Demo 3：QFT 振幅相位

第 3 章 量子算法基础 · 本章演示 · 代码演示总索引

可视化：双图：振幅+相位 —— QFT_4|0010⟩ 16 个分量的极坐标散点

对应教程：第 3 章 量子算法基础 中相关小节 · 依赖安装：pip install "unified-quantum[visualization,simulation]"

实现要点

完整源码

demos/ch03-algo-basics/03_qft_4qubit.py

"""
ch03 演示 3：QFT_4 作用在 |0010⟩ 上，画振幅-相位极坐标散点

对应教程：chenzhaoyun.com/quantum-tutorial/ch03-algo-basics/quantum-fourier-transform.html
依赖：pip install unified-quantum[visualization]
运行：python demos/ch03-algo-basics/03_qft_4qubit.py
"""
from uniqc import Circuit, qft_circuit, submit_task, wait_for_result
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
import numpy as np
from pathlib import Path

OUT = Path(__file__).parent
N_QUBITS = 4
INPUT_INT = 2  # |0010>
SHOTS = 2000

# === 1. 构造 |0010> + QFT_4，并跑 dummy backend 拿计数做交叉验证 ===
prep = Circuit()
# 把 qubit 1 翻成 1，得到 |0010>（qubit 0 是最低位）
prep.x(1)
prep.add_circuit(qft_circuit(n_qubits=N_QUBITS, swaps=True))
for q in range(N_QUBITS):
    prep.measure(q)

result = wait_for_result(
    submit_task(circuit=prep, backend="dummy:local:simulator", shots=SHOTS)
)
counts = result.counts
total = sum(counts.values())

# === 2. 用 Circuit.get_matrix() 解析地拿 4-qubit 状态向量（拿相位） ===
qft = qft_circuit(n_qubits=N_QUBITS, swaps=True)
U = qft.get_matrix()
state_in = np.zeros(2 ** N_QUBITS, dtype=complex)
state_in[INPUT_INT] = 1.0
state_out = U @ state_in  # QFT|0010> 的精确振幅

amps = np.abs(state_out) ** 2
phases = np.angle(state_out)  # in (-pi, pi]
labels = [f"{i:0{N_QUBITS}b}" for i in range(2 ** N_QUBITS)]

# 模拟测量的频率（仅用来显示在标注里）
empirical = np.array([counts.get(lbl, 0) / total for lbl in labels])

# === 3. matplotlib 极坐标散点（振幅半径、相位角度） ===
fig, axes = plt.subplots(1, 2, figsize=(13, 6),
                         subplot_kw={"projection": "polar"})

# 左图：所有 16 个分量的振幅-相位
ax = axes[0]
ax.scatter(phases, amps, s=120, c="#1f77b4", edgecolor="black", zorder=3)
ax.set_theta_zero_location("E")
ax.set_theta_direction(1)
ax.set_ylim(0, 0.30)
ax.set_yticks([0.0625, 0.125, 0.1875])
ax.set_yticklabels(["1/16", "1/8", "3/16"], fontsize=8)
ax.set_xlabel("phase (rad)")
ax.set_title(
    f"QFT_4 |{INPUT_INT:0{N_QUBITS}b}> : all 16 amplitudes\n"
    f"|amplitude|^2 = {amps.mean():.3f} (uniform → 1/{2**N_QUBITS})",
    fontsize=10,
    pad=18,
)

# 右图：仅展示相位（半径固定），突出 QFT 相位编码
ax = axes[1]
fixed_r = 0.25
ax.scatter(phases, np.full_like(phases, fixed_r), s=140, c="#d62728",
           edgecolor="black", zorder=3)
# 用线连接相邻角度方便观察
order = np.argsort(phases)
ax.plot(phases[order], np.full_like(phases[order], fixed_r),
        color="#888", alpha=0.5, lw=1)
ax.set_theta_zero_location("E")
ax.set_theta_direction(1)
ax.set_ylim(0, 0.30)
ax.set_yticks([fixed_r])
ax.set_yticklabels([f"r={fixed_r}"], fontsize=8)
ax.set_xlabel("phase (rad)")
ax.set_title(
    f"Phase pattern of QFT_4 |{INPUT_INT:0{N_QUBITS}b}>\n"
    f"phase(k) = 2π·k·x / 2^n   (x={INPUT_INT}, n={N_QUBITS})",
    fontsize=10,
    pad=18,
)

# 在散点旁标注 3 个有代表性的标签
for k in (0, 4, 8):
    ax = axes[0]
    ax.annotate(
        f"|{k:0{N_QUBITS}b}>",
        xy=(phases[k], amps[k]),
        xytext=(phases[k] + 0.15, amps[k] + 0.04),
        fontsize=8, arrowprops=dict(arrowstyle="-", color="gray", lw=0.5),
    )

fig.suptitle(
    f"Quantum Fourier Transform on {N_QUBITS} qubits, input |{INPUT_INT:0{N_QUBITS}b}>  "
    f"({SHOTS} shots cross-checked, dummy:local:simulator)",
    fontsize=12, y=1.02,
)
plt.tight_layout()
plt.savefig(OUT / "03_qft_4qubit.png", dpi=120, bbox_inches="tight")
plt.close()

# === 4. stdout 关键统计 ===
print("ch03-algo-basics demo 3: QFT_4 amplitude + phase of |0010>")
print(f"  total shots: {total}")
print(f"  unique outcomes: {int((empirical > 0).sum())} / {2**N_QUBITS}  "
      f"(all 16 expected for a uniform Fourier image)")
print(f"  all probabilities ≈ 1/{2**N_QUBITS} = {1/2**N_QUBITS:.4f}  "
      f"(max-min range = {empirical.max()-empirical.min():.3f})")
print(f"  empirical mean |amp|^2 = {empirical.mean():.4f}")
print(f"  phase spread (rad): [{phases.min():+.3f}, {phases.max():+.3f}]")
print(f"  saved: {OUT / '03_qft_4qubit.png'}")

运行结果

运行方式

# 仓库根目录执行；输出 PNG 到 demos/ch03-algo-basics/03_qft_4qubit.png，关键统计到 stdout
.venv/bin/python3 demos/ch03-algo-basics/03_qft_4qubit.py

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