Better MCX Decomposition with logarithmic Toffoli Depth

[patelvyom]

Context:
The recent paper Rise of conditionally clean ancillae for optimizing quantum circuits presents an improved decomposition of MCX gates using conditionally clean ancilla qubits. The proposed method achieves the following trade-offs in terms of Toffoli gate count and circuit depth:

• 2n − 3 Toffoli and O(n) depth using 1 clean ancilla
• 2n − 3 Toffoli and O(log(n)) depth using 2 clean ancilla
• 4n − 8 Toffoli and O(n) depth using 1 dirty ancilla
• 4n - 8 Toffoli and O(log(n)) depth using 2 dirty ancilla

This approach is currently optimal regarding Toffoli count (for 1 and 2 ancillae) and significantly improves circuit depth.

Description of the Change:
Added _decompose_mcx_with_two_workers and _decompose_mcx_with_one_worker_kg24 decompositions.

Benefits:
Extend support to:

• len(work_wires) == 2, enabling a more efficient decomposition
• Improved decomposition for len(work_wires) == 1, reducing circuit depth.

Related GitHub Issues:
Implements #6997

[patelvyom]

@albi3ro I have a draft ready and am currently working on the tests. The implementation for dirty and clean only differs in terms of uncomputing certain gates.

Long term, it would be great to have the compiler find dirty wires and call the appropriate decomposition function because finding clean wires is much stricter condition than reusing other wires of larger circuit as dirty wires.

[albi3ro]

Thanks for opening this @patelvyom 👍 I'll try and take a look at it on Monday.

[patelvyom]

This pull request is nearly ready to merge. The main open question is whether decompose_mcx should assume it is always called within a qml.tape context. Currently, it returns a List[Operation], but if operations are expected to be taped, additional conditional logic (if qml.QueuingManager.recording()) will be needed in both functions.

Summary of Changes:

• Implemented two new decomposition functions, along with helper functions, based on the referenced paper.
• Added four test cases for each new decomposition function.
• Introduced a new argument to decompose_mcx(): work_wire_type: Literal["clean", "dirty"].
• Updated decompose_mcx to select the decomposition method in order: many_workers, two_workers, one_worker based on no. of available work wires.
• Ensured _decompose_recursive sets work_wire_type="dirty" when called recursively, aligning with the assumptions of the cited lemma.
• Renamed _decompose_mcx_with_one_worker to _decompose_mcx_with_one_worker_b95.