Quantum Mechanics Primer (for Computing)

Summary

Quantum computing is mostly linear algebra + probability, with a few quantum-specific rules. This note is the “minimum viable QM” you need for circuits, algorithms, and error correction.

Notes

States are vectors (kets)

A pure state is a normalized vector often written as a “ket” like |ψ⟩ in a complex Hilbert space.
For one qubit, a common basis is |0⟩, |1⟩.
For n qubits, the space dimension is 2^n.

Superposition is linear combination

You can write a state as

|ψ⟩ = Σ_x α_x |x⟩ with Σ_x |α_x|^2 = 1.

The α_x are amplitudes; probabilities come from squared magnitudes.

Global phase doesn’t matter (relative phase does)

|ψ⟩ and e^{iθ}|ψ⟩ represent the same physical state.
But phases between components (relative phase) affect interference.

Evolution (closed systems) is unitary

Quantum gates are represented by unitary matrices U where U† U = I.
Applying a gate updates the state: |ψ'⟩ = U|ψ⟩.

Measurement = probabilities + state update

Measuring in the computational basis |0⟩, |1⟩:

Outcome 0 occurs with probability |⟨0|ψ⟩|^2
Outcome 1 occurs with probability |⟨1|ψ⟩|^2

After measurement, the post-measurement state is the normalized projection onto the observed outcome’s subspace.

Composite systems: tensor products

Two qubits: |ψ⟩ ⊗ |φ⟩ (often written |ψ⟩|φ⟩).
Basis states: |00⟩, |01⟩, |10⟩, |11⟩.

Entanglement

A state is entangled if it cannot be written as a simple product state.

Example Bell state:

|Φ⁺⟩ = (|00⟩ + |11⟩) / √2

Measuring one qubit gives a correlated result in the other; importantly, this correlation can’t be reproduced by classical shared randomness for all measurement choices.

Interference

Because amplitudes add, paths can cancel.

If two computational paths lead to the same basis state with opposite phase, their amplitudes can cancel.
Most quantum algorithms are “interference engines”: design (U) so the amplitude mass concentrates on good outcomes.

Mixed states and density matrices (why you care)

Real devices interact with the environment → noise → you can’t assume a pure state.

A mixed state is represented by a density matrix ρ.
Pure state: ρ = |ψ⟩⟨ψ|
Noise and measurement can be expressed as quantum channels (CPTP maps).

You’ll revisit this for error correction and benchmarking.

No-cloning (practically important)

There is no universal operation that takes an unknown |ψ⟩ and outputs |ψ⟩|ψ⟩ for all |ψ⟩. This is why:

quantum “backup copies” don’t exist
many classical fault-tolerance intuitions don’t transfer directly

References

04-qubits-and-measurement.md (Bloch sphere + measurement intuition)