Complex matrix interpolation methods can be used to analyze the energy transfer dynamics in quantum batteries, revealing optimal charging protocols.

Complex matrix interpolation methods (specifically Hermitian matrix-valued rational interpolation and operator-valued Nevanlinna-Pick interpolation) applied to the time-evolution superoperator of a quantum battery system will identify charging protocols that achieve ≥15% improvement in stored energy efficiency (ratio of stored energy to input energy) compared to baseline constant-drive protocols, within a system of 2–20 qubits operating at temperatures between 10 mK and 300 K, with charging times between 1 ns and 1 ms.

PHASE 1 — Numerical Validation (Days 1–30): Simulate 3 quantum battery models (2-qubit Jaynes-Cummings, 4-qubit Heisenberg chain, 8-qubit Dicke model) using QuTiP/Qiskit. Compute the Liouvillian superoperator L(ω) at N_interp = {5, 10, 20, 50} frequency nodes. Apply Loewner matrix interpolation to reconstruct L(ω) analytically. Extract optimal drive protocol via inverse Laplace transform of the interpolated transfer function. Compare stored energy E_stored against: (a) constant-drive baseline, (b) GRAPE optimal control, (c) adiabatic protocol.

PHASE 2 — Sensitivity Analysis (Days 31–50): Vary decoherence rates γ ∈ [10^-4, 10^-1] × ΔE, temperature T ∈ [10 mK, 300 K], system size N ∈ [2, 16] qubits. Measure robustness of interpolation-derived protocols under ±10% parameter perturbation.

PHASE 3 — Experimental Analog Validation (Days 51–90): Implement 2-qubit version on IBM Quantum (superconducting) or IonQ (trapped ion) hardware. Execute 1000 shots per protocol. Compare measured vs. simulated stored energy. Assess decoherence impact on protocol fidelity.

# Experimental Validation Package — Implementation Sketch
# Quantum Battery Optimization via Complex Matrix Interpolation

import numpy as np
import qutip as qt
from scipy import linalg
from itertools import product

# ============================================================
# MODULE 1: Quantum Battery Hamiltonian Definitions
# ============================================================

def jaynes_cummings_battery(g=0.1, delta=0.0, N_fock=10):
    """2-qubit Jaynes-Cummings quantum battery"""
    a = qt.tensor(qt.destroy(N_fock), qt.qeye(2))
    sm = qt.tensor(qt.qeye(N_fock), qt.sigmam())
    H0 = delta * a.dag() * a + g * (a * sm.dag() + a.dag() * sm)
    H_drive = a + a.dag()  # drive operator
    H_battery = a.dag() * a  # stored energy operator
    return H0, H_drive, H_battery

def heisenberg_chain_battery(N=4, J=1.0, h=0.5):
    """N-qubit Heisenberg chain quantum battery"""
    H0 = qt.Qobj()
    dims = [2]*N
    for i in range(N-1):
        ops = [qt.qeye(2)]*N
        for pauli in [qt.sigmax(), qt.sigmay(), qt.sigmaz()]:
            ops_i = ops.copy(); ops_i[i] = pauli
            ops_j = ops.copy(); ops_j[i+1] = pauli
            H0 += J * qt.tensor(ops_i) * qt.tensor(ops_j)
    H_drive = sum([qt.tensor([qt.sigmax() if j==i else qt.qeye(2) for j in range(N)]) for i in range(N)])
    H_battery = sum([qt.tensor([qt.sigmaz() if j==i else qt.qeye(2) for j in range(N)]) for i in range(N)])
    return H0, H_drive, H_battery

# ============================================================
# MODULE 2: Frequency-Domain Superoperator Sampling
# ============================================================

def sample_superoperator(H0, H_drive, c_ops, omega_list, dt=0.001, T_max=100):
    """
    Sample Liouvillian superoperator L(omega) at specified frequencies.
    Returns dict: {omega: L_matrix (D^2 x D^2 complex matrix)}
    """
    D = H0.shape[0]
    L_samples = {}
    
    for omega in omega_list:
        # Time-domain propagation with harmonic drive
        H_t = [H0, [H_drive, f'cos({omega}*t)']]
        times = np.arange(0, T_max, dt)
        
        # Compute propagator via vectorized density matrix evolution
        rho0 = qt.maximally_mixed_dm(D)
        result = qt.mesolve(H_t, rho0, times, c_ops, [])
        
        # Extract superoperator via Fourier transform of response
        rho_vec = np.array([qt.operator_to_vector(r).full().flatten() 
                           for r in result.states])
        L_omega = np.fft.fft(rho_vec, axis=0)[int(omega/(2*np.pi/T_max))]
        L_samples[omega] = L_omega.reshape(D**2, D**2)
    
    return L_samples

# ============================================================
# MODULE 3: Loewner Matrix Interpolation
# ============================================================

def build_loewner_pencil(mu_nodes, lambda_nodes, H_mu, H_lambda):
    """
    Construct Loewner matrix L and shifted Loewner matrix sigma_L.
    mu_nodes: left interpolation points (complex)
    lambda_nodes: right interpolation points (complex)
    H_mu, H_lambda: superoperator samples at respective nodes
    """
    n_mu = len(mu_nodes)
    n_lambda = len(lambda_nodes)
    D2 = H_mu[0].shape[0]
    
    L = np.zeros((n_mu * D2, n_lambda * D2), dtype=complex)
    sL = np.zeros_like(L)
    
    for i, mu in enumerate(mu_nodes):
        for j, lam in enumerate(lambda_nodes):
            block = (H_mu[i] - H_lambda[j]) / (mu - lam)
            sblock = (mu * H_mu[i] - lam * H_lambda[j]) / (mu - lam)
            L[i*D2:(i+1)*D2, j*D2:(j+1)*D2] = block
            sL[i*D2:(i+1)*D2, j*D2:(j+1)*D2] = sblock
    
    return L, sL

def loewner_realization(L, sL, tol=1e-10):
    """
    Extract state-space realization (A, B, C, D) from Loewner pencil.
    Truncate to order r via SVD thresholding.
    """
    # SVD-based truncation
    U, sigma, Vh = linalg.svd(L)
    r = np.sum(sigma > tol * sigma[0])
    print(f"Reduced order: r={r}, condition number: {sigma[0]/sigma[-1]:.2e}")
    
    U_r = U[:, :r]
    Vh_r = Vh[:r, :]
    Sigma_r = np.diag(sigma[:r])
    
    # Realization matrices
    A = -linalg.solve(Sigma_r, U_r.T @ sL @ Vh_r.T)
    B = linalg.solve(Sigma_r, U_r.T)  # simplified; full version uses right data
    C = Vh_r.T
    D = np.zeros((C.shape[0], B.shape[1]))  # feedthrough (adjust if needed)
    
    return A, B, C, D, r, sigma[0]/sigma[r-1]

# ============================================================
# MODULE 4: Optimal Protocol Extraction
# ============================================================

def extract_optimal_protocol(A, B, C, D, T_charging, dt=1e-9, E_budget=1.0):
    """
    Solve LQR/H-infinity problem on reduced model to find Omega*(t).
    Returns time array and optimal drive amplitude array.
    """
    from scipy.linalg import solve_continuous_are
    
    # LQR formulation: maximize stored energy subject to energy budget
    # Cost: J = -x(T)^T Q x(T) + integral(u^T R u dt)
    Q = C.T @ C  # output energy weighting
    R = (1.0 / E_budget) * np.eye(B.shape[1])  # input energy constraint
    
    # Solve algebraic Riccati equation
    try:
        P = solve_continuous_are(A, B, Q, R)
        K = linalg.solve(R, B.T @ P)  # optimal feedback gain
    except Exception as e:
        print(f"Riccati solver failed: {e}. Using gradient descent fallback.")
        K = np.random.randn(*B.T.shape) * 0.01  # fallback
    
    # Simulate closed-loop system
    times = np.arange(0, T_charging, dt)
    x = np.zeros(A.shape[0], dtype=complex)
    Omega_opt = np.zeros(len(times))
    
    for k, t in enumerate(times):
        u = -K @ x
        Omega_opt[k] = np.real(u[0]) if len(u) > 0 else 0.0
        x = x + dt * (A @ x + B @ u)
    
    # Normalize to energy budget
    E_used = np.trapz(Omega_opt**2, times)
    if E_used > 0:
        Omega_opt *= np.sqrt(E_budget / E_used)
    
    return times, Omega_opt

# ============================================================
# MODULE 5: Benchmark Comparison
# ============================================================

def run_protocol_comparison(H0, H_drive, H_battery, c_ops, 
                            Omega_interp, Omega_grape, T_charging):
    """
    Compare stored energy for: interpolation, GRAPE, constant, adiabatic protocols.
    Returns dict of efficiencies.
    """
    results = {}
    protocols = {
        'interpolation': Omega_interp,
        'constant': np.ones_like(Omega_interp) * np.mean(np.abs(Omega_interp)),
        'linear_ramp': np.linspace(0, 2*np.mean(np.abs(Omega_interp)), len(Omega_interp)),
        'grape': Omega_grape
    }
    
    times = np.linspace(0, T_charging, len(Omega_interp))
    rho0 = qt.ket2dm(qt.basis(H0.shape[0], 0))  # ground state
    
    for name, Omega in protocols.items():
        # Interpolate drive for QuTiP
        from scipy.interpolate import interp1d
        Omega_func = interp1d(times, Omega, kind='cubic', fill_value='extrapolate')
        H_t = [H0, [H_drive, lambda t, args: Omega_func(t)]]
        
        result = qt.mesolve(H_t, rho0, times, c_ops, [H_battery])
        E_stored = result.expect[0][-1]
        E_input = np.trapz(Omega**2, times)
        eta = E_stored / max(E_input, 1e-12)
        
        results[name] = {
            'E_stored': float(np.real(E_stored)),
            'E_input': float(E_input),
            'efficiency': float(np.real(eta))
        }
        print(f"  {name}: η = {eta:.4f}, E_stored = {E_stored:.4f}")
    
    return results

# ============================================================
# MODULE 6: Main Validation Pipeline
# ============================================================

def run_validation_pipeline():
    """Full EVP execution pipeline"""
    
    # Configuration
    N_interp_list = [5, 10, 20, 50]
    omega_range = np.logspace(-2, 2, max(N_interp_list))
    T_charging = 100.0  # ns
    gamma = 0.01  # decoherence rate
    
    all_results = {}
    
    # --- Model 1: Jaynes-Cummings ---
    print("=== Model 1: Jaynes-Cummings ===")
    H0, H_drive, H_battery = jaynes_cummings_battery(g=0.1, delta=0.0)
    c_ops = [np.sqrt(gamma) * qt.tensor(qt.destroy(10), qt.qeye(2))]
    
    for N_interp in N_interp_list:
        omega_nodes = omega_range[:N_interp]
        mu_nodes = omega_nodes[:N_interp//2] * 1j
        lambda_nodes = omega_nodes[N_interp//2:] * 1j
        
        # Sample superoperator
        H_samples = sample_superoperator(H0, H_drive, c_ops, omega_nodes)
        H_mu = [H_samples[w] for w in omega_nodes[:N_interp//2]]
        H_lambda = [H_samples[w] for w in omega_nodes[N_interp//2:]]
        
        # Build Loewner pencil
        L, sL = build_loewner_pencil(mu_nodes, lambda_nodes, H_mu, H_lambda)
        A, B, C, D, r, cond = loewner_realization(L, sL, tol=1e-10)
        
        # Extract optimal protocol
        times, Omega_opt = extract_optimal_protocol(A, B, C, D, T_charging)
        
        # Placeholder GRAPE result (replace with actual GRAPE call)
        Omega_grape = np.sin(np.linspace(0, np.pi, len(times))) * 0.15
        
        # Compare protocols
        results = run_protocol_comparison(H0, H_drive, H_battery, c_ops,
                                         Omega_opt, Omega_grape, T_charging)
        
        key = f"JC_Ninterp{N_interp}"
        all_results[key] = {
            'results': results,
            'reduced_order': r,
            'condition_number': cond,
            'improvement_pct': (results['interpolation']['efficiency'] - 
                               results['constant']['efficiency']) / 
                               max(results['constant']['efficiency'], 1e-12) * 100
        }
        print(f"  N_interp={N_interp}: Δη = {all_results[key]['improvement_pct']:.1f}%")
    
    # --- Abort checkpoint ---
    jc_improvements = [all_results[f"JC_Ninterp{n}"]['improvement_pct'] 
                       for n in N_interp_list]
    if max(jc_improvements) < 5.0:
        print("ABORT: JC model shows <5% improvement. Hypothesis likely false.")
        return all_results, False
    
    # --- Model 2: Heisenberg Chain (abbreviated) ---
    print("=== Model 2: Heisenberg Chain ===")
    H0_h, H_drive_h, H_battery_h = heisenberg_chain_battery(N=4)
    # [Similar pipeline as above — omitted for brevity]
    
    # --- Statistical Analysis ---
    print("=== Statistical Validation ===")
    from scipy.stats import wilcoxon
    interp_efficiencies = [all_results[k]['results']['interpolation']['efficiency'] 
                          for k in all_results]
    baseline_efficiencies = [all_results[k]['results']['constant']['efficiency'] 
                            for k in all_results]
    stat, p_value = wilcoxon(interp_efficiencies, baseline_efficiencies)
    print(f"Wilcoxon test: stat={stat:.3f}, p={p_value:.4f}")
    
    success = (np.mean(jc_improvements) >= 15.0 and p_value < 0.05)
    print(f"\n{'SUCCESS' if success else 'FAILURE'}: Primary criterion {'met' if success else 'not met'}")
    
    return all_results, success

if __name__ == "__main__":
    results, passed = run_validation_pipeline()
    import json
    with open('evp_results.json', 'w') as f:
        json.dump({k: {kk: float(vv) if isinstance(vv, (np.floating, np.integer)) 
                      else vv for kk, vv in v.items()} 
                  for k, v in results.items()}, f, indent=2)