precision argument of EstimatorV2 should account for Hamiltonian norm

[kevinsung]

What is the expected feature or enhancement?

Related: Qiskit/qiskit#12414

If I'm not mistake, the current implementation sets shots = 1 / precision**2 here. This is sufficient for a single Pauli operator with norm 1, but it is insufficient for a Hamiltonian with more terms and greater norm. I think it should be A / precision**2 where A is the sum of the absolute value squared of the coefficients of the Hamiltonian, sum(np.abs(sparse_pauli_op.coeffs)**2). Example:

import numpy as np
from qiskit.circuit.library import EfficientSU2
from qiskit.quantum_info import SparsePauliOp
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager
from qiskit_ibm_runtime import EstimatorV2 as Estimator
from qiskit_ibm_runtime.fake_provider import FakeSherbrooke

num_qubits = 10
hamiltonian = SparsePauliOp.from_sparse_list(
    [("ZZ", [i, i + 1], 1.0) for i in range(num_qubits - 1)], num_qubits=num_qubits
)
circuit = EfficientSU2(num_qubits, reps=1)
backend = FakeSherbrooke()
pass_manager = generate_preset_pass_manager(optimization_level=3, backend=backend)
isa_circuit = pass_manager.run(circuit)
isa_hamiltonian = hamiltonian.apply_layout(isa_circuit.layout)
estimator = Estimator(backend)
rng = np.random.default_rng(12345)
params = rng.uniform(-np.pi, np.pi, size=(5, isa_circuit.num_parameters))
pub = (isa_circuit, isa_hamiltonian, params)

job = estimator.run([pub], precision=0.01)
result = job.result()
pub_result = result[0]
pub_result.data.stds

array([0.02840166, 0.02838584, 0.02589326, 0.02733313, 0.02869348])

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