Controlled-Phase Gate

The Controlled-Phase Gate $CR_k$

The controlled-phase gate $CR_k$ is fundamental to the Quantum Fourier Transform. It applies a phase $e^{2\pi i/2^k}$ to the $|11\rangle$ component only, leaving all other basis states unchanged:

$CR_k = \begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&e^{2\pi i/2^k}\end{pmatrix}$

The state vector for a 2-qubit system is represented as a 4-element list ordered by basis state:

$|\psi\rangle = c_{00}|00\rangle + c_{01}|01\rangle + c_{10}|10\rangle + c_{11}|11\rangle$

So the list [c00, c01, c10, c11] maps index 0 → $|00\rangle$, index 1 → $|01\rangle$, index 2 → $|10\rangle$, index 3 → $|11\rangle$.

Special Cases

• $k=1$: phase $= e^{i\pi} = -1$ (a $Z$ gate on the target, conditioned on control)
• $k=2$: phase $= e^{i\pi/2} = i$ (the $S$ gate, conditioned)
• $k=3$: phase $= e^{i\pi/4} = \frac{1+i}{\sqrt{2}}$ (the $T$ gate, conditioned)

import cmath, math

def cphase(k, state):
    phase = cmath.exp(2j * math.pi / (2**k))
    result = list(state)
    result[3] = state[3] * phase
    return result

# k=1 applies e^(iπ) = -1 to |11⟩ amplitude
result = cphase(1, [0, 0, 0, 1.0])
print(round(abs(result[3]), 4))    # 1.0
print(round(result[3].real, 4))    # -1.0

Why CR_k Matters

In the QFT circuit, $CR_k$ gates with increasing $k$ build up finer phase increments. The $k$-th gate contributes a phase of $2\pi / 2^k$ radians, so larger $k$ values give smaller phase kicks — encoding higher-frequency information into the transform.

Your Task

Implement cphase(k, state) where:

• k is a positive integer (the phase order)
• state is a 4-element list representing amplitudes $[c_{00}, c_{01}, c_{10}, c_{11}]$

The function returns a new list with the $|11\rangle$ amplitude multiplied by $e^{2\pi i/2^k}$.