Normalize two_qubit_decomposition for odd CNOTs to preserve determinant 1

[Qottmann]

The determinant is changed from 1 to -1 when the returned circuit uses an odd number of CNOT gates. This can be normalized by a global phase $\xi^4 = -1$, e.g. $e^{-i \pi /4}$ as indicated in https://arxiv.org/pdf/quant-ph/0308033 in the paragraph below Lemma II.1

Not sure we actually want this change, as it adds GlobalPhases that on a quantum computational level don't matter
Actually might have implications when doing a controlled op like this.

TODO

• decide if we actually want this, then
• changelog
• tests

An alternative approach could be to remove the _convert_to_su4 step in the first place (not sure this works)

[github-actions]

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Please edit doc/releases/changelog-dev.md with:

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[Qottmann]

When doing controlled ops the global phase missmatch becomes important so perhaps it would indeed be better to fix this

import numpy as np
import pennylane as qml

def U():
    qml.exp(-1j * qml.dot(np.arange(16)[1:], list(qml.pauli.pauli_group(2))[1:]))

Um = qml.matrix(U, wire_order=range(2))()

ops = qml.ops.two_qubit_decomposition(Um, range(2))
U_re = qml.QubitUnitary(qml.matrix(qml.tape.QuantumScript(ops, []), wire_order=range(2)), range(2))

ctrl_op_m = qml.matrix(qml.ctrl(U, [2]), wire_order=range(3))()
ctrl_op_m2 = qml.matrix(qml.ctrl(U_re, [2]), wire_order=range(3))
np.abs(np.trace(ctrl_op_m @ ctrl_op_m2.conj().T))/8 # not unitarily equivalent
# 0.9238795325112874

[JerryChen97]

unitarily equivalent

I'm a bit confused here. What is the unitary equivalence we want to bootstrap here? Is it that there exists a unitary U such that $A = U^\dagger B U$ between equivalent A and B, or we only allow A and B to differ in global phase?

[JerryChen97]

When doing controlled ops the global phase missmatch becomes important so perhaps it would indeed be better to fix this

import numpy as np
import pennylane as qml

def U():
    qml.exp(-1j * qml.dot(np.arange(16)[1:], list(qml.pauli.pauli_group(2))[1:]))

Um = qml.matrix(U, wire_order=range(2))()
ops = qml.ops.two_qubit_decomposition(Um, range(2))

def U_re():
    ops

ctrl_op_m = qml.matrix(qml.ctrl(U, [2]), wire_order=range(3))()
ctrl_op_m2 = qml.matrix(qml.ctrl(U_re, [2]), wire_order=range(3))()
np.abs(np.trace(ctrl_op_m @ ctrl_op_m2.conj().T))/8 # not unitarily equivalent
# 0.8261951643289711

Also, when I try this example, I see ctrol_op_m2 is an Identity 😱 maybe quite some trouble with our 2-site decomp

[Qottmann]

Also, when I try this example, I see ctrol_op_m2 is an Identity 😱 maybe quite some trouble with our 2-site decomp

There was a bug in my code snippet in that the decomposition ops were not properly recorded, updated it. The result is the same - as in not unitarily equivalent, which I might not be using correctly here, but I just meant that they are the same up to a global phase. So they are the same unitary operation, but are not element-wise the same.