Partial Trace¶

The partial trace operation performs a measurement or a selective collapse on the specified register(s). It is the key means of extracting classical information in quantum algorithms.

Overview ¶

Partial trace operators overview¶
Operator	Operation	Return value
`PartialTrace`	Randomly measure the specified register(s)	`(measured_values, probability)`
`PartialTraceSelect`	Collapse to specified values	`probability`
`PartialTraceSelectRange`	Collapse to within a specified range	`probability`

Physically, a partial trace “measures away” a register from the quantum state — after the measurement that register collapses to a definite value, while the remaining registers enter the corresponding conditional state according to the measurement outcome.

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PartialTrace (random measurement)¶

Operation: Performs a random measurement on the specified register(s) (sampling from the probability distribution), returning the measurement outcome and the corresponding probability.

Parameters: Register identifier(s) (a name string, a list of names, an ID, or a list of IDs).

Returns: (measured_values, probability)

measured_values — the list of register values obtained from the measurement
probability — the probability of this measurement outcome

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)

# Randomly measure the addr register
values, prob = ps.PartialTrace("addr")(state)
print(f"Measurement result: {values}, probability: {prob:.4f}")
# Example: Measurement result: [3], probability: 0.1250

# After the measurement addr has collapsed; only the corresponding basis state remains in state

When measuring multiple registers, the returned value list is ordered according to the parameter order:

values, prob = ps.PartialTrace(["addr", "data"])(state)

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PartialTraceSelect (selective collapse)¶

Operation: Collapses the specified register(s) to the given values instead of measuring randomly.

Parameters: A register-to-value mapping (dictionary or lists).

Returns: probability — the probability of this collapse outcome.

ps.System.clear()
ps.System.add_register("a", ps.UnsignedInteger, 2)
ps.System.add_register("b", ps.UnsignedInteger, 2)

state = ps.SparseState()
ps.Hadamard_Int("a", 2)(state)

# Collapse a to the value 1
prob = ps.PartialTraceSelect({"a": 1})(state)
print(f"Collapse probability: {prob:.4f}")
# Collapse probability: 0.2500

# A list form also works
prob = ps.PartialTraceSelect(["a"], [1])(state)

—

PartialTraceSelectRange (range collapse)¶

Operation: Collapses the specified register to within a given value range, keeping the basis states that satisfy the condition and renormalizing.

Parameters: Register identifier and value range (min, max).

Returns: probability — the total probability within the range.

ps.System.clear()
ps.System.add_register("x", ps.UnsignedInteger, 4)

state = ps.SparseState()
ps.Hadamard_Int("x", 4)(state)

# Keep only the basis states with x ∈ [2, 5]
prob = ps.PartialTraceSelectRange("x", (2, 5))(state)
print(f"Probability within range: {prob:.4f}")
# Probability within range: 0.2500 (4 out of 16 basis states)

中文版 ===

部分追迹¶

部分追迹（Partial Trace）操作对指定寄存器执行测量或选择性坍缩。这是量子算法中提取经典信息的关键手段。

概述 ¶

部分追迹算子总览¶
算子	操作	返回值
`PartialTrace`	随机测量指定寄存器	`(measured_values, probability)`
`PartialTraceSelect`	坍缩到指定值	`probability`
`PartialTraceSelectRange`	坍缩到指定范围内	`probability`

部分追迹的物理含义是将某个寄存器从量子态中”测量掉”——测量后该寄存器坍缩到确定值，其余寄存器根据测量结果进入相应的条件态。

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PartialTrace（随机测量）¶

操作: 对指定寄存器执行随机测量（按概率分布抽样），返回测量结果和对应概率。

参数: 寄存器标识（名称字符串、名称列表、ID 或 ID 列表）。

返回: (measured_values, probability)

measured_values — 测量得到的寄存器值列表
probability — 该测量结果的概率

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)

# 随机测量 addr 寄存器
values, prob = ps.PartialTrace("addr")(state)
print(f"测量结果: {values}, 概率: {prob:.4f}")
# 例如: 测量结果: [3], 概率: 0.1250

# 测量后 addr 已坍缩，state 中仅保留对应基态

测量多个寄存器时，返回值列表按参数顺序排列：

values, prob = ps.PartialTrace(["addr", "data"])(state)

—

PartialTraceSelect（选择性坍缩）¶

操作: 将指定寄存器坍缩到给定的值，而非随机测量。

参数: 寄存器-值映射（字典或列表）。

返回: probability — 该坍缩结果的概率。

ps.System.clear()
ps.System.add_register("a", ps.UnsignedInteger, 2)
ps.System.add_register("b", ps.UnsignedInteger, 2)

state = ps.SparseState()
ps.Hadamard_Int("a", 2)(state)

# 将 a 坍缩到值 1
prob = ps.PartialTraceSelect({"a": 1})(state)
print(f"坍缩概率: {prob:.4f}")
# 坍缩概率: 0.2500

# 也可以用列表形式
prob = ps.PartialTraceSelect(["a"], [1])(state)

—

PartialTraceSelectRange（范围坍缩）¶

操作: 将指定寄存器坍缩到给定值范围内，保留满足条件的基态并重新归一化。

参数: 寄存器标识和值范围 (min, max)。

返回: probability — 范围内的总概率。

ps.System.clear()
ps.System.add_register("x", ps.UnsignedInteger, 4)

state = ps.SparseState()
ps.Hadamard_Int("x", 4)(state)

# 仅保留 x ∈ [2, 5] 的基态
prob = ps.PartialTraceSelectRange("x", (2, 5))(state)
print(f"范围内概率: {prob:.4f}")
# 范围内概率: 0.2500（4/16 个基态）