Python實例題：基于量子計算的優化算法實現（量子計算、優化理論）

Python實例題

題目

基于量子計算的優化算法實現（量子計算、優化理論）

問題描述

開發一個基于量子計算的優化算法實現，包含以下功能：

量子計算基礎：實現量子比特、量子門和量子電路
量子優化算法：實現量子近似優化算法 (QAOA) 和變分量子特征求解器 (VQE)
組合優化問題：解決旅行商問題 (TSP)、最大割問題 (Max-Cut) 等
量子 - 經典混合算法：設計并實現量子與經典算法的混合架構
性能評估：比較量子算法與經典算法在不同問題上的性能

解題思路

使用量子計算框架 (如 Qiskit、PennyLane) 構建量子電路
設計適合量子計算的優化問題表示方法
實現參數化量子電路和經典優化器的混合訓練
針對具體問題設計量子算法的損失函數和測量方法
在模擬環境和真實量子設備上測試算法性能

關鍵代碼框架

# 量子計算基礎實現
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import minimize
import networkx as nx# 量子比特狀態表示
class Qubit:def __init__(self, alpha=1, beta=0):# 初始化量子比特狀態 |ψ? = α|0? + β|1?norm = np.sqrt(alpha**2 + beta**2)self.state = np.array([alpha/norm, beta/norm], dtype=complex)def measure(self):# 測量量子比特，返回0或1probs = np.abs(self.state)**2return np.random.choice([0, 1], p=probs)def apply_gate(self, gate):# 應用量子門self.state = np.dot(gate, self.state)return self# 量子門定義
class QuantumGate:@staticmethoddef X():# Pauli-X門 (NOT門)return np.array([[0, 1], [1, 0]], dtype=complex)@staticmethoddef Y():# Pauli-Y門return np.array([[0, -1j], [1j, 0]], dtype=complex)@staticmethoddef Z():# Pauli-Z門return np.array([[1, 0], [0, -1]], dtype=complex)@staticmethoddef H():# Hadamard門return np.array([[1, 1], [1, -1]], dtype=complex) / np.sqrt(2)@staticmethoddef Rx(theta):# X旋轉門return np.array([[np.cos(theta/2), -1j*np.sin(theta/2)],[-1j*np.sin(theta/2), np.cos(theta/2)]], dtype=complex)@staticmethoddef Ry(theta):# Y旋轉門return np.array([[np.cos(theta/2), -np.sin(theta/2)],[np.sin(theta/2), np.cos(theta/2)]], dtype=complex)@staticmethoddef Rz(theta):# Z旋轉門return np.array([[np.exp(-1j*theta/2), 0],[0, np.exp(1j*theta/2)]], dtype=complex)@staticmethoddef CNOT():# 受控NOT門return np.array([[1, 0, 0, 0],[0, 1, 0, 0],[0, 0, 0, 1],[0, 0, 1, 0]], dtype=complex)# 量子電路
class QuantumCircuit:def __init__(self, num_qubits):self.num_qubits = num_qubits# 初始化全零狀態self.state = np.zeros(2**num_qubits, dtype=complex)self.state[0] = 1def apply_single_qubit_gate(self, gate, qubit_index):# 應用單量子比特門if qubit_index >= self.num_qubits:raise ValueError(f"Qubit index {qubit_index} out of range for {self.num_qubits} qubits")# 構建完整門矩陣full_gate = Nonefor i in range(self.num_qubits):if i == 0:if i == qubit_index:current_gate = gateelse:current_gate = np.eye(2, dtype=complex)else:if i == qubit_index:current_gate = np.kron(current_gate, gate)else:current_gate = np.kron(current_gate, np.eye(2, dtype=complex))# 應用門self.state = np.dot(current_gate, self.state)return selfdef apply_two_qubit_gate(self, gate, control_qubit, target_qubit):# 應用雙量子比特門if control_qubit >= self.num_qubits or target_qubit >= self.num_qubits:raise ValueError(f"Qubit index out of range for {self.num_qubits} qubits")# 這里簡化處理，僅支持CNOT門if not np.array_equal(gate, QuantumGate.CNOT()):raise ValueError("Only CNOT gate is supported for two-qubit operations in this implementation")# 構建完整CNOT門矩陣 (簡化實現，實際應使用更高效的方法)# 此處省略具體實現...return selfdef measure(self, shots=1):# 測量所有量子比特probs = np.abs(self.state)**2outcomes = []for _ in range(shots):outcome = np.random.choice(range(2**self.num_qubits), p=probs)# 轉換為二進制字符串表示binary = format(outcome, '0'+str(self.num_qubits)+'b')outcomes.append(binary)# 統計結果counts = {}for outcome in outcomes:if outcome in counts:counts[outcome] += 1else:counts[outcome] = 1return counts

# 量子近似優化算法(QAOA)
class QAOA:def __init__(self, graph, p=1, backend='simulator'):"""初始化QAOA算法參數:graph: 要解決的問題對應的圖p: QAOA電路深度backend: 量子計算后端 ('simulator' 或 'real')"""self.graph = graphself.p = pself.backend = backendself.num_qubits = graph.number_of_nodes()# 如果使用真實量子設備，這里需要配置相應的后端if backend == 'real':# 配置真實量子設備連接# 這里省略具體實現...passdef create_cost_hamiltonian(self, gamma):"""創建代價哈密頓量對應的量子門"""# 針對Max-Cut問題的代價哈密頓量gates = []for u, v in self.graph.edges():# 每個邊對應的ZZ門gates.append(('ZZ', u, v, gamma))return gatesdef create_mixer_hamiltonian(self, beta):"""創建混合哈密頓量對應的量子門"""# 混合哈密頓量使用X門gates = []for i in range(self.num_qubits):gates.append(('X', i, beta))return gatesdef create_qaoa_circuit(self, params):"""創建完整的QAOA量子電路"""# 分離參數gammas = params[:self.p]betas = params[self.p:]# 初始化量子電路circuit = QuantumCircuit(self.num_qubits)# 應用初始哈達瑪門，制備均勻疊加態for i in range(self.num_qubits):circuit.apply_single_qubit_gate(QuantumGate.H(), i)# 交替應用代價和混合哈密頓量for i in range(self.p):# 應用代價哈密頓量cost_gates = self.create_cost_hamiltonian(gammas[i])for gate_type, u, v, param in cost_gates:if gate_type == 'ZZ':# 簡化實現，實際應構建完整的ZZ門# 這里使用兩個CNOT和Rz門實現circuit.apply_two_qubit_gate(QuantumGate.CNOT(), u, v)circuit.apply_single_qubit_gate(QuantumGate.Rz(2*param), v)circuit.apply_two_qubit_gate(QuantumGate.CNOT(), u, v)# 應用混合哈密頓量mixer_gates = self.create_mixer_hamiltonian(betas[i])for gate_type, qubit, param in mixer_gates:if gate_type == 'X':circuit.apply_single_qubit_gate(QuantumGate.Rx(2*param), qubit)return circuitdef evaluate_cost(self, bitstring):"""評估給定比特串的代價函數值"""cost = 0for u, v in self.graph.edges():# 如果u和v的比特值不同，則邊被切割if bitstring[u] != bitstring[v]:cost += 1return costdef expectation_value(self, params, shots=1000):"""計算代價函數的期望值"""# 創建QAOA電路circuit = self.create_qaoa_circuit(params)# 執行測量counts = circuit.measure(shots)# 計算期望值exp_val = 0total_shots = sum(counts.values())for bitstring, count in counts.items():cost = self.evaluate_cost(bitstring)exp_val += cost * (count / total_shots)return -exp_val  # 因為我們要最大化代價函數，所以取負值進行最小化def optimize_parameters(self, initial_params=None, method='COBYLA'):"""優化QAOA參數"""if initial_params is None:# 隨機初始化參數initial_params = np.random.uniform(0, np.pi, 2 * self.p)# 使用經典優化器優化參數result = minimize(self.expectation_value,initial_params,method=method,options={'maxiter': 1000})return result.x, -result.fun  # 返回優化后的參數和最大期望值def find_max_cut(self, shots=1000):"""找到最大割"""# 優化參數optimal_params, max_exp_val = self.optimize_parameters()# 使用優化后的參數運行電路circuit = self.create_qaoa_circuit(optimal_params)counts = circuit.measure(shots)# 找到最可能的比特串max_cut_bitstring = max(counts, key=counts.get)return max_cut_bitstring, counts, max_exp_val

# 主程序：解決最大割問題
def main():# 創建一個圖n = 5  # 節點數graph = nx.complete_graph(n)# 可視化圖plt.figure(figsize=(8, 6))pos = nx.spring_layout(graph)nx.draw(graph, pos, with_labels=True, node_color='lightblue', node_size=500)plt.title("Original Graph")plt.show()# 初始化QAOAp = 3  # QAOA電路深度qaoa = QAOA(graph, p=p)# 找到最大割max_cut_bitstring, counts, max_exp_val = qaoa.find_max_cut(shots=1000)print(f"最大割比特串: {max_cut_bitstring}")print(f"估計的最大割值: {max_exp_val}")# 可視化結果plt.figure(figsize=(8, 6))# 分離節點為兩個集合partition0 = [i for i, bit in enumerate(max_cut_bitstring) if bit == '0']partition1 = [i for i, bit in enumerate(max_cut_bitstring) if bit == '1']# 繪制節點nx.draw_networkx_nodes(graph, pos, nodelist=partition0, node_color='lightblue', node_size=500)nx.draw_networkx_nodes(graph, pos, nodelist=partition1, node_color='pink', node_size=500)# 繪制邊edges_cut = [(u, v) for u, v in graph.edges() if (u in partition0 and v in partition1) or (u in partition1 and v in partition0)]edges_not_cut = [(u, v) for u, v in graph.edges() if (u in partition0 and v in partition0) or (u in partition1 and v in partition1)]nx.draw_networkx_edges(graph, pos, edgelist=edges_cut, edge_color='red', width=2)nx.draw_networkx_edges(graph, pos, edgelist=edges_not_cut, edge_color='gray', width=1)# 繪制節點標簽nx.draw_networkx_labels(graph, pos)plt.title(f"Max-Cut Partition (Value: {max_exp_val:.2f})")plt.axis('off')plt.show()# 可視化測量結果分布plt.figure(figsize=(10, 6))bitstrings = list(counts.keys())probabilities = [count / 1000 for count in counts.values()]# 按概率排序sorted_indices = np.argsort(probabilities)[::-1]bitstrings = [bitstrings[i] for i in sorted_indices]probabilities = [probabilities[i] for i in sorted_indices]# 只顯示前10個最可能的結果if len(bitstrings) > 10:bitstrings = bitstrings[:10]probabilities = probabilities[:10]plt.bar(bitstrings, probabilities)plt.xlabel('Bitstrings')plt.ylabel('Probability')plt.title('Measurement Outcomes')plt.xticks(rotation=90)plt.tight_layout()plt.show()if __name__ == "__main__":main()