10.3 學習資源與實戰建議#

import numpy as np

class QuantumComputer:
    """本書累積成果的總結：狀態向量模擬器 + 閘庫 + 測量 + 雜訊"""
    H  = np.array([[1,1],[1,-1]], dtype=complex)/np.sqrt(2)
    X  = np.array([[0,1],[1,0]], dtype=complex)
    Z  = np.array([[1,0],[0,-1]], dtype=complex)
    S  = np.array([[1,0],[0,1j]], dtype=complex)
    T  = np.array([[1,0],[0,np.exp(1j*np.pi/4)]], dtype=complex)
    CX = np.array([[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]], dtype=complex)

    def __init__(self, n, seed=None):
        self.n = n
        self.state = np.zeros(2**n, dtype=complex); self.state[0] = 1
        self.rng = np.random.default_rng(seed)
        self.gate_count = 0

    def _apply_1q(self, U, k):
        for i in range(2**self.n):
            if (i >> k) & 1 == 0:
                j = i | (1 << k)
                a, b = self.state[i], self.state[j]
                self.state[i]   = U[0,0]*a + U[0,1]*b
                self.state[j]   = U[1,0]*a + U[1,1]*b
        self.gate_count += 1

    def cx(self, c, t):
        for i in range(2**self.n):
            if (i >> c) & 1 and (i >> t) & 1 == 0:
                j = i | (1 << t)
                self.state[i], self.state[j] = self.state[j], self.state[i]
        self.gate_count += 1

    def h(self, k): self._apply_1q(self.H, k)
    def x(self, k): self._apply_1q(self.X, k)
    def z(self, k): self._apply_1q(self.Z, k)

    def depolarize(self, p):
        """每個位元獨立的退極化雜訊（密度矩陣的簡化：隨機套 Pauli）"""
        for k in range(self.n):
            r = self.rng.random()
            if r < p/3:   self.x(k)
            elif r < 2*p/3: self.z(k)
            elif r < p:   self._apply_1q(self.X @ self.Z, k)

    def measure(self, shots=1):
        p = np.abs(self.state)**2
        return self.rng.choice(2**self.n, size=shots, p=p)

# 專案示範：帶雜訊的 Grover（n=4）
qc = QuantumComputer(4, seed=42)
w = 11
f = lambda x: 1 if x == w else 0
O = np.diag(np.array([(-1)**f(x) for x in range(16)], dtype=complex))

def grover_iter(qc, O):
    qc._apply_1q(O, 0); qc._apply_1q(O, 1); qc._apply_1q(O, 2); qc._apply_1q(O, 3)
    # 擴散（簡化：用矩陣一次性完成）
    H4 = np.eye(1)
    for _ in range(4): H4 = np.kron(H4, qc.H)
    psi0 = H4 @ np.eye(16, dtype=complex)[:, 0]
    D = 2*np.outer(psi0, psi0.conj()) - np.eye(16)
    qc.state = D @ qc.state
    qc.gate_count += 1

for label, noise in [("無雜訊", 0.0), ("輕度雜訊 p=0.005", 0.005), ("重度雜訊 p=0.02", 0.02)]:
    qc = QuantumComputer(4, seed=1)
    for k in range(4): qc.h(k)
    for _ in range(3):                       # r ≈ π/4·√16 ≈ 3
        qc.state = O @ qc.state
        grover_iter(qc, O)
        qc.depolarize(noise)
    hits = np.mean(qc.measure(1000) == w)
    print(f"{label}: 找到解的頻率 = {hits:.1%}, 閘數 = {qc.gate_count}")