Measuring Dynamical Quantities

Evolving a state is only half the job — the physics lives in the observables you measure afterward. This lesson shows how to extract expectation values like $\langle Z\rangle$ and correlation functions from a circuit, using nothing but computational-basis measurement statistics.

Expectation values from the $Z$ -basis

The simulator measures in the computational ( $Z$ ) basis, returning the probabilities $p_x$ of each bitstring $x$ . For a single qubit the magnetisation $\langle Z\rangle$ follows directly from the two outcome probabilities, because $Z$ has eigenvalue $+1$ on $|0\rangle$ and $-1$ on $|1\rangle$ :

\langle Z\rangle = (+1)\,p_0 + (-1)\,p_1 = p_0 - p_1.

No extra gates are needed: any observable diagonal in the computational basis is just a weighted sum of the measured probabilities, with weights equal to its eigenvalues.

Measuring off-diagonal observables

What about $\langle X\rangle$ or $\langle Y\rangle$ ? They are not diagonal in the $Z$ -basis, so we rotate the measurement basis first:

For $\langle X\rangle$ : apply $H$ before measuring (since $H Z H = X$ ), then read $p_0 - p_1$ .
For $\langle Y\rangle$ : apply $S^\dagger$ then $H$ before measuring, then read $p_0 - p_1$ .

The general recipe: to measure a Pauli $P$ , conjugate by the single-qubit gate that maps $Z \to P$ , measure in the $Z$ -basis, and combine with the $\pm 1$ eigenvalue weights.

Correlation functions

A two-point correlation function such as $\langle Z_i Z_j\rangle$ measures how the spins at sites $i$ and $j$ are statistically linked. Since $Z_i Z_j$ is diagonal, it is again a sum over basis probabilities weighted by the parity $(-1)^{x_i + x_j}$ of the two relevant bits:

\langle Z_i Z_j\rangle = \sum_x (-1)^{x_i \oplus x_j}\, p_x.

Tracking $\langle Z_i Z_j\rangle$ as a function of evolution time gives a dynamical correlation function, the central object in studies of transport, thermalisation, and information spreading (light cones) in quantum many-body systems.

A clean analytic check

Prepare a single qubit with a $Y$ -rotation $R_Y(\theta)$ applied to $|0\rangle$ :

R_Y(\theta)|0\rangle = \cos\tfrac{\theta}{2}\,|0\rangle + \sin\tfrac{\theta}{2}\,|1\rangle.

The outcome probabilities are $p_0 = \cos^2\tfrac{\theta}{2}$ and $p_1 = \sin^2\tfrac{\theta}{2}$ , so

\langle Z\rangle = p_0 - p_1 = \cos^2\tfrac{\theta}{2} - \sin^2\tfrac{\theta}{2} = \cos\theta.

For $\theta = \pi/3$ this gives $\langle Z\rangle = \cos(\pi/3) = \tfrac12$ . The exercise checks exactly this: that the measured $p_0 - p_1$ reproduces the analytic $\cos\theta$ .

Try it

Prepare the state $R_Y(\pi/3)|0\rangle$ and return the circuit. The grader reads the measured probabilities and checks that $\langle Z\rangle = p_0 - p_1$ equals $\cos(\pi/3) = \tfrac12$ .

Run your code to see the quantum state.