Ansatz Circuits

In VQE the quantum computer's job is to prepare a trial state $|\psi(\boldsymbol{\theta})\rangle$ whose energy $\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$ we want to minimize. The circuit that prepares this trial state is called the ansatz (German: "initial attempt" or "approach"). Choosing a good ansatz is the central design decision in any VQE experiment.

What makes a good ansatz?

An ansatz must balance two competing demands:

Expressibility — the set of states it can reach must include (or closely approximate) the true ground state.
Efficiency — the circuit must be shallow enough to run on near-term hardware without accumulating too much noise.

The two are in tension: a circuit with more layers can reach more states but also incurs more error.

Hardware-efficient ansatz

The most common near-term choice is the hardware-efficient ansatz (HEA): apply single-qubit rotation gates (typically $R_Y$ or $R_Y R_Z$ pairs) to every qubit, then add a layer of two-qubit entangling gates (CNOT or CZ) that match the device's native connectivity. Repeat for $L$ layers.

For two qubits, one layer looks like this:

|\psi(\theta_0, \theta_1)\rangle = \mathrm{CNOT}_{01}\,\bigl[R_Y(\theta_0)\otimes R_Y(\theta_1)\bigr]\,|00\rangle.

The $R_Y$ rotation is

R_Y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix},

Entanglement is essential

Without the CNOT layer the ansatz is a product state $|\psi_0\rangle \otimes |\psi_1\rangle$ , which can never reach an entangled ground state. Even a single entangling gate can dramatically increase expressibility. For a 2-qubit system, one CNOT layer is often enough to reach any state in the two-dimensional subspace of interest.

Try it

Build a two-qubit hardware-efficient ansatz with the angles fixed to $\theta_0 = \pi/3$ and $\theta_1 = \pi/4$ . The grader checks the full state vector, so the circuit must produce the correct amplitudes.

Run your code to see the quantum state.

After running, inspect the amplitudes: the CNOT entangles the two qubits so you cannot write the output as a simple tensor product. That entanglement is exactly what lets VQE explore correlated ground states.