🌀 Grover’s Algorithm Tutorial
Grover’s algorithm is a quantum search algorithm that finds a marked item in an unsorted database of size
𝑁
N in roughly
𝑂
(
𝑁
)
O(
N
) steps, faster than classical
𝑂
(
𝑁
)
O(N).
We’ll implement it step by step for a small example.
1️⃣ Prerequisites
You need Python and Qiskit installed:
pip install qiskit matplotlib
Then import libraries:
from qiskit import QuantumCircuit, Aer, execute
from qiskit.visualization import plot_histogram
import numpy as np
2️⃣ Define the Problem
Suppose we have a 2-qubit database with 4 states: 00, 01, 10, 11. We want to find the marked element |11⟩.
Number of qubits =
𝑛
=
2
n=2
Number of iterations ≈
𝜋
4
𝑁
4
π
N
n = 2
N = 2 ** n
marked_state = '11'
iterations = int(np.floor(np.pi/4 * np.sqrt(N)))
print(f"Iterations needed: {iterations}")
3️⃣ Step 1: Initialize Qubits in Superposition
Grover starts with uniform superposition using Hadamard gates:
qc = QuantumCircuit(n)
# Apply Hadamard to all qubits
qc.h(range(n))
qc.barrier()
qc.draw('mpl')
Now all states have equal probability
1
𝑁
N
1
.
4️⃣ Step 2: Define the Oracle
The oracle flips the phase of the marked state.
For |11⟩, we can implement it with a controlled-Z gate:
def oracle(qc):
qc.cz(0, 1) # Phase flip for |11>
qc.barrier()
If the marked state was different, we’d adjust controls/NOTs.
5️⃣ Step 3: Implement the Diffusion Operator
The diffusion operator amplifies the probability of the marked state:
Apply Hadamard to all qubits
Apply X to all qubits
Apply multi-controlled Z (phase flip for |00⟩)
Apply X and Hadamard again
def diffuser(n):
qc = QuantumCircuit(n)
qc.h(range(n))
qc.x(range(n))
qc.h(n-1)
qc.mcx(list(range(n-1)), n-1) # multi-controlled X
qc.h(n-1)
qc.x(range(n))
qc.h(range(n))
qc.barrier()
return qc
6️⃣ Step 4: Apply Grover Iterations
Now we combine oracle + diffuser for the required number of iterations:
for _ in range(iterations):
oracle(qc)
qc += diffuser(n)
7️⃣ Step 5: Measure Qubits
qc.measure_all()
qc.draw('mpl')
8️⃣ Step 6: Simulate the Circuit
backend = Aer.get_backend('qasm_simulator')
result = execute(qc, backend, shots=1024).result()
counts = result.get_counts()
print(counts)
plot_histogram(counts)
✅ You should see the marked state 11 appears most of the time, demonstrating Grover’s quadratic speedup.
9️⃣ Optional: Run on a Real Device
from qiskit import IBMQ
# Load your IBM Quantum account
IBMQ.load_account()
provider = IBMQ.get_provider(hub='ibm-q')
backend = provider.get_backend('ibmq_qasm_simulator')
result = execute(qc, backend, shots=1024).result()
counts = result.get_counts()
plot_histogram(counts)
Summary of Steps
Initialize qubits in superposition
Define the oracle for the marked state
Implement the diffusion operator
Apply oracle + diffuser iterations
Measure the qubits
Run on simulator or real quantum device
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