【量子计算探秘】QAOA：用量子魔法破解优化难题

import java.util.*;

// 量子比特状态
class Qubit {
    private Complex amplitude0; // |0>态的振幅
    private Complex amplitude1; // |1>态的振幅

    public Qubit() {
        // 初始化为|0>态
        this.amplitude0 = new Complex(1, 0);
        this.amplitude1 = new Complex(0, 0);
    }

    public Qubit(Complex amplitude0, Complex amplitude1) {
        this.amplitude0 = amplitude0;
        this.amplitude1 = amplitude1;
    }

    // 应用Hadamard门
    public void applyHadamard() {
        Complex newAmp0 = amplitude0.add(amplitude1).scale(1.0 / Math.sqrt(2));
        Complex newAmp1 = amplitude0.subtract(amplitude1).scale(1.0 / Math.sqrt(2));
        amplitude0 = newAmp0;
        amplitude1 = newAmp1;
    }

    // 应用Pauli-X门
    public void applyPauliX() {
        Complex temp = amplitude0;
        amplitude0 = amplitude1;
        amplitude1 = temp;
    }

    // 应用参数化X旋转门
    public void applyRX(double theta) {
        Complex cos = new Complex(Math.cos(theta / 2), 0);
        Complex isin = new Complex(0, Math.sin(theta / 2));
        
        Complex newAmp0 = amplitude0.multiply(cos).subtract(amplitude1.multiply(isin));
        Complex newAmp1 = amplitude1.multiply(cos).subtract(amplitude0.multiply(isin));
        
        amplitude0 = newAmp0;
        amplitude1 = newAmp1;
    }

    // 应用参数化Z旋转门
    public void applyRZ(double gamma) {
        Complex phase = new Complex(Math.cos(gamma), Math.sin(gamma));
        amplitude1 = amplitude1.multiply(phase);
    }

    // 测量量子比特
    public int measure() {
        double prob0 = amplitude0.modSquared();
        return Math.random() < prob0 ? 0 : 1;
    }

    @Override
    public String toString() {
        return amplitude0 + "|0> + " + amplitude1 + "|1>";
    }
}

// 复数类
class Complex {
    private double real;
    private double imag;

    public Complex(double real, double imag) {
        this.real = real;
        this.imag = imag;
    }

    public Complex add(Complex other) {
        return new Complex(real + other.real, imag + other.imag);
    }

    public Complex subtract(Complex other) {
        return new Complex(real - other.real, imag - other.imag);
    }

    public Complex multiply(Complex other) {
        return new Complex(
            real * other.real - imag * other.imag,
            real * other.imag + imag * other.real
        );
    }

    public Complex scale(double factor) {
        return new Complex(real * factor, imag * factor);
    }

    public double modSquared() {
        return real * real + imag * imag;
    }

    @Override
    public String toString() {
        if (imag == 0) return String.format("%.4f", real);
        if (real == 0) return String.format("%.4fi", imag);
        if (imag > 0) return String.format("%.4f + %.4fi", real, imag);
        return String.format("%.4f - %.4fi", real, Math.abs(imag));
    }
}

// 量子系统
class QuantumSystem {
    private List<Qubit> qubits;

    public QuantumSystem(int numQubits) {
        this.qubits = new ArrayList<>();
        for (int i = 0; i < numQubits; i++) {
            qubits.add(new Qubit());
        }
    }

    public void applyHadamard(int qubitIndex) {
        qubits.get(qubitIndex).applyHadamard();
    }

    public void applyRX(int qubitIndex, double theta) {
        qubits.get(qubitIndex).applyRX(theta);
    }

    public void applyRZ(int qubitIndex, double gamma) {
        qubits.get(qubitIndex).applyRZ(gamma);
    }

    public void applyCZ(int qubitIndex1, int qubitIndex2) {
        // 简化实现，实际需要两比特操作
        // 这里仅用于演示，不反映真实量子计算
    }

    public List<Integer> measureAll() {
        List<Integer> results = new ArrayList<>();
        for (Qubit qubit : qubits) {
            results.add(qubit.measure());
        }
        return results;
    }

    public int getNumQubits() {
        return qubits.size();
    }
}

// 图结构，用于表示最大割问题
class Graph {
    private int numVertices;
    private List<List<Integer>> adjacencyList;

    public Graph(int numVertices) {
        this.numVertices = numVertices;
        this.adjacencyList = new ArrayList<>();
        for (int i = 0; i < numVertices; i++) {
            adjacencyList.add(new ArrayList<>());
        }
    }

    public void addEdge(int u, int v) {
        adjacencyList.get(u).add(v);
        adjacencyList.get(v).add(u);
    }

    public int getNumVertices() {
        return numVertices;
    }

    public List<Integer> getNeighbors(int vertex) {
        return adjacencyList.get(vertex);
    }
}

// QAOA算法实现
class QAOA {
    private Graph graph;
    private int p; // QAOA深度参数

    public QAOA(Graph graph, int p) {
        this.graph = graph;
        this.p = p;
    }

    // 运行QAOA算法
    public List<Integer> run(int numShots) {
        // 初始化量子系统
        QuantumSystem qs = new QuantumSystem(graph.getNumVertices());
        
        // 应用初始Hadamard门，创建均匀叠加态
        for (int i = 0; i < qs.getNumQubits(); i++) {
            qs.applyHadamard(i);
        }
        
        // 随机初始化参数
        double[] gammas = new double[p];
        double[] betas = new double[p];
        for (int i = 0; i < p; i++) {
            gammas[i] = Math.random() * Math.PI;
            betas[i] = Math.random() * Math.PI / 2;
        }
        
        // 应用p层QAOA电路
        for (int layer = 0; layer < p; layer++) {
            // 应用问题哈密顿量演化
            for (int u = 0; u < graph.getNumVertices(); u++) {
                for (int v : graph.getNeighbors(u)) {
                    if (u < v) { // 避免重复
                        // 应用参数化Z-Z门
                        qs.applyCZ(u, v);
                        qs.applyRZ(u, gammas[layer]);
                        qs.applyRZ(v, gammas[layer]);
                        qs.applyCZ(u, v);
                    }
                }
            }
            
            // 应用混合哈密顿量演化
            for (int i = 0; i < qs.getNumQubits(); i++) {
                qs.applyRX(i, betas[layer]);
            }
        }
        
        // 测量并收集结果
        Map<String, Integer> results = new HashMap<>();
        for (int shot = 0; shot < numShots; shot++) {
            List<Integer> measurement = qs.measureAll();
            String resultStr = measurement.toString();
            results.put(resultStr, results.getOrDefault(resultStr, 0) + 1);
        }
        
        // 找到最频繁出现的结果
        String mostFrequentResult = "";
        int maxCount = 0;
        for (Map.Entry<String, Integer> entry : results.entrySet()) {
            if (entry.getValue() > maxCount) {
                maxCount = entry.getValue();
                mostFrequentResult = entry.getKey();
            }
        }
        
        // 解析结果
        List<Integer> solution = new ArrayList<>();
        mostFrequentResult = mostFrequentResult.substring(1, mostFrequentResult.length() - 1);
        String[] parts = mostFrequentResult.split(", ");
        for (String part : parts) {
            solution.add(Integer.parseInt(part));
        }
        
        return solution;
    }
    
    // 计算割值
    public int computeCutValue(List<Integer> solution) {
        int cutValue = 0;
        for (int u = 0; u < graph.getNumVertices(); u++) {
            for (int v : graph.getNeighbors(u)) {
                if (u < v && solution.get(u) != solution.get(v)) {
                    cutValue++;
                }
            }
        }
        return cutValue;
    }
}

// 示例使用
public class QAOADemo {
    public static void main(String[] args) {
        // 创建一个简单的图
        Graph graph = new Graph(4);
        graph.addEdge(0, 1);
        graph.addEdge(1, 2);
        graph.addEdge(2, 3);
        graph.addEdge(3, 0);
        graph.addEdge(0, 2);
        
        // 创建并运行QAOA
        QAOA qaoa = new QAOA(graph, 2); // p=2
        List<Integer> solution = qaoa.run(1000);
        
        System.out.println("QAOA找到的解: " + solution);
        System.out.println("割值: " + qaoa.computeCutValue(solution));
        
        // 对比经典解法
        List<Integer> classicalSolution = findMaxCutClassical(graph);
        System.out.println("经典解法找到的解: " + classicalSolution);
        System.out.println("割值: " + qaoa.computeCutValue(classicalSolution));
    }
    
    // 简单的经典解法（仅用于演示，实际应使用更高效的算法）
    private static List<Integer> findMaxCutClassical(Graph graph) {
        int numVertices = graph.getNumVertices();
        int maxCut = 0;
        List<Integer> bestSolution = new ArrayList<>();
        
        // 枚举所有可能的划分
        for (int mask = 0; mask < (1 << numVertices); mask++) {
            List<Integer> solution = new ArrayList<>();
            for (int i = 0; i < numVertices; i++) {
                solution.add((mask >> i) & 1);
            }
            
            int cutValue = 0;
            for (int u = 0; u < numVertices; u++) {
                for (int v : graph.getNeighbors(u)) {
                    if (u < v && solution.get(u) != solution.get(v)) {
                        cutValue++;
                    }
                }
            }
            
            if (cutValue > maxCut) {
                maxCut = cutValue;
                bestSolution = solution;
            }
        }
        
        return bestSolution;
    }
}