Allow multi controlled x decomposition without working wires

[EmilianoG-byte]

Context:
MultiControlled.decomposition() gives an error when no work_wire is provided.

Description of the Change:

Implement decomposition given in Lemma 7.5 of [1] to decompose a general multi controlled unitary gate. MultiControlledX being a particular case.

The function ctrl_decomp_zyz now returns the correct decomposition including a controlled global phase. It now checks that this decomposition only works for single control wire.

A decomposition for single controlled GlobalPhase is now available using Lemma 5.2 of [1], i.e. adds a PhaseShift to the control wire. Raise an exception when used with more than one control.

Benefits:

Decomposition of a MultiControlledX (and in general any multi controlled unitary) is now possible even if work wires are not provided.

If controlled gate is SU(2), this resorts to the specialized algorithm from Lemma 7.9, yielding a decomposition with $\mathcal{O}(\text{len(control wires)})$ gates. For general unitaries, the decomposition yields $\mathcal{O}(\text{len(control wires)}^2)$ gates.

Sources:

[1] Barenco et al. (1995). https://arxiv.org/abs/quant-ph/9503016

[SC-38954]

[github-actions]

Hello. You may have forgotten to update the changelog!
Please edit doc/releases/changelog-dev.md with:

• A one-to-two sentence description of the change. You may include a small working example for new features.
• A link back to this PR.
• Your name (or GitHub username) in the contributors section.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.66%. Comparing base (38b3e74) to head (9cbbec3).
Report is 251 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #5735      +/-   ##
==========================================
- Coverage   99.66%   99.66%   -0.01%     
==========================================
  Files         415      415              
  Lines       39601    39352     -249     
==========================================
- Hits        39469    39219     -250     
- Misses        132      133       +1     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[EmilianoG-byte]

I think the code in line:

pennylane/pennylane/ops/op_math/controlled.py

Line 799 in 7a216b6

if _is_single_qubit_special_unitary(op.base):

should be modified since ctrl_decomp_zyz can now be used for general unitaries. Is this necessary? Currently the code returns the controlled version of each of the operators in the op.base decomposition. In cases like:

base = qml.SX(0)
op = qml.ops.Controlled(base, control_wires=1)
op.decomposition()
>>> [CRZ(1.5707963267948966, wires=<Wires = [1, 0]>),
 CRY(1.5707963267948966, wires=[1, 0])),
 CRZ(-3.141592653589793, wires=<Wires = [1, 0]>),
 ControlledPhaseShift(1.5707963267948966, wires=<Wires = [1, 0]>)]

shouldn't this be further decomposed since we know how to handle the decomposition of each of the controlled operators?

[mudit2812]

Looks great! Just one blocking comment about the RuntimeWarning, but otherwise I'm happy to get this in.

[albi3ro]

This is going to make a lot of people very happy 🎉