The Variational Principle

Every quantum algorithm that finds a ground state energy must start from one fundamental inequality — the variational principle — which guarantees that any state you prepare gives an upper bound on the true ground energy.

The variational bound

Let $H$ be a Hamiltonian with eigenstates $|E_k\rangle$ and eigenvalues $E_0 \leq E_1 \leq \cdots$ ordered from lowest to highest. For an arbitrary normalized trial state $|\psi\rangle$ the expectation value satisfies

\langle \psi | H | \psi \rangle \;\geq\; E_0.

Proof. Expand $|\psi\rangle = \sum_k c_k |E_k\rangle$ with $\sum_k |c_k|^2 = 1$ . Then

\langle \psi | H | \psi \rangle = \sum_k |c_k|^2 E_k \geq E_0 \sum_k |c_k|^2 = E_0.

Equality holds if and only if $|\psi\rangle = |E_0\rangle$ — the true ground state.

A concrete example: $H = Z$

The single-qubit Pauli- $Z$ operator has matrix

Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Its eigenvalues are $+1$ (eigenvector $|0\rangle$ ) and $-1$ (eigenvector $|1\rangle$ ), so the ground state energy is $E_0 = -1$ .

The variational principle guarantees that for any single-qubit state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ ,

\langle \psi | Z | \psi \rangle = |\alpha|^2 (1) + |\beta|^2 (-1) = |\alpha|^2 - |\beta|^2 \;\geq\; -1,

with equality only when $\alpha = 0$ , i.e.\ $|\psi\rangle = |1\rangle$ .

Energy landscape across trial states

E(\theta) = \langle \psi(\theta) | Z | \psi(\theta) \rangle = \cos^2(\theta/2) - \sin^2(\theta/2) = \cos\theta.

This function is minimized at $\theta = \pi$ , where $E(\pi) = -1 = E_0$ . VQE would discover this minimum by evaluating $E(\theta)$ on the quantum device and updating $\theta$ classically (a hybrid loop).

Why this matters for VQE

VQE replaces "solve $H|\psi\rangle = E_0|\psi\rangle$ analytically" with "prepare $|\psi(\theta)\rangle$ on a quantum processor, measure $\langle H \rangle$ , pass that number to a classical optimizer, and repeat." The variational principle is the guarantee that the optimizer's objective — the energy expectation value — is always an upper bound on $E_0$ , so the algorithm can only improve.

Try it

The Pauli- $Z$ Hamiltonian has eigenvalues $+1$ and $-1$ . The variational principle says $\langle \psi | Z | \psi \rangle \geq -1$ for all $|\psi\rangle$ . Return the expectation value of $Z$ in the state $|1\rangle$ to confirm that the bound is tight for the true ground state.

Run your code to see the quantum state.