[Unified compiler] Convert to MBQC formalism

[multiphaseCFD]

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---

Context:
[sc-88794]

This PR adds a pass to lower all gates in the MBQC gate set to a MBQC formalism. Those gates include Hadamard, S, RZ, RotXZX, CNOT. The program structure or control flows (if, for, while) can be also maintained.

The resource auxiliary qubits used as computational resources are statically allocated and deallocated in a gate by gate manner. This PR also assumes that a auxiliary qubit can be swapped with a target qubit in the global register.

This pass can lower a circuit with 1k wires, 10 k gates and 100k shots on a null.qubit device, given that autograph is set as True. It's also compatible with the measurements_from_samples_pass, which is tested in the unit test as well.

Here is a sample script (pennylane_catalyst==0.13.0.dev18 and xdsl==0.46.0):

import xdsl
import catalyst

from catalyst.ftqc import mbqc_pipeline

from catalyst.passes.xdsl_plugin import getXDSLPluginAbsolutePath

import pennylane as qml
from pennylane.compiler.python_compiler.transforms import (
    convert_to_mbqc_formalism_pass,
    measurements_from_samples_pass,
)

from pennylane.ftqc import RotXZX

qml.capture.enable()

dev = qml.device("null.qubit", wires=1000, shots=100000)

@qml.qjit(
    target="mlir",
    pass_plugins=[getXDSLPluginAbsolutePath()],
    pipelines=mbqc_pipeline(),
    autograph=True,
)
@convert_to_mbqc_formalism_pass
@measurements_from_samples_pass
@qml.qnode(dev)
def circuit():
    for i in range(1000):
        qml.H(i)
        qml.S(i)
        RotXZX(0.1, 0.2, 0.3, wires=[i])
        qml.RZ(phi=0.1, wires=[i])
    qml.CNOT(wires=[0, 1])
    return qml.expval(qml.Z(wires=0))

res = circuit()

To capture the program structure without unrolling loops:

@qml.qjit(target="mlir", pass_plugins=[xdsl_plugin.getXDSLPluginAbsolutePath()], autograph=True)
@convert_to_mbqc_formalism_pass
@qml.qnode(dev)
def circuit_ref():
    @qml.for_loop(0, 100, 1)
    def loop_func(i):
        qml.H(i)
        qml.S(i)
        RotXZX(0.1, 0.2, 0.3, wires=[i])
        qml.RZ(phi=0.1, wires=[i])
    loop_func()
    qml.CNOT(wires=[0, 1])
    return qml.expval(qml.Z(wires=0))

Description of the Change:

Benefits:

Possible Drawbacks:

Related GitHub Issues:

[joeycarter]

Looks good, thanks @multiphaseCFD! Happy to approve once these few remaining items are resolved.

[joeycarter]

Looks good! 💯

[lillian542]

Thanks @multiphaseCFD, looks great! I left a few non-blocking comments 🚀