Grover Search (Toy Example)

Overview

This page demonstrates a small, concrete version of Grover’s algorithm:

• a very small search space (two qubits)
• an oracle that “marks” one answer
• one round of amplitude amplification

This is intentionally a toy. The goal is to see amplitude amplification in code and in measurement results, not to claim scalability.

Conceptual Mapping

From the Algorithms overview and Grover intuition:

• you start in a uniform superposition over candidates
• an oracle flips the phase of the “marked” answer
• a diffuser amplifies the marked answer’s probability

From Foundations:

• probabilities come from amplitudes
• gates shape amplitudes through interference

In code:

• h gates create the uniform superposition
• the oracle is a small subcircuit
• the diffuser is another subcircuit
• measurement shows whether probability mass moved toward the marked state

Code Walkthrough

We will search over 4 items (2 qubits) and mark |11⟩ as the target.

from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
 
qc = QuantumCircuit(2, 2)
 
# 1) Start in uniform superposition
qc.h(0)
qc.h(1)
 
# 2) Oracle: mark |11> by flipping its phase (CZ does that)
qc.cz(0, 1)
 
# 3) Diffuser (inversion about the mean) for 2 qubits
qc.h(0)
qc.h(1)
qc.x(0)
qc.x(1)
qc.cz(0, 1)
qc.x(0)
qc.x(1)
qc.h(0)
qc.h(1)
 
# 4) Measure
qc.measure(0, 0)
qc.measure(1, 1)
 
sim = AerSimulator()
counts = sim.run(qc, shots=1000).result().get_counts()
print(counts)

Line by line (conceptual):

• The first two h gates create an equal-weight superposition over 00, 01, 10, 11.
• cz(0, 1) is the oracle here: it marks |11⟩ by applying a phase flip only to that basis state.
• The diffuser is a fixed pattern of gates that increases the marked state’s amplitude (for a small system, one iteration is enough to see the effect).
• Measurement shows where probability mass ended up.

Results & Interpretation

Without Grover steps (just h then measure), counts would be roughly uniform across four outcomes.

With one oracle + diffuser round, you should see the marked state 11 appear more often. Interpretation:

• the oracle does not “tell you the answer” directly
• it changes the phase of the marked state
• the diffuser turns that phase information into amplitude change
• measurement frequencies reflect those changed amplitudes

Why this is a toy:

• real Grover setups require carefully defined oracles
• the number of iterations depends on problem size
• hardware noise can overwhelm amplitude amplification for larger circuits

The learning goal is: interference can concentrate probability on a desired answer.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

You’ve now run experiments that show core quantum behaviors in code: sampling, amplitudes vs probabilities, entanglement correlations, and amplitude amplification. Next we can start mini-projects that combine these ideas (for example, small algorithm demonstrations with careful result interpretation and sanity checks).