Support for arbitrary inputs in Pauli preprocessing

[thierry-martinez]

Is your feature request related to a problem? Please describe.
Pauli preprocessing is currently only available when inputs are initialized in |+> state.

Describe the feature you'd like
Pauli preprocessing should support arbitrary inputs.

Additional context
Support for arbitrary inputs in state vector and density matrix simulations has been handled in #135 .

[masa10-f]

Could you share the error case? I couldn't find it in #135
I'm suspecting that local complementation or pivot flips measurement plane of input nodes.

[matulni]

I'll mark this as a bug because current implementation doesn't warn the user that only $|+\rangle$ input states are supported.
Minimal example showcasing the bug:

import numpy as np
from graphix.fundamentals import Plane
from graphix.states import PlanarState
from graphix.transpiler import Circuit

# alpha = 0 # -> Works
alpha = 2 * np.pi * np.random.random()

# Circuit implementing H and CZ
circuit = Circuit(2)
circuit.h(0)
circuit.h(1)
circuit.cnot(0,1)
circuit.h(1)

p_circuit = circuit.transpile().pattern
s = p_circuit.simulate_pattern(input_state=PlanarState(Plane.XY, alpha))


p_circuit.perform_pauli_measurements()
s_pauli = p_circuit.simulate_pattern(input_state=PlanarState(Plane.XY, alpha))

assert np.isclose(np.abs(np.dot(s.flatten().conjugate(), s_pauli.flatten())), 1)

An immediate solution could be to support Pauli simulation on patterns without inputs, while we investigate how to improve the Pauli simulator.