Wave-based dispatch methods from hybrid HPC-quantum systems can be leveraged to simulate and predict the onset of ergodicity in disordered quantum many-body systems on digital quantum processors.

Wave-based dispatch methods—specifically, quantum circuit scheduling algorithms that propagate task assignments as wavefronts across hybrid HPC-quantum processor topologies—can predict the critical disorder strength W_c at which a disordered quantum many-body system transitions from ergodic (thermalizing) to many-body localized (MBL) behavior on digital quantum processors, with prediction accuracy ≥85% relative to exact diagonalization (ED) benchmarks for system sizes N ≤ 20 qubits, and with a computational speedup factor ≥2× over purely classical tensor-network or ED methods for N ≥ 16 qubits.

Minimum Viable Test (MVT): Implement wave-based dispatch on a 12-qubit chain with random-field Heisenberg Hamiltonian. Measure level-spacing ratio r and half-chain entanglement entropy S across 10 disorder strengths W ∈ {0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 10.0} with 200 disorder realizations each. Compare predicted W_c to ED ground truth (N=12 is tractable classically). Run on IBM Eagle (127-qubit) or IonQ Forte using wave-dispatch scheduler. Full validation extends to N=16, 20 with 500 realizations and cross-backend comparison.

# Wave-Based Dispatch MBL Simulation — Architecture Outline

# === LAYER 1: DISORDER ENSEMBLE GENERATOR ===
def generate_disorder_ensemble(N, W, n_realizations, seed=42):
    """Generate random field configurations h_i ~ Uniform[-W, W]"""
    rng = np.random.default_rng(seed)
    return rng.uniform(-W, W, size=(n_realizations, N))  # shape: (R, N)

# === LAYER 2: CIRCUIT BUILDER ===
def build_heisenberg_trotter_circuit(N, h_fields, J=1.0, dt=0.1, p_steps=4):
    """
    Trotterized time evolution: U(t) = [exp(-i H_even dt/2) exp(-i H_odd dt/2)]^p
    Returns: Qiskit QuantumCircuit object
    """
    qc = QuantumCircuit(N, N)
    for step in range(p_steps):
        # Even bonds: (0,1), (2,3), ...
        for i in range(0, N-1, 2):
            qc.append(XXZGate(J, dt), [i, i+1])
        # Odd bonds: (1,2), (3,4), ...
        for i in range(1, N-1, 2):
            qc.append(XXZGate(J, dt), [i, i+1])
        # Disorder fields
        for i in range(N):
            qc.rz(2 * h_fields[i] * dt, i)
    qc.measure_all()
    return qc

# === LAYER 3: WAVE-DISPATCH SCHEDULER ===
class WaveDispatchScheduler:
    """
    Wavefront propagation: assigns circuit layers to QPU or HPC-MPS
    based on estimated entanglement entropy gradient.
    """
    def __init__(self, qpu_backend, hpc_mps_engine, entropy_threshold=1.5):
        self.qpu = qpu_backend
        self.hpc = hpc_mps_engine
        self.S_threshold = entropy_threshold  # bits
        self.wave_front = []  # active task queue
    
    def estimate_layer_entropy(self, circuit_layer, current_state_mps):
        """Classical pre-estimation of entanglement growth per layer"""
        S_est = current_state_mps.apply_layer(circuit_layer).entanglement_entropy(N//2)
        return S_est
    
    def dispatch(self, circuits_batch):
        """
        Wave propagation: process circuits in dependency order,
        routing high-entropy layers to QPU, low-entropy to HPC.
        Returns: list of (circuit_id, backend, scheduled_time)
        """
        schedule = []
        wave = deque(circuits_batch)
        while wave:
            task = wave.popleft()
            S_est = self.estimate_layer_entropy(task.current_layer, task.mps_state)
            if S_est > self.S_threshold:
                backend = self.qpu
                task.flag = 'QPU'
            else:
                backend = self.hpc
                task.flag = 'HPC-MPS'
            schedule.append((task.id, backend, current_time()))
            # Propagate wave: enqueue dependent tasks
            for dep in task.dependents:
                dep.mps_state = task.output_state  # pass state forward
                wave.append(dep)
        return schedule

# === LAYER 4: OBSERVABLE COMPUTATION ===
def compute_level_spacing_ratio(energy_levels):
    """r = <min(delta_n, delta_{n+1}) / max(delta_n, delta_{n+1})>"""
    deltas = np.diff(np.sort(energy_levels))
    r_n = np.minimum(deltas[:-1], deltas[1:]) / np.maximum(deltas[:-1], deltas[1:])
    return np.mean(r_n)  # GOE: ~0.53, Poisson: ~0.39

def compute_entanglement_entropy(bitstring_counts, N):
    """Estimate S from bitstring distribution via shadow tomography"""
    # Use classical shadow protocol (Huang et al. 2020)
    shadows = ClassicalShadow(bitstring_counts, N)
    rho_A = shadows.reduced_density_matrix(subsystem=range(N//2))
    eigenvalues = np.linalg.eigvalsh(rho_A)
    eigenvalues = eigenvalues[eigenvalues > 1e-12]
    return -np.sum(eigenvalues * np.log(eigenvalues))

# === LAYER 5: FINITE-SIZE SCALING ===
def finite_size_scaling_collapse(r_data, W_values, N_values):
    """
    Fit: r(W, N) = f((W - W_c) * N^(1/nu))
    Returns: W_c, nu (critical exponent)
    """
    from scipy.optimize import curve_fit
    def scaling_ansatz(X, W_c, nu, r_c, a, b):
        W, N = X
        x = (W - W_c) * N**(1/nu)
        return r_c + a * x + b * x**2
    popt, pcov = curve_fit(scaling_ansatz, 
                           (W_values.flatten(), N_values.flatten()),
                           r_data.flatten(),
                           p0=[3.5, 1.0, 0.46, 0.01, 0.001],
                           bounds=([1.0, 0.3, 0.39, -1, -1], [8.0, 3.0, 0.53, 1, 1]))
    W_c, nu = popt[0], popt[1]
    W_c_err = np.sqrt(pcov[0,0])
    return W_c, nu, W_c_err

# === LAYER 6: MAIN ORCHESTRATION ===
def run_mbl_experiment(N_list=[12,16,20], W_list=np.linspace(0.5,10,10),
                       n_realizations=200, shots=2000):
    scheduler = WaveDispatchScheduler(qpu_backend=IBMBackend('ibm_eagle'),
                                      hpc_mps_engine=ITensorMPS(chi=512))
    results = {}
    for N in N_list:
        results[N] = {}
        for W in W_list:
            fields = generate_disorder_ensemble(N, W, n_realizations)
            circuits = [build_heisenberg_trotter_circuit(N, h, p_steps=4) 
                       for h in fields]
            schedule = scheduler.dispatch(circuits)
            counts = execute_scheduled(schedule, shots=shots)
            # Apply ZNE error mitigation
            counts_mitigated = apply_zne(counts, noise_factors=[1,2,3])
            r_vals = [compute_level_spacing_ratio(c) for c in counts_mitigated]
            S_vals = [compute_entanglement_entropy(c, N) for c in counts_mitigated]
            results[N][W] = {'r': np.mean(r_vals), 'r_std': np.std(r_vals),
                             'S': np.mean(S_vals), 'S_std': np.std(S_vals)}
    W_c, nu, W_c_err = finite_size_scaling_collapse(results)
    return results, W_c, nu, W_c_err

# === LAYER 7: BENCHMARKING ===
def benchmark_speedup(N, W_list, n_realizations):
    t_hybrid = timeit(lambda: run_mbl_experiment([N], W_list, n_realizations))
    t_classical = timeit(lambda: run_ed_benchmark(N, W_list, n_realizations))
    speedup = t_classical / t_hybrid
    dispatch_idle_reduction = measure_idle_time_reduction(scheduler='wave') / \
                              measure_idle_time_reduction(scheduler='fifo')
    return speedup, dispatch_idle_reduction