Rotation Gates Rx, Ry, Rz

Rotation Gates

The rotation gates $R_x$, $R_y$, and $R_z$ rotate the qubit state around the X, Y, and Z axes of the Bloch sphere by an angle $\theta$.

Rx Gate

$R_x(\theta) = \begin{pmatrix} \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}$

Applied to $[\alpha, \beta]$:

$R_x(\theta)[\alpha, \beta] = \left[\cos\tfrac{\theta}{2}\,\alpha - i\sin\tfrac{\theta}{2}\,\beta,\ -i\sin\tfrac{\theta}{2}\,\alpha + \cos\tfrac{\theta}{2}\,\beta\right]$

Ry Gate

$R_y(\theta) = \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}$

Applied to $[\alpha, \beta]$:

$R_y(\theta)[\alpha, \beta] = \left[\cos\tfrac{\theta}{2}\,\alpha - \sin\tfrac{\theta}{2}\,\beta,\ \sin\tfrac{\theta}{2}\,\alpha + \cos\tfrac{\theta}{2}\,\beta\right]$

Rz Gate

$R_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$

Applied to $[\alpha, \beta]$:

$R_z(\theta)[\alpha, \beta] = \left[e^{-i\theta/2}\,\alpha,\ e^{i\theta/2}\,\beta\right]$

Special Cases

At $\theta = \pi$, these gates recover familiar Pauli gates (up to global phase):

• $R_x(\pi) \propto X$ (bit flip with imaginary phase)
• $R_y(\pi) \propto Y$ (bit flip, real entries)
• $R_z(\pi) \propto Z$ (phase flip, no amplitude change)

Note that $R_z$ only changes phases, never measurement probabilities.

Implementation

import cmath, math

def rx(theta, state):
    c = math.cos(theta / 2)
    s = math.sin(theta / 2)
    return [
        c * complex(state[0]) + (-1j * s) * complex(state[1]),
        (-1j * s) * complex(state[0]) + c * complex(state[1])
    ]

def ry(theta, state):
    c = math.cos(theta / 2)
    s = math.sin(theta / 2)
    return [
        c * complex(state[0]) + (-s) * complex(state[1]),
        s * complex(state[0]) + c * complex(state[1])
    ]

def rz(theta, state):
    return [
        cmath.exp(-1j * theta / 2) * complex(state[0]),
        cmath.exp(1j * theta / 2) * complex(state[1])
    ]

Your Task

Implement rx(theta, state), ry(theta, state), and rz(theta, state).