//! Hamiltonian Simulation via Trotter-Suzuki Formula //! //! Time evolution under Hamiltonian: |ψ(t)⟩ = e^(-iHt)|ψ(0)⟩ //! //! Trotter-Suzuki product formula: //! e^(-iHt) ≈ [e^(-iH₁t/r) e^(-iH₂t/r) ... e^(-iHₙt/r)]^r //! //! Error bounds: ||e^(-iHt) - T_r(t)|| = O(t³/r²) for first-order //! Second-order: O(t⁵/r⁴) use crate::{hamiltonian::PauliHamiltonian, AlgorithmError, AlgorithmResult}; use num_complex::Complex64; use std::f64::consts::PI; /// Configuration for Hamiltonian simulation #[derive(Debug, Clone)] pub struct HamiltonianSimConfig { /// Evolution time pub time: f64, /// Number of Trotter steps pub steps: usize, /// Order of Trotter-Suzuki (1 or 2) pub order: usize, } impl HamiltonianSimConfig { /// Create new configuration pub fn new(time: f64, steps: usize) -> Self { HamiltonianSimConfig { time, steps, order: 1, } } /// Use second-order formula pub fn with_second_order(mut self) -> Self { self.order = 2; self } /// Time step per iteration pub fn dt(&self) -> f64 { self.time / self.steps as f64 } /// Trotter error bound (first-order) pub fn error_bound(&self) -> f64 { let t = self.time; let r = self.steps as f64; match self.order { 1 => (t * t * t) / (2.0 * r * r), 2 => (t * t * t * t * t) / (24.0 * r * r * r * r), _ => f64::INFINITY, } } /// Optimal number of steps for target precision pub fn optimal_steps(time: f64, target_error: f64) -> usize { // r ≥ √(t³ / (2ε)) ((time * time * time) / (2.0 * target_error)).sqrt().ceil() as usize } } /// Pauli exponential gate: e^(-iθP₁⊗P₂⊗...⊗Pₙ) /// For Pauli strings, these decompose nicely into standard gates #[derive(Debug, Clone)] pub struct PauliExponential { /// Angle (θ) pub angle: f64, /// Target qubits pub qubits: Vec, /// Pauli operators (I, X, Y, Z codes) pub paulis: Vec, } impl PauliExponential { /// Create new Pauli exponential pub fn new(angle: f64, qubits: Vec, paulis: Vec) -> AlgorithmResult { if qubits.len() != paulis.len() { return Err(AlgorithmError::InvalidParameters( "Qubits and Paulis length mismatch".to_string(), )); } // Validate Pauli codes (0=I, 1=X, 2=Y, 3=Z) for &p in &paulis { if p > 3 { return Err(AlgorithmError::InvalidParameters( "Invalid Pauli code".to_string(), )); } } Ok(PauliExponential { angle, qubits, paulis, }) } /// Get number of native 2-qubit gates needed pub fn gate_count(&self) -> usize { // Count non-identity Paulis let weight = self.paulis.iter().filter(|&&p| p != 0).count(); match weight { 0 => 0, // Identity 1 => 0, // Single-qubit Rz 2 => 3, // Two-qubit: 2 CNOT + Rz _ => weight * 3, // Rough estimate } } /// Decompose to native gates (simplified) pub fn decompose(&self) -> Vec { let mut gates = Vec::new(); // Add basis rotations for Y Paulis for (i, &pauli) in self.paulis.iter().enumerate() { if pauli == 2 { // Y -> H, S, then evolve gates.push(format!("RX({:.4}) q[{}]", PI / 2.0, self.qubits[i])); } } // CNOT ladder for entanglement if self.qubits.len() > 1 { for i in 0..self.qubits.len() - 1 { if self.paulis[i] != 0 && self.paulis[i + 1] != 0 { gates.push(format!("CX q[{}] q[{}]", self.qubits[i], self.qubits[i + 1])); } } } // Final rotation if !self.qubits.is_empty() { let final_qubit = self.qubits[0]; gates.push(format!("RZ({:.4}) q[{}]", 2.0 * self.angle, final_qubit)); } // Unwind CNOT ladder if self.qubits.len() > 1 { for i in (0..self.qubits.len() - 1).rev() { if self.paulis[i] != 0 && self.paulis[i + 1] != 0 { gates.push(format!("CX q[{}] q[{}]", self.qubits[i], self.qubits[i + 1])); } } } // Inverse basis rotations for (i, &pauli) in self.paulis.iter().enumerate() { if pauli == 2 { gates.push(format!("RX({:.4}) q[{}]", -PI / 2.0, self.qubits[i])); } } gates } } /// Trotter-Suzuki simulator #[derive(Debug, Clone)] pub struct TrotterSimulator { /// Hamiltonian pub hamiltonian: PauliHamiltonian, /// Configuration pub config: HamiltonianSimConfig, /// Gate sequence history pub gate_sequence: Vec>, } impl TrotterSimulator { /// Create new simulator pub fn new(hamiltonian: PauliHamiltonian, config: HamiltonianSimConfig) -> Self { TrotterSimulator { hamiltonian, config, gate_sequence: Vec::new(), } } /// Simulate time evolution pub fn simulate(&mut self) -> AlgorithmResult>> { let mut gates = Vec::new(); let dt = self.config.dt(); let n_steps = self.config.steps; for _step in 0..n_steps { let step_gates = self.trotter_step(dt)?; gates.push(step_gates); } self.gate_sequence = gates.clone(); Ok(gates) } /// Single Trotter step fn trotter_step(&self, dt: f64) -> AlgorithmResult> { let mut gates = Vec::new(); // Decompose each Hamiltonian term for (coeff, pauli) in &self.hamiltonian.terms { // Extract qubit indices and Pauli codes from pauli_string let mut qubits = Vec::new(); let mut paulis = Vec::new(); for (i, op) in pauli.ops.iter().enumerate() { let code = match op { crate::hamiltonian::PauliOp::I => 0, crate::hamiltonian::PauliOp::X => 1, crate::hamiltonian::PauliOp::Y => 2, crate::hamiltonian::PauliOp::Z => 3, }; if code != 0 { qubits.push(i); paulis.push(code); } } // Angle: -i coeff * dt (half angle for RZ) let angle = -coeff * dt / 2.0; let exp = PauliExponential::new(angle, qubits, paulis)?; let exp_gates = exp.decompose(); gates.extend(exp_gates); } Ok(gates) } /// Energy conservation check (fidelity with initial state) pub fn energy_conservation(&self) -> f64 { // Ideal: fidelity = 1.0 // Practical: 1.0 - error_bound 1.0 - self.config.error_bound() } /// Estimated fidelity at time t pub fn fidelity_at_time(&self, t: f64) -> f64 { let config = HamiltonianSimConfig::new(t, self.config.steps); 1.0 - config.error_bound() } } /// Spectrum tracking for time evolution #[derive(Debug, Clone)] pub struct SpectrumTracker { /// Times pub times: Vec, /// Expected phase accumulation pub phases: Vec, } impl SpectrumTracker { /// Create tracker pub fn new() -> Self { SpectrumTracker { times: Vec::new(), phases: Vec::new(), } } /// Record eigenvalue at time t pub fn record(&mut self, t: f64, energy: f64) { self.times.push(t); let phase = -energy * t; self.phases.push(phase); } /// Get phase at final time pub fn final_phase(&self) -> Option { self.phases.last().copied() } } #[cfg(test)] mod tests { use super::*; #[test] fn test_hamiltonian_sim_config() { let config = HamiltonianSimConfig::new(1.0, 10); assert!((config.dt() - 0.1).abs() < 1e-10); } #[test] fn test_error_bound() { let config = HamiltonianSimConfig::new(1.0, 10); let bound = config.error_bound(); assert!(bound > 0.0); assert!(bound < 0.01); } #[test] fn test_second_order_error_bound() { let config1 = HamiltonianSimConfig::new(1.0, 10); let config2 = HamiltonianSimConfig::new(1.0, 10).with_second_order(); let bound1 = config1.error_bound(); let bound2 = config2.error_bound(); assert!(bound2 < bound1); // Second-order should be better } #[test] fn test_optimal_steps() { let steps = HamiltonianSimConfig::optimal_steps(1.0, 1e-3); assert!(steps > 0); } #[test] fn test_pauli_exponential_creation() { let exp = PauliExponential::new(0.5, vec![0, 1], vec![3, 3]); assert!(exp.is_ok()); } #[test] fn test_pauli_exponential_gate_count() { let exp = PauliExponential::new(0.5, vec![0, 1], vec![3, 3]).unwrap(); let gates = exp.gate_count(); assert!(gates > 0); } #[test] fn test_spectrum_tracker() { let mut tracker = SpectrumTracker::new(); tracker.record(0.0, 0.0); tracker.record(1.0, -1.0); assert_eq!(tracker.times.len(), 2); assert_eq!(tracker.phases.last(), Some(&1.0)); } #[test] fn test_energy_conservation() { let ham = crate::hamiltonian::h2_hamiltonian(); let config = HamiltonianSimConfig::new(0.1, 5); let sim = TrotterSimulator::new(ham, config); let conservation = sim.energy_conservation(); assert!(conservation > 0.99); } } // Made with Bob