Paper Napkin Notes on Quantum Computing

How bits translate to qubits to open a universe of possibilities for computer scientists

I started my masters in computer science earlier this year. Quantum computing is one of the subjects on the pathway course for admission. Though I love the topic and was excited about the implications of the subject, I must confess that this isn’t all intuitive to me.

So, I decided to write about it. That usually helps me find out what I understand about something. Or if I really understand it.

The concepts start simple enough.

Classical bit

\(\begin{aligned} |0\rangle \;\xrightarrow{\text{voltage} < 1.8\text{V}}\; &\text{physical state A} \\[0.4em] |1\rangle \;\xrightarrow{\text{voltage} > 3.2\text{V}}\; &\text{physical state B} \end{aligned}\)

Classical computer bits are always one or the other. Never both.

Qubit

\(\begin{aligned} |0\rangle \;\xrightarrow{\text{electron spin up}}\; &\text{physical state A} \\[0.4em] |1\rangle \;\xrightarrow{\text{electron spin down}}\; &\text{physical state B} \end{aligned}\)

Now this is where things start to get interesting.

1. Superposition of states

A qubit can be in both states at once, enabling a computation technique known as quantum parallelism.

Unlike a classical bit, which holds a single value at any moment, a qubit is in both states 0 and 1 simultaneously.

\(|\text{state}\rangle = \alpha|0\rangle + \beta|1\rangle\)

Alpha and beta are complex numbers. They represent probability amplitudes. When you measure it, the system picks one and collapses to its value, 0 or 1. You get 0 with probability ∣α∣² and you get 1 with probability ∣β∣².

Superposition is the first concept where intuition starts to fail. We have no everyday experience of something being two mutually exclusive things at once.

Heard about Schrödinger’s cat? Yep, that’s a thought experiment that mocks the idea of scaling superposition up to everyday objects, it does help some of us grasp the concept, but be careful. People get excited about theoretical possibilities based on quantum mechanics principles, which is a dangerous and misleading route.

2. Measurement and collapse

Measurement collapses the state. A qubit in superposition holds a blend of 0 and 1, but the instant you measure it, it snaps to a definite 0 or 1 and the superposition is gone.

Why, you ask? Not intuitive, but here’s the analogy I like: measuring means interacting with it, and in the quantum world you can’t interact without leaving a mark.

In the classical world observing feels harmless: you glance at a coffee cup and nothing happens to the cup. At the quantum scale there’s no free glance. To “see” a particle, something has to physically interact with it, a photon bouncing off it, for example, and that interaction is enough to force the qubit to pick a side. (more here)

Careful though: the photon-bouncing picture is a useful analogy, not the whole story. It’s not that measurement is clumsy and “bumps” the answer. Any interaction that reveals which state the qubit is in counts as a measurement, whether or not a person is watching. That leads people to derive the wrong philosophical conclusions from measurement. Physicists call the fuller mechanism decoherence, but the touching analogy will get you 90% of the way.

3. Entanglement

Two qubits become correlated. How that happens would need another article, maybe more. But once they are, measuring one instantly tells you the state of the other, no matter the distance between them.

Careful here, because this is where people go off the rails: nothing is actually sent from one qubit to the other. There’s no faster-than-light message. The outcomes are correlated, but you only see the correlation once you compare the two results over an ordinary channel, phone, email, a cable, all of it slower than light. Einstein called it “spooky action at a distance,” and the spookiness is real. The signaling is not.

What entanglement does buy you is scale. The state of N qubits takes 2ⁿ numbers to describe, and that exponential is where quantum computing’s power hides. Just don’t read it as 2ⁿ of free storage: when you measure N qubits, you get back N bits, not 2ⁿ.

All of this is still conceptually graspable. You can follow the implications.

The wall appears when you need to understand what entanglement means, not just what it enables. How do unitary matrices preserve probability? Why must ∣α∣²+ ∣β∣² equal 1? How does the Hadamard gate actually transform a state?

Well, let’s leave it as a cliffhanger for my next article:

\(\begin{aligned} H|0\rangle &= \tfrac{1}{\sqrt{2}}|0\rangle + \tfrac{1}{\sqrt{2}}|1\rangle \\[0.6em] H|1\rangle &= \tfrac{1}{\sqrt{2}}|0\rangle - \tfrac{1}{\sqrt{2}}|1\rangle \end{aligned}\)

These two lines show how it does that: take a definite 0 or 1 and spread it into an even superposition. Gates like this are all unitary, and that property is what keeps quantum algorithms valid, which is exactly where I'll pick up next time.

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This is the real value of writing about what you learn. Not teaching others. Not building an audience. Finding the parts you thought you understood but did not.

If no one read this, at least I got a better understanding of the basics of quantum computing out of this exercise.