Conditional Rotation Operators¶

Conditional rotation operators rotate a target register based on the value of an input register.

Overview ¶

Conditional rotation operators overview¶
Operator	Operation	Unitarity class
`CondRot_Fixed_Bool`	Fixed conditional rotation driven by a Rational angle register	BaseOperator

Rotation Mechanism ¶

Conditional rotation converts the value of the input register into a rotation angle and then applies it to the output register.

For Rational-type input:

\[\theta = \frac{value}{2^n} \cdot 2\pi\]

Rotation matrix:

\[\begin{split}R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\end{split}\]

—

CondRot_Fixed_Bool ¶

Operation: Rotates a Boolean register according to the value of a Rational register

Type constraints: - Input register: Rational - Output register: Boolean (size must be 1)

Dagger: Uses the inverse rotation matrix

import pysparq as ps
import numpy as np

ps.System.clear()

# Rational is used for angle encoding
ps.System.add_register("angle", ps.Rational, 16)  # High-precision angle
ps.System.add_register("target", ps.Boolean, 1)

state = ps.SparseState()

# Set the angle (e.g. π/4 → value = 2^16 / 8 = 8192)
angle_value = int(2**16 * 0.125)  # 0.125 = 1/8 of a turn
ps.Init_Unsafe("angle", angle_value)(state)

# Initialize the target to |0⟩
# Initial state: |angle=8192, target=0⟩

# Conditional rotation
op = ps.CondRot_Fixed_Bool("angle", "target")
op(state)

ps.pprint(state)
# The target state is rotated by an angle ≈ π/4

# Undo
op.dag(state)

Use Cases ¶

Phase estimation support ¶

# The angle register stores the estimated phase
ps.System.add_register("phase", ps.Rational, 16)
ps.System.add_register("ancilla", ps.Boolean, 1)

# Initialize the ancilla to |+⟩
ps.Hadamard_Bool("ancilla")(state)

# The conditional rotation encodes the phase information
ps.CondRot_Fixed_Bool("phase", "ancilla")(state)

Hamiltonian simulation ¶

# Time-step angle
dt = 0.01
angle_val = int(dt * 2**16)

ps.Init_Unsafe("dt_angle", angle_val)(state)
ps.CondRot_Fixed_Bool("dt_angle", "qubit")(state)

Quantum amplitude encoding ¶

# Rotate the target bit according to a data value
def amplitude_encoding(value: int) -> np.ndarray:
    theta = np.arccos(value / 255.0)  # Assume 8-bit data
    return np.array([
        [np.cos(theta), -np.sin(theta)],
        [np.sin(theta), np.cos(theta)]
    ])

# Preferred pattern:
# 1. compute angle into a Rational register using an arithmetic adapter
# 2. rotate with CondRot_Fixed_Bool
# 3. apply the same adapter again to uncompute
ps.CondRot_Fixed_Bool("angle", "qubit")(state)

中文版 ===

条件旋转算子¶

条件旋转算子根据输入寄存器的值对目标寄存器进行旋转操作。

概述 ¶

条件旋转算子总览¶
算子	操作	幺正类
`CondRot_Fixed_Bool`	基于 Rational 角度寄存器的固定条件旋转	BaseOperator

旋转机制 ¶

条件旋转将输入寄存器的值转换为旋转角度，然后应用于输出寄存器。

对于 Rational 类型输入：

\[\theta = \frac{value}{2^n} \cdot 2\pi\]

旋转矩阵：

\[\begin{split}R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\end{split}\]

—

CondRot_Fixed_Bool ¶

操作: 根据 Rational 寄存器值旋转 Boolean 寄存器

类型约束: - 输入寄存器: Rational - 输出寄存器: ``Boolean``（大小必须为 1）

Dagger: 使用反向旋转矩阵

import pysparq as ps
import numpy as np

ps.System.clear()

# Rational 用于角度编码
ps.System.add_register("angle", ps.Rational, 16)  # 高精度角度
ps.System.add_register("target", ps.Boolean, 1)

state = ps.SparseState()

# 设置角度（如 π/4 → value = 2^16 / 8 = 8192）
angle_value = int(2**16 * 0.125)  # 0.125 = 1/8 圈
ps.Init_Unsafe("angle", angle_value)(state)

# 初始化目标为 |0⟩
# 初始状态: |angle=8192, target=0⟩

# 条件旋转
op = ps.CondRot_Fixed_Bool("angle", "target")
op(state)

ps.pprint(state)
# target 状态被旋转角度 ≈ π/4

# 撤销
op.dag(state)

使用场景 ¶

相位估计辅助 ¶

# 角度寄存器存储估计相位
ps.System.add_register("phase", ps.Rational, 16)
ps.System.add_register("ancilla", ps.Boolean, 1)

# 初始化 ancilla 为 |+⟩
ps.Hadamard_Bool("ancilla")(state)

# 条件旋转编码相位信息
ps.CondRot_Fixed_Bool("phase", "ancilla")(state)

哈密顿模拟 ¶

# 时间步长角度
dt = 0.01
angle_val = int(dt * 2**16)

ps.Init_Unsafe("dt_angle", angle_val)(state)
ps.CondRot_Fixed_Bool("dt_angle", "qubit")(state)

量子振幅编码 ¶

# 根据数据值旋转目标比特
def amplitude_encoding(value: int) -> np.ndarray:
    theta = np.arccos(value / 255.0)  # 假设 8 位数据
    return np.array([
        [np.cos(theta), -np.sin(theta)],
        [np.sin(theta), np.cos(theta)]
    ])

# Preferred pattern:
# 1. compute angle into a Rational register using an arithmetic adapter
# 2. rotate with CondRot_Fixed_Bool
# 3. apply the same adapter again to uncompute
ps.CondRot_Fixed_Bool("angle", "qubit")(state)