Pauli Expectation Values

Pauli Expectation Values

The expectation value of an observable $H$ in quantum state $|\psi\rangle$ is:

$\langle H \rangle = \langle\psi|\, H\, |\psi\rangle$

This is the average outcome of measuring $H$ many times when the system is in state $|\psi\rangle$. Expectation values are the fundamental output of variational quantum algorithms such as VQE and QAOA.

The Pauli Matrices

Every single-qubit Hermitian observable can be written as a linear combination of the three Pauli matrices and the identity:

$X = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}, \qquad Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}, \qquad Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$

Each has eigenvalues $\pm 1$.

Computing Expectation Values

For a qubit state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, write out the matrix products:

Z expectation:

$\langle Z \rangle = \langle\psi|Z|\psi\rangle = |\alpha|^2 - |\beta|^2$

This is simply the probability of measuring $|0\rangle$ minus the probability of measuring $|1\rangle$.

X expectation:

$\langle X \rangle = \langle\psi|X|\psi\rangle = \alpha^* \beta + \beta^* \alpha = 2\,\text{Re}(\alpha^*\beta)$

Y expectation:

$\langle Y \rangle = \langle\psi|Y|\psi\rangle = i(\beta^*\alpha - \alpha^*\beta) = 2\,\text{Im}(\alpha^*\beta)$

Special States

These correspond exactly to the six face-centres of the Bloch sphere: states aligned with the positive and negative $X$, $Y$, and $Z$ axes have expectation value $+1$ or $-1$ along their axis and $0$ along the others.

Bloch Sphere Connection

A pure state on the Bloch sphere at polar angle $\theta$ and azimuthal angle $\phi$ is:

$|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle$

Its Pauli expectation values give the Cartesian coordinates of the Bloch vector:

$\langle X \rangle = \sin\theta\cos\phi,\qquad \langle Y \rangle = \sin\theta\sin\phi,\qquad \langle Z \rangle = \cos\theta$

Your Task

Implement pauli_expectation(state, pauli) where state is a 2-element list $[\alpha, \beta]$ (possibly complex) and pauli is 'X', 'Y', or 'Z'. Return the real-valued expectation value.