M1 — Complex Numbers, Vectors, Dirac

Slide 10

Anchor · retrieve · master

ARM drill

Three quick M0 questions. Three-in-a-row correct unlocks the module.

ARM drillstreak 0/3

0/0

The histogram in M0 showed peaks on |00⟩ and |11⟩. What made that happen?

Slide 11

The renaming table

reference

Every quantum notation on the right is a NumPy operation on the left. Screenshot this. Keep it open.

Math	NumPy
\|0⟩ (ket, column vector)	`np.array([[1], [0]])`
⟨0\| (bra, row vector)	`np.array([[1, 0]])`
⟨A\|B⟩ (inner product)	`np.dot(A_conj, B)`
\|A⟩⊗\|B⟩ (tensor product)	`np.kron(A, B)`
A† (dagger / conjugate transpose)	`np.conj(A).T`
‖ψ‖ = √⟨ψ\|ψ⟩ (norm)	`np.linalg.norm(psi)`

The intimidation of Dirac notation collapses when you see it’s just a coordinate-change of things you already know.

Slide 12

Inner product, live

predict + run

inner.py python

import numpy as np

A = np.array([[1], [4-5j], [5], [3]])       # ket |A⟩
B = np.array([[1, 5, -4j, -1j]])            # bra ⟨B|

inner = (B @ A).item()
print(inner)  # → (21-48j)

Will ⟨B|A⟩ come out real, imaginary, or complex?

predict

Slide 13

Tensor products, live

RRSS-solve

The tensor product is the operation that stacks qubits. Pick two states and watch ⊗ build the 4-vector.

Tensor product|A⟩ ⊗ |B⟩ ∈ ℂ⁴

|A⟩ (left)

[1.000, 0.000]

⊗

|B⟩ (right)

[0.000, 1.000]

If |A⟩ = [α, β] and |B⟩ = [γ, δ], what's the fourth entry of |A⟩⊗|B⟩?

solve

Slide 14

Why this matters

skillopt · scale

n qubits → 2ⁿ complex amplitudes. This is the whole game.

n	Dim	Bytes (complex64)	Fits in
1	2	16 B	anything
10	1 024	16 KB	L1 cache
20	1 M	16 MB	RAM easy
30	1 B	16 GB	RAM tight
40	1 T	16 TB	supercomputer
50	10¹⁵	16 PB	nothing

Classical simulators wall out around n = 40. That’s the floor of possible quantum advantage — and it’s why VQE and QAOA (the algorithms you’ll build in M9–M10) aim for 50–100 qubits, not a million.

Linear algebra fluency40 / 100

60 points to unlock next module.

Next module: Bloch geometry. You’ll see |ψ⟩ as a point, gates as rotations.