Quantum Computing - Deutsch-Jozsa Algorithm

Quantum Circuit:

Let's represent the function f(x) as a black box oracle. In the quantum circuit, the oracle is implemented as a unitary transformation (a reversible quantum operation) that depends on the function's properties. The oracle takes n qubits as input and produces the result in an additional output qubit.

The quantum circuit consists of the following steps:

• Step 1: Initialization: Prepare the n+1 qubits in the state |0?|1?. The first n qubits are the input qubits, and the last qubit is the output qubit.
• Step 2: Hadamard Transform: Apply Hadamard gates (H) to all qubits. This puts all the qubits into an equal superposition of all possible binary strings.

# Quantum Circuit for n = 3 qubits

from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister

# Create a quantum register with 4 qubits (n+1)

qr = QuantumRegister(4)

# Create a classical register to store the measurement outcomes of the first n qubits

cr = ClassicalRegister(3)

# Create a quantum circuit

circuit = QuantumCircuit(qr, cr)

# Apply Hadamard gates to all qubits

circuit.h(qr)

Step 3: Oracle (Function Evaluation): Implement the function f(x) as a quantum oracle. Depending on whether the function is constant or balanced, the oracle will perform different transformations. For a constant function, the oracle does nothing, and for a balanced function, it applies a controlled-phase shift (cz) to the output qubit.

# Define the quantum oracle for the Deutsch-Jozsa algorithm

def deutsch_jozsa_oracle(circuit, qr):

??# Implement the oracle for a balanced function (f(x) = x[0] XOR x[1] XOR x[2])

??circuit.cx(qr[0], qr[3])

??circuit.cx(qr[1], qr[3])

??circuit.cx(qr[2], qr[3])

# Apply the oracle to the circuit

deutsch_jozsa_oracle(circuit, qr)

Step 4: Hadamard Transform Again: Apply Hadamard gates (H) to the first n qubits again.

# Apply Hadamard gates to the first n qubits again

circuit.h(qr[0:3])

Step 5: Measurement: Finally, measure the first n qubits and store the outcomes in the classical register.

# Measure the first n qubits and store the results in the classical register

circuit.measure(qr[0:3], cr)

Explanation:

The quantum algorithm takes advantage of quantum parallelism. Before the final measurement, the Hadamard gates create a superposition of all possible inputs. The oracle then performs a transformation based on the function f(x). If the function is balanced, the constructive interference caused by the superposition results in destructive interference for the |0? state and constructive interference for the |1? state. As a result, when the first n qubits are measured, they are more likely to collapse into the |1? state. However, if the function is constant, the interference patterns cancel out, and the probability of measuring |1? is nearly zero.

In a classical algorithm, you would need to evaluate the function multiple times to determine if it's constant or balanced. But in the quantum algorithm, thanks to superposition and interference, a single evaluation is enough to provide the answer with certainty.

Complexity:

The Deutsch-Jozsa algorithm solves the problem in just one function evaluation, providing a quadratic speedup compared to classical algorithms. The classical approach would require evaluating the function for at least half of all possible inputs (2^(n-1) + 1 evaluations) to distinguish between constant and balanced functions. In contrast, the quantum algorithm can do this in one shot, regardless of the value of n.

Applications:

The Deutsch-Jozsa algorithm itself may not have many direct practical applications beyond being a fundamental building block in quantum computing. However, it serves as a stepping stone to more complex quantum algorithms that leverage similar techniques, such as the Grover's search algorithm and Shor's factoring algorithm, which have broader real-world applications in searching unsorted databases and breaking certain cryptographic protocols, respectively.