Project: Mini-VQE

The Variational Quantum Eigensolver (VQE) is a hybrid algorithm that finds the lowest eigenvalue of a Hamiltonian $H$ — its ground-state energy $E_0$ — using a classical optimization loop wrapped around a parameterized quantum circuit.

The variational principle

The foundation of VQE is the quantum mechanical variational principle: for any normalized state $|\psi\rangle$ ,

$E_0 \;\leq\; \langle\psi|H|\psi\rangle,$

with equality only when $|\psi\rangle$ is the true ground state. This means that if we can minimize $\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$ over the parameter vector $\boldsymbol{\theta}$ , the lowest value we reach is an upper bound on $E_0$ — and, with a rich enough ansatz, it equals $E_0$ exactly.

The VQE loop

Choose a Hamiltonian $H$ and a parameterized ansatz circuit $U(\boldsymbol{\theta})$ .
Prepare $|\psi(\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})|0\rangle^{\otimes n}$ .
Measure the energy $E(\boldsymbol{\theta}) = \langle\psi|H|\psi\rangle$ by decomposing $H$ into a sum of Pauli strings and measuring each term.
Update $\boldsymbol{\theta}$ using a classical optimizer (gradient descent, parameter-shift gradients, COBYLA, etc.).
Repeat steps 2–4 until convergence.

Hamiltonian: $Z \otimes Z$

For this project we work with the two-qubit Hamiltonian

$H = Z_0 \otimes Z_1.$

Its eigenvalues come from the product of the $Z$ eigenvalues $\pm 1$ . Explicitly:

| State | $\langle Z_0 \rangle$ | $\langle Z_1 \rangle$ | $\langle Z_0 Z_1 \rangle$ | |-------|-------|-------|-------| | $\|00\rangle$ | $+1$ | $+1$ | $+1$ | | $\|01\rangle$ | $+1$ | $-1$ | $-1$ | | $\|10\rangle$ | $-1$ | $+1$ | $-1$ | | $\|11\rangle$ | $-1$ | $-1$ | $+1$ |

The ground-state energy is $E_0 = -1$ , achieved by $|01\rangle$ or $|10\rangle$ .

Ansatz and optimal angles

We use the hardware-efficient ansatz with independent $R_y$ rotations on each qubit:

$|\psi(\theta_0, \theta_1)\rangle = R_y(\theta_0)|0\rangle \otimes R_y(\theta_1)|0\rangle$

where $R_y(\theta) = e^{-i\theta Y/2}$ . Since the two qubits are unentangled, the expectation value factorizes:

$\langle Z_0 Z_1 \rangle = \langle Z_0 \rangle \cdot \langle Z_1 \rangle = \cos\theta_0 \cdot \cos\theta_1.$

To minimize $\cos\theta_0 \cos\theta_1$ we want one factor to be $+1$ and the other $-1$ . The choice $\theta_0 = 0,\; \theta_1 = \pi$ gives

$\cos(0) \cdot \cos(\pi) = 1 \cdot (-1) = -1 = E_0.$

Try it

The starter code provides the optimal angles from a completed VQE run ( $\theta_0 = 0$ , $\theta_1 = \pi$ ). Apply the ansatz $R_y(\theta_0)$ on qubit 0 and $R_y(\theta_1)$ on qubit 1. The grader checks the resulting statevector, which should be exactly $|01\rangle$ — the ground state of $Z \otimes Z$ .

Run your code to see the quantum state.