Capstok — learn by doing

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.

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad \alpha,\beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1, \quad |0\rangle = \begin{pmatrix}1\\0\end{pmatrix},\ |1\rangle = \begin{pmatrix}0\\1\end{pmatrix}

Try it yourself

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.

References

// main.go
// go run main.go
package main

import (
	"fmt"
	"math"
	"math/cmplx"
)

func main() {
	ket0 := [2]complex128{1, 0}
	ket1 := [2]complex128{0, 1}

	// A normalised superposition: |psi> = (|0> + i|1>) / sqrt(2)
	alpha := complex(1/math.Sqrt(2), 0)
	beta := complex(0, 1/math.Sqrt(2))
	psi := [2]complex128{
		alpha*ket0[0] + beta*ket1[0],
		alpha*ket0[1] + beta*ket1[1],
	}

	fmt.Printf("|psi> = [%v %v]\n", psi[0], psi[1])
	fmt.Printf("|alpha|^2 + |beta|^2 = %g\n", cmplx.Abs(alpha)*cmplx.Abs(alpha)+cmplx.Abs(beta)*cmplx.Abs(beta))
	inner := psi[0]*cmplx.Conj(psi[0]) + psi[1]*cmplx.Conj(psi[1])
	fmt.Printf("inner product <psi|psi> = %g\n", real(inner))
}

Run: go run main.go

What a Qubit Really Is

Why this matters

Demo

Try it yourself

References