Qubits and quantum states

Why this matters

Every conversation about post-quantum cryptography eventually bottoms out at the question 'what can a quantum computer actually compute, and on what?' The answer starts with the qubit. A qubit is not a probabilistic bit — it is a unit vector in a two-dimensional complex Hilbert space, and that single change in the state space is what makes Shor's algorithm possible and RSA fragile. If you treat $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ as 'a coin that's 70% heads' you will arrive at every wrong intuition about quantum speedups. Get the linear-algebraic picture concrete now and the rest of this course — superposition, entanglement, Shor, lattice schemes that survive — will land cleanly. Choosing which cryptographic primitives survive quantum computing is fundamentally a question of which mathematical problems remain hard for this model of computation, and you cannot judge hardness against a model you do not understand.

Demo

A qubit is a unit vector in $\mathbb{C}^2$. The two computational basis states are $|0\rangle = \begin{pmatrix}1\\0\end{pmatrix}$ and $|1\rangle = \begin{pmatrix}0\\1\end{pmatrix}$. Any pure qubit state is a complex linear combination $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ with the normalisation constraint $|\alpha|^2 + |\beta|^2 = 1$. The numbers $\alpha, \beta \in \mathbb{C}$ are called probability amplitudes, not probabilities — they can be negative, complex, and interfere.

Try it yourself

1. Run the snippet and confirm $|\alpha|^2 + |\beta|^2 = 1$ and $\langle\psi|\psi\rangle = 1$. Both must equal 1 to one part in — the second is the actual definition of normalisation.

Prompt your AI

Use these three in order. Each builds on the one before.

1. Basics & terminology

In one paragraph, explain what a qubit is to someone who knows what a classical bit is. Be precise about what's in $\mathbb{C}^2$, what 'normalised' means, and why $\alpha, \beta$ are called amplitudes rather than probabilities.

2. Why it works (the mechanism)

Walk me step by step through how a single-qubit state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ encodes information. What does the column vector look like? What does the constraint $|\alpha|^2 + |\beta|^2 = 1$ geometrically mean? Why are amplitudes complex, not real?

3. Advanced — application & what's next

Given two single-qubit states $|\psi_1\rangle = \alpha|0\rangle + \beta|1\rangle$ and $|\psi_2\rangle = \alpha|0\rangle - \beta|1\rangle$, they are *not* the same state but produce identical statistics if I measure in the $\{|0\rangle, |1\rangle\}$ basis. Explain how a quantum algorithm like Shor's exploits this distinction — i.e. why classical probabilistic computers cannot simulate this efficiently.

References

• Nielsen & Chuang — Quantum Computation and Quantum Information (Ch. 1.2)
• Qiskit Textbook — The atoms of computation
• NIST Post-Quantum Cryptography overview