Create a circuit that implements Deutsch's algorithm for a constant function and verify your results.

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Deutsch's algorithm is a simple quantum algorithm that demonstrates the quantum parallelism and interference phenomena. It's designed to determine whether a given quantum oracle represents a constant or balanced (non-constant) function. In the case of a constant function, the oracle maps all inputs to the same output value.

Here's how you can implement Deutsch's algorithm for a constant function using Qiskit and then verify the results:

Code: Select all

from qiskit import QuantumCircuit, Aer, execute

# Create a circuit with two qubits and two classical bits
n = 2
circuit = QuantumCircuit(n+1, n)

# Apply Hadamard gates to the first qubit
circuit.h(0)

# Apply the oracle for a constant function (identity gate)
# This oracle maps |x, y⟩ to |x, y⨁f(x)⟩ where ⨁ is bitwise XOR
# Since we're implementing a constant function, f(x) = 0
# Applying identity gate (I) on the second qubit
circuit.i(1)

# Apply Hadamard gates again to the first qubit
circuit.h(0)

# Measure the first qubit
circuit.measure(0, 0)

# Simulate the circuit
simulator = Aer.get_backend('qasm_simulator')
job = execute(circuit, simulator, shots=1)

# Get and print the measurement result
result = job.result().get_counts()
print(result)

In this example, the oracle is implemented using the identity gate, which means the oracle maps all inputs to themselves (i.e., no change). As a result, the algorithm should output the state {'0': 1} for a constant function.

You can run the above code in a Qiskit environment to verify the results. Keep in mind that due to the nature of quantum systems, running the algorithm multiple times may result in different measurement outcomes. However, for a constant function, all measurement outcomes should be '0'.