Example: the Bell state

The Bell state is the smallest example of quantum entanglement, and the "hello world" of quantum computing. This example is complete and you can copy and run it.

The circuit

Two qubits, two gates:

import vqubit as vq

# 1. Two qubits on a line.
q0, q1 = vq.LineQubit.range(2)

# 2. A Hadamard gate on q0 and a CNOT controlled by q0.
circuit = vq.Circuit(
    vq.H(q0),
    vq.CNOT(q0, q1),
)

print(circuit)

Output:

0: ───H───@───
          │
1: ───────X───

The H gate puts q0 into superposition. Then CNOT flips q1 only if q0 is 1. The result is that the two qubits become entangled: their fates are no longer independent.

Add a measurement and simulate

# 3. Measure both qubits into an outcome called "result".
circuit.append(vq.measure(q0, q1, key="result"))

# 4. Run the circuit 1000 times.
result = vq.Simulator().run(circuit, repetitions=1000)

# 5. Count how often each outcome came up.
print(result.histogram(key="result"))

Typical output:

Counter({0: 498, 3: 502})

What those numbers mean

Each measured qubit produces a 0 or a 1. With two qubits there are four possible outcomes: 00, 01, 10 and 11. vqubit encodes them as a single integer, reading q0 as the most significant bit:

Outcome	Integer
00	0
01	1
10	2
11	3

In the output only 0 (00) and 3 (11) appear, never 1 or 2. That is, the two qubits always agree. This is entanglement. If you measure q0 and get 0, you know for certain that q1 will also give 0, without looking.

Variations to experiment with

• Replace CNOT with CZ and note that the outcome set is the same.
• Add an X(q1) gate before the measurement and watch the results change.
• Replace H with vq.Ry(math.pi / 8)(q0) and watch the probabilities become uneven.