Implement QFT on a five-qubit state—for example, '10110'.

[quantumadmin]

The Quantum Fourier Transform (QFT) is a fundamental operation in quantum computing that converts a quantum state into a superposition of its frequency components. To perform the QFT on a five-qubit state '10110', we need to apply a series of quantum gates that implement the QFT algorithm. Here's how you can implement the QFT on the given state:

Assuming qubit 0 is the least significant qubit and qubit 4 is the most significant qubit (big-endian notation), the QFT on a five-qubit state '10110' can be implemented as follows:

Apply Hadamard gates to all qubits:

Code: Select all

|10110⟩ -> (H ⊗ H ⊗ H ⊗ H ⊗ H) |10110⟩

Apply controlled-phase gates:

Code: Select all

(H ⊗ H ⊗ H ⊗ H) Cφ1 --- Cφ2 --- Cφ3 --- Cφ4 (H ⊗ H ⊗ H ⊗ H) |10110⟩

Where:

Cφ1 applies a controlled-phase gate with a rotation of 2π/2^1 between qubit 0 and qubit 1.
Cφ2 applies a controlled-phase gate with a rotation of 2π/2^2 between qubit 0 and qubit 2.
Cφ3 applies a controlled-phase gate with a rotation of 2π/2^3 between qubit 0 and qubit 3.
Cφ4 applies a controlled-phase gate with a rotation of 2π/2^4 between qubit 0 and qubit 4.

Apply additional Hadamard gates:

Code: Select all

(H ⊗ H ⊗ H ⊗ H) Cφ1 --- Cφ2 --- Cφ3 --- Cφ4 (H ⊗ H ⊗ H ⊗ H) |10110⟩
        |            |            |            |
(H ⊗ H ⊗ H ⊗ H) H       H       H       H       |10110⟩

This completes the Quantum Fourier Transform on the given five-qubit state '10110'. Please note that the actual implementation might vary depending on the quantum programming framework you are using and the specific gates available on the quantum hardware you are targeting.