Chapter 03 / 20

Vectors, matrices and Hilbert spaces

The mathematical language of finite quantum systems.

1. Core idea

A qubit is a unit vector in a two-dimensional complex inner-product space. Linear operators map states to states; a unitary operator preserves inner products and therefore probability. Eigenvectors and projectors make measurement and dynamics precise.

2. Mathematical foundation

The notation describes a precise relationship among states, transformations or probabilities. Identify each symbol, check the dimensions of the objects, and distinguish amplitudes from the probabilities produced by a measurement.

3. Worked example

For U=diag(1,e^{iθ}), applying U to |+⟩ changes only a relative phase. A subsequent Hadamard converts that phase difference into observable probabilities.

4. Reasoning flow

Define the initial register and its basis states.
Apply the transformations in their intended order.
Track amplitude magnitudes and relative phases.
Measure in the basis that reveals the desired property.
Interpret the result and check the resource assumptions.

Hands-on activity

Use a two- or three-qubit circuit to model the stated example. The worked calculation above gives a reference against which to check the simulated output.

5. Check your understanding

Practice question

Prove that the Pauli X matrix satisfies X†X=I.

6. What to watch out for

Ideal-state calculations assume perfect gates and measurements. When assessing an algorithm, separate its mathematical guarantee from the cost of building oracles, preparing inputs, repeating experiments and correcting physical noise.

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