Modeling the energy storage processes in dual-use quantum batteries with ergodicity onset parameters derived from digital quantum processors will uncover new regimes of collective quantum behavior relevant for scalable quantum technologies.

Quantum battery systems composed of N ≥ 4 coupled two-level systems (qubits), when initialized and driven using ergodicity onset parameters extracted from digital quantum processors (e.g., gate-based superconducting or trapped-ion platforms), will exhibit at least one measurable new collective charging regime—characterized by superextensive energy storage scaling (E_stored ∝ N^α, α > 1) or anomalous ergotropy dynamics—that is absent in classically parameterized or mean-field models. Specifically, the ergodicity onset threshold (measured via level-spacing statistics transitioning from Poisson to Wigner-Dyson distributions) will correlate (r² > 0.75) with the onset of enhanced collective charging power, and at least one regime will be identified that is not predicted by existing Tavis-Cummings or Dicke model frameworks.

PHASE 1 — Classical Simulation Baseline (Weeks 1–4): Simulate quantum battery charging dynamics for N = 4, 8, 12, 16, 20 qubits using exact diagonalization (ED) and time-dependent Schrödinger equation integration. Compute ergotropy W = Tr[ρH] - min_{U} Tr[UρU†H] as the primary energy storage metric. Establish baseline scaling without ergodicity parameters.

PHASE 2 — Ergodicity Parameter Extraction (Weeks 3–6): Deploy circuits on IBM Quantum (127-qubit Eagle or 433-qubit Osprey) or IonQ Aria to measure level-spacing statistics of the system Hamiltonian. Extract ⟨r⟩ parameter as a function of coupling strength J and system size N. Map the ergodicity onset boundary in (J, N) parameter space.

PHASE 3 — Ergodicity-Informed Model (Weeks 5–10): Incorporate extracted ergodicity parameters into the quantum battery Hamiltonian as renormalized coupling constants or effective interaction terms. Re-simulate charging dynamics. Compare ergotropy scaling, charging power P = W/τ_ch, and quantum advantage ratio Q_adv = W_quantum/W_classical.

PHASE 4 — Regime Identification and Validation (Weeks 9–14): Identify candidate new regimes via clustering in (α, ⟨r⟩, J/ω, N) parameter space. Validate each regime against Tavis-Cummings and Dicke model predictions. Perform sensitivity analysis on noise parameters.

# Quantum Battery EVP — Implementation Sketch
# Dependencies: QuTiP, Qiskit, NumPy, SciPy, scikit-learn

import numpy as np
from qutip import *
from qiskit import QuantumCircuit, transpile
from qiskit_ibm_runtime import QiskitRuntimeService
from scipy.stats import pearsonr
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score

# ============================================================
# MODULE 1: Hamiltonian Construction
# ============================================================
def build_battery_hamiltonian(N, omega, J, V0=0.0):
    """
    H = omega * sum_i sigma_z_i / 2
      + J * sum_{<i,j>} (sigma_x_i sigma_x_j + sigma_y_i sigma_y_j)
      + V0 * sum_i sigma_x_i  [charging field, time-dependent externally]
    Returns: QuTiP Qobj
    """
    H0 = sum([omega/2 * tensor([sigmaz() if k==i else qeye(2)
                                 for k in range(N)]) for i in range(N)])
    H_int = sum([J * (tensor([sigmax() if k in [i,(i+1)%N] else qeye(2)
                               for k in range(N)]) +
                      tensor([sigmay() if k in [i,(i+1)%N] else qeye(2)
                               for k in range(N)])) for i in range(N)])
    H_charge = V0 * sum([tensor([sigmax() if k==i else qeye(2)
                                  for k in range(N)]) for i in range(N)])
    return H0 + H_int + H_charge

# ============================================================
# MODULE 2: Ergotropy Computation
# ============================================================
def compute_ergotropy(rho, H):
    """
    W = Tr[rho H] - min_U Tr[U rho U† H]
    Minimum achieved by passive state (eigenvalues sorted oppositely)
    """
    rho_evals, rho_evecs = rho.eigenstates()
    H_evals, H_evecs = H.eigenstates()
    # Sort: rho descending, H ascending
    rho_evals_sorted = np.sort(rho_evals)[::-1]
    H_evals_sorted = np.sort(H_evals)
    E_passive = np.dot(rho_evals_sorted, H_evals_sorted)
    E_current = expect(H, rho)
    return float(E_current - E_passive)

# ============================================================
# MODULE 3: Time Evolution and Ergotropy Scaling
# ============================================================
def run_charging_simulation(N_list, omega, J_list, tau_ch, dt=0.001):
    results = {}
    for N in N_list:
        for J in J_list:
            H0 = build_battery_hamiltonian(N, omega, J, V0=0.0)
            H_drive = build_battery_hamiltonian(N, omega, J=0, V0=1.0)
            # Charging: H(t) = H0 + sin(omega*t) * H_drive
            H_td = [H0, [H_drive, lambda t, args: np.sin(omega*t)]]
            psi0 = tensor([basis(2,1)]*N)  # all qubits in ground state
            tlist = np.arange(0, tau_ch, dt)
            output = mesolve(H_td, psi0, tlist, [], [])
            rho_final = ket2dm(output.states[-1])
            W = compute_ergotropy(rho_final, H0)
            results[(N, J)] = W
    return results

# ============================================================
# MODULE 4: Level-Spacing Statistics
# ============================================================
def compute_level_spacing_ratio(H):
    evals = np.sort(H.eigenenergies())
    spacings = np.diff(evals)
    r_vals = [min(spacings[i], spacings[i+1]) / max(spacings[i], spacings[i+1])
              for i in range(len(spacings)-1)]
    return np.mean(r_vals)

def map_ergodicity_onset(N_list, omega, J_range, n_disorder=500):
    onset_map = {}
    for N in N_list:
        r_means = []
        for J in J_range:
            r_batch = []
            for _ in range(n_disorder):
                # Add small random disorder to probe level statistics
                delta = np.random.uniform(-0.05*omega, 0.05*omega, N)
                H = build_battery_hamiltonian(N, omega, J) + \
                    sum([delta[i]*tensor([sigmaz() if k==i else qeye(2)
                                          for k in range(N)]) for i in range(N)])
                r_batch.append(compute_level_spacing_ratio(H))
            r_means.append(np.mean(r_batch))
        onset_map[N] = (J_range, np.array(r_means))
    return onset_map

# ============================================================
# MODULE 5: Ergodicity-Informed Renormalization
# ============================================================
def ergodicity_renormalize(J, r_mean, r_poisson=0.386, r_goe=0.530):
    """Scale J by ergodicity fraction: 0 (Poisson) to 1 (GOE)"""
    frac = np.clip((r_mean - r_poisson) / (r_goe - r_poisson), 0, 1)
    J_eff = J * (1 + 0.5 * frac)  # enhancement factor; to be fit from data
    return J_eff

# ============================================================
# MODULE 6: Scaling Exponent Extraction
# ============================================================
def extract_scaling_exponent(N_list, ergotropy_dict, J):
    W_vals = [ergotropy_dict[(N, J)] for N in N_list]
    log_N = np.log(N_list)
    log_W = np.log(W_vals)
    alpha, intercept = np.polyfit(log_N, log_W, 1)
    return alpha

# ============================================================
# MODULE 7: Regime Clustering
# ============================================================
def identify_regimes(feature_matrix, k_range=range(2,7)):
    best_k, best_score = 2, -1
    for k in k_range:
        km = KMeans(n_clusters=k, random_state=42, n_init=20)
        labels = km.fit_predict(feature_matrix)
        score = silhouette_score(feature_matrix, labels)
        if score > best_score:
            best_score, best_k = score, k
    km_final = KMeans(n_clusters=best_k, random_state=42, n_init=20)
    labels = km_final.fit_predict(feature_matrix)
    return labels, best_score, best_k

# ============================================================
# MODULE 8: IBM Quantum Hardware Interface
# ============================================================
def run_hardware_level_statistics(N, J, omega, n_shots=10000):
    """
    Construct Trotterized circuit for Hamiltonian simulation,
    measure in energy eigenbasis via QPE or VQE landscape.
    Returns estimated level-spacing ratio from hardware.
    """
    service = QiskitRuntimeService(channel="ibm_quantum")
    backend = service.least_busy(min_num_qubits=N, simulator=False)
    # [Circuit construction omitted for brevity — Trotterized H simulation]
    # Apply readout error mitigation
    # Return ⟨r⟩ estimate
    pass  # Placeholder for full circuit implementation

# ============================================================
# MODULE 9: Main Validation Pipeline
# ============================================================
def main():
    omega = 1.0
    N_list = [4, 8, 12, 16, 20]
    J_list = [0.01, 0.05, 0.1, 0.2, 0.3, 0.5]
    tau_ch = np.pi / omega

    # Step 1: Baseline simulation
    baseline_results = run_charging_simulation(N_list, omega, J_list, tau_ch)

    # Step 2: Ergodicity onset mapping
    J_range = np.linspace(0.01, 0.5, 30)
    onset_map = map_ergodicity_onset(N_list, omega, J_range, n_disorder=500)

    # Step 3: Ergodicity-informed simulation
    informed_results = {}
    for N in N_list:
        J_arr, r_arr = onset_map[N]
        for J in J_list:
            r_interp = np.interp(J, J_arr, r_arr)
            J_eff = ergodicity_renormalize(J, r_interp)
            H_eff = build_battery_hamiltonian(N, omega, J_eff)
            # Re-run charging with J_eff
            # [abbreviated — same as run_charging_simulation]
            pass

    # Step 4: Scaling exponents
    alphas = {J: extract_scaling_exponent(N_list, baseline_results, J)
              for J in J_list}
    alphas_informed = {J: extract_scaling_exponent(N_list, informed_results, J)
                       for J in J_list}

    # Step 5: Correlation analysis
    r_means_flat = [np.interp(J, onset_map[12][0], onset_map[12][1])
                    for J in J_list]
    W_enhancements = [informed_results.get((12,J),1) /
                      max(baseline_results.get((12,J),1), 1e-10)
                      for J in J_list]
    r2, pval = pearsonr(r_means_flat, W_enhancements)
    print(f"Pearson r²={r2**2:.3f}, p={pval:.4f}")

    # Step 6: Regime clustering
    features = np.array([[alphas_informed[J], np.interp(J, onset_map[N][0],
                          onset_map[N][1]), J/omega,
                          informed_results.get((N,J),0)/
                          max(baseline_results.get((N,J),1e-10),1e-10), N]
                         for N in N_list for J in J_list])
    labels, sil_score, n_clusters = identify_regimes(features)
    print(f"Identified {n_clusters} regimes, silhouette={sil_score:.3f}")

    # Step 7: Success/failure assessment
    success = (
        any(a > 1.05 for a in alphas_informed.values()) and
        r2**2 >= 0.75 and pval < 0.01 and
        sil_score >= 0.6
    )
    print(f"Hypothesis {'SUPPORTED' if success else 'NOT SUPPORTED'}")

if __name__ == "__main__":
    main()