Dynamical decoupling: Support pulling through the whole circuit with non-clifford moments

[babacry]

Previously, we only insert dd for split Clifford pieces. While, some use cases require

1. insert into consecutive idle moments cross Clifford pieces.
2. pull through clifford op even the moment contains non-clifford ops.
3. etc

Example: schema=(X, Y, X, Y)

Input:
                                                              ┌────────────────────────┐                             ┌──┐
0: ─────────────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@─────PhXZ(a=0.2,x=0.2,z=0.1)───H───
                                                               │                                                      │
1: ─────────────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@─────PhXZ(a=0.2,x=0.2,z=0.1)───H───

2: ───PhXZ(a=0.2,x=0.2,z=0.1)───@───PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────────H───
                                │                              │                                                      │
3: ─────────────────────────────┼───PhXZ(a=0.2,x=0.2,z=0.1)────┼──────────────────────@───────────────────────────────┼@──────────────────────────────H───
                                │                              │                      │                               ││
4: ─────────────────────────────┼──────────────────────────────┼──────────────────────@───────────────────────────────┼┼──────────────────────────────H───
                                │                              │                                                      ││
5: ───PhXZ(a=0.2,x=0.2,z=0.1)───@───PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@┼────PhXZ(a=0.2,x=0.2,z=0.1)───H───
                                                                                                                       │
6: ────────────────────────────────────────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)─────────────────────────────────@────PhXZ(a=0.2,x=0.2,z=0.1)───H───
                                                              └────────────────────────┘                             └──┘
Output:
                                                              ┌────────────────────────┐                             ┌──┐
0: ─────────────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@─────PhXZ(a=0.2,x=0.2,z=0.1)─────H────────────────────────
                                                               │                                                      │
1: ─────────────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@─────PhXZ(a=0.2,x=0.2,z=0.1)─────H────────────────────────

2: ───PhXZ(a=0.2,x=0.2,z=0.1)───@───PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@─────X───────────────────────────PhXZ(a=-0.5,x=0.5,z=0)───
                                │                              │                                                      │
3: ─────────────────────────────┼───PhXZ(a=0.2,x=0.2,z=0.1)────┼──────────────────────@────X──────────────────────────┼@────Y───────────────────────────H────────────────────────
                                │                              │                      │                               ││
4: ─────────────────────────────┼──────────────────────────────┼──────────────────────@────X──────────────────────────┼┼────Y───────────────────────────PhXZ(a=0.5,x=0.5,z=0)────
                                │                              │                                                      ││
5: ───PhXZ(a=0.2,x=0.2,z=0.1)───@───PhXZ(a=0.2,x=0.2,z=0.1)────@───────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)────@┼────PhXZ(a=0.2,x=0.2,z=0.1)─────H────────────────────────
                                                                                                                       │
6: ────────────────────────────────────────────────────────────PhXZ(a=0.2,x=0.2,z=0.1)─────X───────────────────────────@────PhXZ(a=-0.2,x=0.8,z=-0.5)───H────────────────────────
                                                              └────────────────────────┘                             └──┘
    """

[codecov[bot]]

Codecov Report

Attention: Patch coverage is 99.31034% with 1 line in your changes missing coverage. Please review.

Project coverage is 97.83%. Comparing base (bb24dae) to head (0b42839). Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
...irq-core/cirq/transformers/dynamical_decoupling.py	99.21%	1 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## main #6718 +/- ## ========================================== - Coverage 97.83% 97.83% -0.01% ========================================== Files 1077 1077 Lines 92537 92558 +21 ========================================== + Hits 90535 90555 +20 - Misses 2002 2003 +1 ```

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[babacry]

I have simplified the code with some abstractions.

Also added comments in a couple of places, especially pulling through mechanisms.

[babacry]

Thanks for the careful review Nour, will submit the code shortly.

[eliottrosenberg]

Thank you both!!