Measuring Energy Expectation

The Variational Quantum Eigensolver (VQE) finds the ground-state energy of a molecule by minimizing $\langle \psi(\theta) | H | \psi(\theta) \rangle$ over circuit parameters $\theta$. On real quantum hardware you cannot read off the full state vector; instead you must measure each Pauli term separately and add up the contributions.

Decomposing the Hamiltonian

Any Hermitian operator on $n$ qubits can be written as a weighted sum of Pauli tensor-products:

$H = \sum_k c_k \, P_k, \qquad P_k \in \{I, X, Y, Z\}^{\otimes n}.$

By linearity,

$\langle H \rangle = \sum_k c_k \langle P_k \rangle.$

This means you only need to measure $\langle P_k \rangle$ for each Pauli string and then form the weighted sum classically.

Measuring a single Pauli term

Z terms are the easiest: measure in the computational basis and compute

$\langle Z \rangle = P(0) - P(1).$

For a qubit in state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ this gives $|\alpha|^2 - |\beta|^2$, which is exactly the diagonal matrix element $\langle \psi | Z | \psi \rangle$.

X and Y terms require a basis rotation before measurement. To measure $\langle X \rangle$ apply a Hadamard $H$ to rotate $|+\rangle \to |0\rangle$ and $|-\rangle \to |1\rangle$, then measure in the $Z$ basis. To measure $\langle Y \rangle$ apply $S^\dagger H$ first.

Multi-qubit Pauli strings

For a product operator like $Z_0 \otimes Z_1$ (often written $ZZ$) the expectation value factors only when the state is a product state. In general,

$\langle Z_0 Z_1 \rangle = P(00) - P(01) - P(10) + P(11),$

where $P(xy)$ is the probability of measuring the bit string $xy$. The sign rule follows from the fact that $Z|0\rangle = +|0\rangle$ and $Z|1\rangle = -|1\rangle$: the contribution of a bit string is the product of the signs of each individual bit under its Pauli.

Putting it together in VQE

A single VQE energy estimate proceeds as follows:

1. Prepare $|\psi(\theta)\rangle$ with the parametrized ansatz circuit.
2. For each Pauli term $P_k$: rotate the state to the measurement basis for $P_k$, measure, collect shot statistics, compute $\langle P_k \rangle$.
3. Return $E(\theta) = \sum_k c_k \langle P_k \rangle$ to the classical optimizer.
4. The optimizer proposes a new $\theta$ and the loop repeats until convergence.

On a simulator the expectation values can be computed exactly from the statevector; on hardware you average over many shots, so statistical noise limits precision.

Try it

For the product state $|{+}\rangle \otimes |0\rangle$ (qubit 0 in $|{+}\rangle$, qubit 1 in $|0\rangle$), what is the expectation value $\langle Z_1 \rangle$ of the Pauli-Z operator acting on qubit 1? Compute the exact value and return it.

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