Simple VQE Demo

Overview

This page teaches a complete, tiny hybrid workflow:

• choose a very small “Hamiltonian” (an observable we want to minimize)
• build a parameterized circuit (ansatz)
• estimate an expectation value from measurement results
• update parameters using a simple classical loop

It matters because this is the core pattern behind many near-term workflows: classical code proposes circuits, quantum execution provides statistics, classical code updates the next step.

This is a learning demo, not chemistry-grade VQE.

Conceptual Mapping

From the Variational & NISQ module:

• VQE minimizes an energy-like objective by tuning circuit parameters
• the objective is an expectation value of an observable
• evaluation is statistical (shots), not exact

From Foundations + Gates:

• measurement is probabilistic (Born rule)
• expectation values come from many samples
• a single-qubit rotation is a simple example of a parameterized circuit

In code, the mapping is:

• Ansatz → a QuantumCircuit with a parameter (like a rotation angle)
• Hamiltonian (toy) → a measurement rule that assigns a number to outcomes
• Expectation value → average of those numbers over many shots
• Optimizer (toy) → try a few parameter values and pick the best

Code Walkthrough

We’ll use a 1‑qubit toy Hamiltonian that is “measure Z” (conceptually: prefer one computational basis state over the other).

Idea:

• prepare a 1‑qubit state with a rotation parameter theta
• measure many shots
• map outcomes to values (+1 for 0, −1 for 1)
• average to estimate the expectation value

import math
from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
 
def estimate_z_expectation(theta, shots=500):
    qc = QuantumCircuit(1, 1)
    qc.ry(theta, 0)
    qc.measure(0, 0)
 
    sim = AerSimulator()
    counts = sim.run(qc, shots=shots).result().get_counts()
 
    # Map bit outcomes to Z eigenvalues: 0 -> +1, 1 -> -1
    p0 = counts.get('0', 0) / shots
    p1 = counts.get('1', 0) / shots
    return (+1) * p0 + (-1) * p1
 
# Toy “optimizer”: try a few angles and keep the best
candidates = [0, math.pi/4, math.pi/2, 3*math.pi/4, math.pi]
 
best_theta = None
best_value = None
 
for theta in candidates:
    value = estimate_z_expectation(theta)
    print(theta, value)
 
    if best_value is None or value < best_value:
        best_value = value
        best_theta = theta
 
print("best", best_theta, best_value)

Line by line (conceptual):

• estimate_z_expectation builds a circuit with a parameter theta.
• ry(theta, 0) is the ansatz: a tunable state preparation.
• shots repeats the experiment to estimate probabilities.
• counts is a histogram of outcomes.
• We convert counts into an expectation value by mapping outcomes to numbers and averaging.
• The “optimizer” is a simple loop that tries a few candidate angles.

Results & Interpretation

What you’re looking for is not a specific number. It’s the workflow behavior:

• different parameters produce different measurement distributions
• those distributions imply different expectation values
• the classical loop can improve the objective by choosing better parameters

How to judge correctness (for a learning demo):

• sanity check 1: results fluctuate a bit if you rerun (shot noise)
• sanity check 2: increasing shots makes estimates more stable
• sanity check 3: the “best” parameter should be reasonably consistent across reruns

Connection back to amplitudes:

• ry(theta) changes amplitudes of |0⟩ and |1⟩
• probabilities come from those amplitudes
• expectation is a weighted average of outcome values

This is the key VQE lesson: you rarely compute an exact value; you estimate it from repeated measurements.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Now you’ve seen an end-to-end hybrid loop. Next we’ll focus on debugging: how to use simulators, intermediate inspection, and sanity checks to understand “wrong” results productively.