QPY fails to load circuits whose first register qubit is after a loose bit

[jakelishman]

Environment

• Qiskit Terra version: 0.22.2
• Python version: 3.10
• Operating system: macOS

What is happening?

QPY fails to load a circuit that has registers, but bit index 0 isn't in a register or at least, isn't in the circuit by the time a later register is read.

How can we reproduce the issue?

import io

from qiskit.circuit import QuantumCircuit, Clbit, ClassicalRegister
from qiskit.qpy import load, dump

qc = QuantumCircuit()
qc.add_bits([Clbit()])
qc.add_register(ClassicalRegister(2))

with io.BytesIO() as f:
    dump(qc, f)
    f.seek(0)
    out, = load(f)

fails with

TypeError                                 Traceback (most recent call last)
<ipython-input-1-ddaf374c6759> in <module>
     11     dump(qc, f)
     12     f.seek(0)
---> 13     out, = load(f)
     14

~/code/qiskit/terra/qiskit/qpy/interface.py in load(file_obj, metadata_deserializer)
    269     for _ in range(data.num_programs):
    270         programs.append(
--> 271             loader(file_obj, data.qpy_version, metadata_deserializer=metadata_deserializer)
    272         )
    273     return programs

~/code/qiskit/terra/qiskit/qpy/binary_io/circuits.py in read_circuit(file_obj, version, metadata_deserializer)
    924                         if start > len(circ.qubits):
    925                             bits = [bit_type() for _ in range(start - bit_len)]
--> 926                             circ.add_bits(bit_len)
    927                         if in_circuit:
    928                             circ.add_register(reg)

~/code/qiskit/terra/qiskit/circuit/quantumcircuit.py in add_bits(self, bits)
   1394     def add_bits(self, bits: Iterable[Bit]) -> None:
   1395         """Add Bits to the circuit."""
-> 1396         duplicate_bits = set(self._qubit_indices).union(self._clbit_indices).intersection(bits)
   1397         if duplicate_bits:
   1398             raise CircuitError(f"Attempted to add bits found already in circuit: {duplicate_bits}")

TypeError: 'int' object is not iterable

What should happen?

Correct load from QPY.

Any suggestions?

qpy/binary_io/circuits.py:926 is obviously wrong: it's passing an int to QuantumCircuit.add_bits rather than the bits that it has defined right above. This isn't the only issue, though - the whole register reconstruction has an incorrect model of bit ownership and bit ordering, so a more complete example like

import io
from qiskit.circuit import QuantumCircuit, Qubit, Clbit, QuantumRegister, ClassicalRegister
from qiskit.qpy import dump, load

qc = QuantumCircuit()
qc.add_bits([Qubit(), Clbit()])
qc.add_register(QuantumRegister(2, name="q1"))
qc.add_register(ClassicalRegister(2, name="c1"))
with io.BytesIO() as fptr:
    dump(qc, fptr)
    fptr.seek(0)
    new_circuit = load(fptr)[0]
assert qc == new_circuit

fails, because in the output circuit, the clbits go [('c1', 0), ('c1', 1), Clbit()], whereas they should be [Clbit(), ('c1', 0), ('c1', 1)].