Phase and Reflection Operators¶

Phase and reflection operators implement conditional phase flips, global phases, Grover reflections and similar operations, and are widely used in quantum search and amplitude amplification algorithms.

Overview ¶

Phase operators overview¶
Operator	Operation	Unitarity class
`ZeroConditionalPhaseFlip`	Flip the phase when the specified registers are all zero	SelfAdjoint
`Reflection_Bool`	Reflection about the `\|0⟩` state (Grover diffusion)	SelfAdjoint
`GlobalPhase`	Multiply by a complex global phase factor	BaseOperator

—

ZeroConditionalPhaseFlip (zero-conditional phase flip)¶

Operation: Applies a \(-1\) phase flip to the basis states when all of the specified registers have value zero.

Parameters: A list of register identifiers (list of names or list of IDs).

Mathematical representation:

\[\begin{split}|x_1, x_2, \ldots, x_n\rangle \rightarrow \begin{cases} -|x_1, x_2, \ldots, x_n\rangle & \text{if } x_1 = x_2 = \cdots = x_n = 0 \\ |x_1, x_2, \ldots, x_n\rangle & \text{otherwise} \end{cases}\end{split}\]

Purpose: Building the Oracle in Grover’s algorithm. When the target state is |0...0⟩, this operator is exactly the phase Oracle.

import pysparq as ps

ps.System.clear()
ps.System.add_register("addr", ps.UnsignedInteger, 3)
ps.System.add_register("data", ps.UnsignedInteger, 4)

state = ps.SparseState()
ps.Hadamard_Int("addr", 3)(state)
# After loading, flip the phase when the data corresponding to addr is 0
ps.ZeroConditionalPhaseFlip(["data"])(state)

# A list of register names also works
ps.ZeroConditionalPhaseFlip(["addr", "data"])(state)

—

Reflection_Bool (reflection)¶

Operation: Reflection about the |0⟩ state, i.e. the core component of the Grover diffusion operator.

Parameters:

reg — target register (name or ID, or a list)
inverse — whether to use the inverse reflection (optional, default False)

Mathematical representation:

\[R = 2|0\rangle\langle 0| - I\]

Purpose: The diffusion operator in Grover’s algorithm is composed of Hadamard + Reflection + Hadamard.

ps.System.clear()
ps.System.add_register("q", ps.Boolean, 1)

state = ps.SparseState()
ps.Hadamard_Bool("q")(state)

# Reflection
ps.Reflection_Bool("q")(state)

—

GlobalPhase (global phase)¶

Operation: Multiplies the entire quantum state by a complex global phase factor.

Parameters: c — the complex phase factor.

Dagger: Multiplies by the complex conjugate.

Note: A global phase is unobservable in quantum mechanics, but in some algorithms (such as amplitude encoding, QDA) it must be tracked exactly.

import numpy as np

# Global phase rotation e^{iπ/4}
op = ps.GlobalPhase(np.exp(1j * np.pi / 4))
op(state)

# Undo
op.dag(state)

中文版 ===

相位与反射算子¶

相位与反射算子实现条件相位翻转、全局相位和 Grover 反射等操作，广泛用于量子搜索和振幅放大算法。

概述 ¶

相位算子总览¶
算子	操作	幺正类
`ZeroConditionalPhaseFlip`	指定寄存器全为零时翻转相位	SelfAdjoint
`Reflection_Bool`	关于 `\|0⟩` 态的反射（Grover 扩散）	SelfAdjoint
`GlobalPhase`	全局相位乘以复数因子	BaseOperator

—

ZeroConditionalPhaseFlip（零条件相位翻转）¶

操作: 当所有指定寄存器的值均为零时，对基态施加 \(-1\) 相位翻转。

参数: 寄存器标识列表（名称列表或 ID 列表）。

数学表示:

用途: 构造 Grover 算法中的 Oracle。当目标态为 |0...0⟩ 时，该算子即为相位 Oracle。

import pysparq as ps

ps.System.clear()
ps.System.add_register("addr", ps.UnsignedInteger, 3)
ps.System.add_register("data", ps.UnsignedInteger, 4)

state = ps.SparseState()
ps.Hadamard_Int("addr", 3)(state)
# 加载后，当 addr 对应的 data 为 0 时翻转相位
ps.ZeroConditionalPhaseFlip(["data"])(state)

# 也可以用寄存器名称列表
ps.ZeroConditionalPhaseFlip(["addr", "data"])(state)

—

Reflection_Bool（反射）¶

操作: 关于 |0⟩ 态的反射操作，即 Grover 扩散算子中的核心组件。

参数:

reg — 目标寄存器（名称或 ID，或列表）
inverse — 是否使用逆反射（可选，默认 False）

数学表示:

\[R = 2|0\rangle\langle 0| - I\]

用途: Grover 算法的扩散算子（Diffusion Operator）由 Hadamard + Reflection + Hadamard 组合而成。

ps.System.clear()
ps.System.add_register("q", ps.Boolean, 1)

state = ps.SparseState()
ps.Hadamard_Bool("q")(state)

# 反射操作
ps.Reflection_Bool("q")(state)

—

GlobalPhase（全局相位）¶

操作: 对整个量子态乘以一个复数全局相位因子。

参数: c — 复数相位因子。

Dagger: 乘以共轭复数。

注意: 全局相位在量子力学中不可观测，但在某些算法（如振幅编码、QDA）中需要精确追踪。

import numpy as np

# 全局相位旋转 e^{iπ/4}
op = ps.GlobalPhase(np.exp(1j * np.pi / 4))
op(state)

# 撤销
op.dag(state)