How to use the blueqat.pauli function in blueqat

def numpartition_qaoa(n_step, nums, minimizer=None, sampler=None):
    """Do the Number partition QAOA.

    :param n_step: The number of step of QAOA
    :param nums: The edges list of the graph.
    :returns Vqe object
    """
    hamiltonian = pauli.Expr.zero()
    for i, x in enumerate(nums):
        hamiltonian += pauli.Z[i] * x
    hamiltonian = (hamiltonian ** 2).simplify()

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)

def mk_expr(offset, n):
        expr = pauli.Expr.from_number(1)
        for i in range(n):
            expr = expr + 2**(i + 1) * q(i + offset)
        return expr

:param n_step: The number of step of QAOA
    :param n_sample: The number of sampling time of each measurement in VQE.
                     If None, use calculated ideal value.
    :param edges: The edges list of the graph.
    :returns Vqe object
    """
    sampler = sampler or vqe.non_sampling_sampler
    minimizer = minimizer or vqe.get_scipy_minimizer(
        method="Powell",
        options={"ftol": 5.0e-2, "xtol": 5.0e-2, "maxiter": 1000, "disp": True}
    )
    hamiltonian = pauli.I() * 0

    for i, j in edges:
        hamiltonian += pauli.Z(i) * pauli.Z(j)

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)

:param n_step: The number of step of QAOA
    :param n_sample: The number of sampling time of each measurement in VQE.
                     If None, use calculated ideal value.
    :param edges: The edges list of the graph.
    :returns Vqe object
    """
    sampler = sampler or vqe.non_sampling_sampler
    minimizer = minimizer or vqe.get_scipy_minimizer(
        method="Powell",
        options={"ftol": 5.0e-2, "xtol": 5.0e-2, "maxiter": 1000, "disp": True}
    )
    hamiltonian = pauli.I() * 0

    for i, j in edges:
        hamiltonian += pauli.Z(i) * pauli.Z(j)

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)

def numpartition_qaoa(n_step, nums, minimizer=None, sampler=None):
    """Do the Number partition QAOA.

    :param n_step: The number of step of QAOA
    :param nums: The edges list of the graph.
    :returns Vqe object
    """
    hamiltonian = pauli.Expr.zero()
    for i, x in enumerate(nums):
        hamiltonian += pauli.Z[i] * x
    hamiltonian = (hamiltonian ** 2).simplify()

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)

:param n_step: The number of step of QAOA
    :param n_sample: The number of sampling time of each measurement in VQE.
                     If None, use calculated ideal value.
    :param edges: The edges list of the graph.
    :returns Vqe object
    """
    sampler = sampler or vqe.non_sampling_sampler
    minimizer = minimizer or vqe.get_scipy_minimizer(
        method="Powell",
        options={"ftol": 5.0e-2, "xtol": 5.0e-2, "maxiter": 1000, "disp": True}
    )
    hamiltonian = pauli.I() * 0

    for i, j in edges:
        hamiltonian += pauli.Z(i) * pauli.Z(j)

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)

def pauli(self, simplify=True) -> blueqat.pauli.Expr:
        expr = sum(reduce(operator.mul, map(blueqat.pauli.qubo_bit, ind)) * val if ind else val
                   for ind, val in self.terms.items())
        if simplify:
            return expr.to_expr().simplify()
        return expr.to_expr()

def maxcut_qaoa(n_step, edges, minimizer=None, sampler=None, verbose=True):
    """Setup QAOA.

    :param n_step: The number of step of QAOA
    :param n_sample: The number of sampling time of each measurement in VQE.
                     If None, use calculated ideal value.
    :param edges: The edges list of the graph.
    :returns Vqe object
    """
    sampler = sampler or vqe.non_sampling_sampler
    minimizer = minimizer or vqe.get_scipy_minimizer(
        method="Powell",
        options={"ftol": 5.0e-2, "xtol": 5.0e-2, "maxiter": 1000, "disp": True}
    )
    hamiltonian = pauli.I() * 0

    for i, j in edges:
        hamiltonian += pauli.Z(i) * pauli.Z(j)

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)

def maxcut_qaoa(n_step, edges, minimizer=None, sampler=None, verbose=True):
    """Setup QAOA.

    :param n_step: The number of step of QAOA
    :param n_sample: The number of sampling time of each measurement in VQE.
                     If None, use calculated ideal value.
    :param edges: The edges list of the graph.
    :returns Vqe object
    """
    sampler = sampler or vqe.non_sampling_sampler
    minimizer = minimizer or vqe.get_scipy_minimizer(
        method="Powell",
        options={"ftol": 5.0e-2, "xtol": 5.0e-2, "maxiter": 1000, "disp": True}
    )
    hamiltonian = pauli.I() * 0

    for i, j in edges:
        hamiltonian += pauli.Z(i) * pauli.Z(j)

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)

def maxcut_qaoa(n_step, edges, minimizer=None, sampler=None, verbose=True):
    """Setup QAOA.

    :param n_step: The number of step of QAOA
    :param n_sample: The number of sampling time of each measurement in VQE.
                     If None, use calculated ideal value.
    :param edges: The edges list of the graph.
    :returns Vqe object
    """
    sampler = sampler or vqe.non_sampling_sampler
    minimizer = minimizer or vqe.get_scipy_minimizer(
        method="Powell",
        options={"ftol": 5.0e-2, "xtol": 5.0e-2, "maxiter": 1000, "disp": True}
    )
    hamiltonian = pauli.I() * 0

    for i, j in edges:
        hamiltonian += pauli.Z(i) * pauli.Z(j)

    return vqe.Vqe(vqe.QaoaAnsatz(hamiltonian, n_step), minimizer, sampler)