ComplexSimulation.qs

namespace Quantum.ComplexSimulation {
    open Microsoft.Quantum.Canon;
    open Microsoft.Quantum.Intrinsic;
    open Microsoft.Quantum.Measurement;

    // Quantum Fourier Transform (QFT) - A key part of the algorithm for frequency analysis
    operation QFT(qubits: Qubit[]) : Unit {
        let n = Length(qubits);
        
        for (i in 0..n-1) {
            // Apply Hadamard gate to each qubit
            H(qubits[i]);
            
            for (j in i+1..n-1) {
                // Apply controlled rotation for each qubit pair
                let angle = 1.0 / (2.0 ^ (j - i + 1));
                CR(qubits[j], qubits[i], angle * PI);
            }
        }
        
        // Apply final Hadamard gate
        ApplyToEach(H, qubits);
        
        // Reverse the qubit order (to make the result readable)
        ApplyToEach(X, qubits);
    }

    // Quantum Simulation of a complex system with differential equations (e.g., Schrödinger equation)
    operation QuantumDifferentialEquationSolver() : Unit {
        // Simulating a simple quantum system (Schrödinger equation)
        mutable qubits = Qubit[5];  // Number of qubits for the system state
        
        using (qubits) {
            // Prepare the system in a superposition state
            ApplyToEach(H, qubits);
            
            // Simulate a quantum evolution (solving the Schrödinger equation)
            for (i in 0..4) {
                // Apply a series of rotations to simulate evolution
                let angle = 2.0 * PI / (2.0 ^ (i + 1));
                Rz(angle, qubits[i]);  // Rotation around the Z-axis
                Ry(angle, qubits[i]);  // Rotation around the Y-axis
            }
            
            // Perform a Quantum Fourier Transform to analyze the frequencies of the system
            QFT(qubits);
            
            // Measure the qubits to observe the final state
            let result = M(qubits[0]);
            if (result == Zero) {
                Message("The system evolved into state 0.");
            } else {
                Message("The system evolved into state 1.");
            }
        }
    }

    // Main function to run the quantum system simulation
    operation RunComplexQuantumSimulation() : Unit {
        // Run the quantum differential equation solver simulation
        QuantumDifferentialEquationSolver();
    }
}