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B4: Quantum Fourier Transform

Time Limit: 3 sec

Memory Limit: 512 MiB

Score: 200

Problem Statement

You are given an integer nn. Implement the quantum Fourier transform (QFT) for nn qubits.

The quantum Fourier transform is defined as a nn-qubit oracle QFT\mathrm{QFT} acting on computational basis states as

∣j⟩n→QFT12n∑k=02n−1exp⁡(2πijk2n)∣k⟩n\begin{equation} \ket{j}_n \xrightarrow{\mathrm{QFT}} \sqrt{\frac{1}{2^n}} \sum_{k=0}^{2^n-1} \exp{\left(\frac{2\pi i j k}{2^n}\right)} \ket{k}_n \nonumber \end{equation}

for any integer jj such that 0≤j<2n0 \leq j < 2^n.

Constraints

  • 1≤n≤101 \leq n \leq 10
  • The circuit depth must not exceed 2525.
  • Integers are encoded by little-endian notation, i.e., ∣100⟩=1≠∣001⟩\ket{100} = 1 \neq \ket{001}.
  • Global phase is ignored in judge.
  • The submitted code must follow the specified format:
from qiskit import QuantumCircuit
 
 
def solve(n: int) -> QuantumCircuit:
    qc = QuantumCircuit(n)
    # Write your code here:
 
    return qc

Sample Input

  • n=2n = 2: Implemented circuit qc\mathrm{qc} should perform the following transformation.
∣10⟩→qc14(∣00⟩+eπi/2∣10⟩+eπi∣01⟩+e3πi/2∣11⟩)\ket{10} \xrightarrow{\mathrm{qc}} \frac{1}{\sqrt{4}} (\ket{00} + e^{\pi i / 2}\ket{10} + e^{\pi i}\ket{01} + e^{3 \pi i / 2}\ket{11})

Hints

Open
  • Please be aware that the circuit must be implemented assuming little-endian.

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