Variational Quantum Eigensolver (VQE) – Overview

Overview

VQE is a variational algorithm designed to estimate the ground-state energy of a quantum system.

In many scientific and engineering problems, you have a Hamiltonian $H$ (an energy operator) and you want:

• the lowest possible energy value (ground-state energy)
• and sometimes an approximation of the corresponding ground state

VQE matters in the NISQ era because many “textbook” quantum approaches to eigenvalues, like full Quantum Phase Estimation (QPE), require deep, precise circuits. VQE instead uses a hybrid quantum–classical loop with relatively shallow circuits and many repeated measurements.

Intuition

Why we care about ground-state energy

Ground states are where many physical systems “live” at low temperature. Knowing the ground-state energy can tell you:

• whether a configuration is stable
• how a system changes when parameters change
• what the lowest-energy arrangement looks like

Why QPE is impractical on NISQ

QPE is powerful, but it tends to require:

• long coherent evolution
• controlled operations that are deep
• high fidelity across many steps

NISQ devices have limited coherence time and noisy gates. So VQE takes a different approach:

• don’t try to compute the eigenvalue directly via long coherent interference
• instead, propose a family of “guess states,” evaluate their energy, and improve the guess

The hybrid loop (the core idea)

VQE repeats a simple cycle:

1. The quantum computer prepares a parameterized state $|\psi(\theta)\rangle$.
2. The quantum computer measures the energy estimate $\langle \psi(\theta)|H|\psi(\theta)\rangle$.
3. A classical optimizer updates $\theta$ to try to reduce the energy.

Over time, the parameters move toward a lower-energy state.

Formal Description

VQE relies on a variational principle:

• For any normalized state $|\psi\rangle$, the expected energy $$\langle\psi|H|\psi\rangle$$ is always at least the true ground-state energy.

So if you can find a state with low expected energy, you have an upper bound on the ground energy.

In VQE you choose an ansatz (a parameterized circuit) that produces states $|\psi(\theta)\rangle$. You then try to solve the optimization problem:

• minimize the cost function $C(\theta) = \langle \psi(\theta)|H|\psi(\theta)\rangle$

Key realities (important for honesty):

• You only minimize over the states your ansatz can reach.
• Noise and finite sampling mean you don’t observe the true expectation value, only an estimate.
• A “low energy” result is evidence, not a proof, that you found the true ground state.

Worked Example

Consider a 1-qubit toy Hamiltonian:

$H = Z.$

The two eigenstates are:

• $|0\rangle$ with energy +1
• $|1\rangle$ with energy −1 (the ground state)

Pick a simple ansatz:

• prepare $|\psi(\theta)\rangle = R_y(\theta)|0\rangle$

As you vary $\theta$, the state moves from $|0\rangle$ toward $|1\rangle$.

Energy measurement:

• measuring $\langle Z \rangle$ tells you the expected energy

A classical optimizer would push $\theta$ toward the value that makes the energy as low as possible. In the ideal case, it converges to $|1\rangle$ with energy −1.

This tiny example shows the full VQE shape:

• parameterized circuit
• expectation-value measurement
• classical optimization

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Next we focus on a make-or-break component of VQE: ansatz design. The choice of parameterized circuit determines which states are reachable and how difficult optimization will be.