qubit gate - one line in and one line out. The digital NOT gate gets its name for its Boolean logic function. Pauli-X gets its name from the physicist Wolfgang Pauli and the effect the gate has in the more complex quantum world.

Controlled NOT gate (Closet to an XOR gate)

If you entangle a qubit with the qubit in a Pauli-X gate, you get a different gate called a Controlled NOT gate, or CNOT.

The | and ⟩ symbols are part of what’s called bra-ket notation. I described it a bit back in newsletter issue 15. In quick review, the | is called a bra and the ⟩ is a ket. The pronunciation is like “bracket” - the thing you put on a wall to hang a shelf. Braket notation is used to indicate a qubit in superposition. Known results values are written as |0⟩ and |1⟩ while unknown values may be shown as letters, as I have in figure 5, to denote a variable.

The symbol |a⟩ in figure 5 represents a qubit in superposition without a known value. I’m calling it qubit a, hence the “a” surrounded by the line and braket. The symbol |b⟩ represents a second qubit in superposition, called qubit “b”. The vertical line between the dot on top and circle with a cross on the bottom shows that the two qubits are entangled. The two horizontal lines represent more of a timeline than a connection. It’s still the same qubit on either side of the horizontal line, but the state may or may not have changed. As drawn, figure 5 shows a CNOT quantum gate.

The CNOT is closest in equivalence to the digital XOR gate. One qubit is the control qubit (solid black dot, figure 5, upper) and the other is a data qubit (circle with cross, figure 5, lower). Actions on |a⟩ have an effect on |b⟩.

Quantum gates are more difficult to represent than the relatively simple digital Boolean logic truth tables. Since qubits are a superposition of both values, they are represented as a matrix of values instead of a single (scalar) value. The symbols in the middle on figure 6 are the schematic diagram signals for the Pauli-X (top) and CNOT (bottom) gates. These are used when drawing out larger quantum circuits.

I will go into more detail explaining the matrixes and the operation of these two gates next time.

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

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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.