Optimizing quantum battery charging protocols by leveraging equilibrium concepts that account for coordinated multi-qubit deviations will increase ergotropy retention compared to protocols based solely on unilateral deviation models.

Given an N-qubit quantum battery (N≥2) coupled to a shared or structured charging Hamiltonian, a charging protocol derived from a coordinated/coalitional equilibrium concept (e.g., correlated equilibrium, coalition-proof Nash equilibrium, or collective passivity/ergotropy over joint states) will yield strictly higher extractable ergotropy retention — defined as E_ergotropy(t_final)/E_input, measured after a fixed charging time T and after N_cycles of charge–partial-discharge — than a protocol optimized under a unilateral (single-qubit, product-state/Nash) deviation model, for the same total input energy, same T, and same local control constraints. The effect must exceed a pre-registered minimum improvement of ≥5% relative ergotropy retention (not merely statistically distinguishable from zero) to count as confirmed, and must persist under realistic decoherence (T1, T2 finite) and at least one non-idealized noise channel.

Minimum viable test (MVT): 2-qubit quantum battery, closed-form comparison.

1. Define H_charge(t) = Σ_i ω_i σ_z^i /2 + g(t)(σ_x^1σ_x^2 + σ_y^1σ_y^2) with tunable coupling g(t).
2. Compute unilateral-deviation-optimal protocol: each qubit's drive optimized assuming the other's strategy fixed (best-response / Nash iteration), converged to fixed point.
3. Compute coordinated-equilibrium-optimal protocol: joint optimization over the 2-qubit unitary group maximizing joint ergotropy (correlated-equilibrium formulation via joint probability distribution over control settings, or direct global-optimal control, e.g. GRAPE/Krotov over the full 2-qubit Hilbert space).
4. Simulate both protocols under (a) ideal unitary evolution, (b) Lindblad noise at T1=50–100 µs, T2=30–70 µs (transmon-realistic), (c) two-qubit gate infidelity 0.1–1%.
5. Extract ergotropy at fixed T and compute retention ratio; repeat 3.–4. for N=3,4,6,8 with scalable ansatz (variational quantum control / tensor-network simulation for N>6).
6. Validate on real hardware (IBM Quantum or IonQ) for N=2,3 as feasibility check.
7. Statistical test: paired comparison (Wilcoxon signed-rank) across 200+ random Hamiltonian instances (varying ω_i, coupling strength, detuning) per N, Bonferroni-corrected across N values.

# Pseudocode: coordinated vs unilateral quantum battery charging

for N in [2,3,4,6,8]:
    for instance in generate_hamiltonian_instances(N, n=200, seed=i):
        H_charge = build_hamiltonian(N, instance)

        # Unilateral (Nash best-response) protocol
        pi_uni = nash_best_response_iterate(
            H_charge, max_iters=500, tol=1e-6,
            local_control_space=per_qubit_drives)

        # Coordinated (joint/correlated equilibrium) protocol
        pi_coord = joint_optimal_control(
            H_charge, method="GRAPE", # or Krotov
            control_space=joint_unitary_group(N),
            objective=lambda U: ergotropy(rho_final(U)))

        for noise_model in [None, lindblad_realistic(T1,T2), hw_calibration]:
            rho_uni = evolve(H_charge, pi_uni, noise_model)
            rho_coord = evolve(H_charge, pi_coord, noise_model)

            E_uni = ergotropy(rho_uni)
            E_coord = ergotropy(rho_coord)
            R = E_coord / E_uni
            record(N, instance, noise_model, R)

# Statistics
for N in [2,3,4,6,8]:
    wilcoxon_test(R_values[N], null=1.0, correction="bonferroni")
    cliffs_delta(R_values[N])

# Hardware validation subset
select_top_instances(R_values, k=3, per_N=[2,3])
compile_to_native_gates(pi_coord_selected, backend="ibm_eagle" or "ionq_aria")
run_hardware(shots=4096, repeats=3_calibration_windows)
compare_hw_vs_sim(tolerance=2x_sim_CI)