Why does the qubit give random results in the circuit with rearranged CNOTs for Steane's seven qubit code in Stim?

Question

The following is a part of the syndrome measurement circuit for Steane's seven qubit code in Stim(For ease of viewing, the TICK is omitted.). Since we are considering the detection of X errors, we use the measurement results of the Z stabilizers' ancilla. Circuit_1 is a circuit that constructs the CNOTs of the Z stabilizer circuit after configuring the CNOTs of the X stabilizer circuit, and since it is a circuit with no errors, the results of DETECTOR in circuit_1 will always be False, which is the desired result. On the other hand, when considering circuit_2, which is a circuit designed to reduce the overall circuit depth by cleverly arranging the order of CNOTs, the CNOT operations are the same as in circuit_1, but the result of qubit 12 becomes random. Why is this happening?

circuit_1=stim.Circuit('''

    #encode into code state
    MPP X3*X4*X5*X6
    MPP X1*X2*X5*X6
    MPP X0*X2*X4*X6
    
    #ancilla of X stabilizers
    RX 7 8 9 
    
    #ancilla of Z stabilizers
    R 10 11 12 
    
    #CNOT of X stabilizers
    CX 7 6 8 5
    CX 7 2 8 1
    CX 7 4 8 6
    CX 7 0 8 2 9 5
    CX 9 6
    CX 9 3
    CX 9 4
    
    #CNOT of Z stabilizers
    CX 6 12
    CX 3 12
    CX 4 12
    CX 6 10 5 11
    CX 2 10 1 11
    CX 4 10 6 11    
    CX 0 10 2 11 5 12
    
    #measurement of stabilizers
    MRX 7 8 9
    MR 10 11 12
    DETECTOR rec[-3]
    DETECTOR rec[-2]
    DETECTOR rec[-1]

''')

circuit_2=stim.Circuit('''

    MPP X3*X4*X5*X6
    MPP X1*X2*X5*X6
    MPP X0*X2*X4*X6

    RX 7 8 9 
    R 10 11 12 

    CX 7 6 8 5
    CX 7 2 8 1 6 12
    CX 7 4 8 6 3 12
    CX 7 0 8 2 9 5 4 12
    CX 6 10 5 11
    CX 9 6 2 10 1 11
    CX 9 3 4 10 6 11    
    CX 9 4 0 10 2 11 5 12

    MRX 7 8 9
    MR 10 11 12
    DETECTOR rec[-3]
    DETECTOR rec[-2]
    DETECTOR rec[-1]

''')
```

Answer 1

I can't read your circuits, unfortunately. But if rearranging changes the outcome of your circuit, then the gates do not all commute.

Answer 2

It doesn't work because you've rearrange your circuit as if the CNOT gates don't interact with each other, but they do.

Here is what you should do:

1. Add the missing three detectors to your circuit, comparing the X stabilizer measurements at the end to the ones at the start. You also want to be checking the X stabilizers, to prevent mistakes.
2. Run circuit.detector_error_model() on the circuit. It checks that all the detectors are actually deterministic. It should finish without throwing an exception.
3. Move ONE CNOT gate ONE step.
4. Run circuit.detector_error_model() on the circuit. If it throws an exception, you just made a mistake. Don't move that CNOT that way. Put it back. Otherwise it's okay.
5. Goto step (2).
6. Start making multi-step moves. Some things can only be achieved by travelling through wrongspace; making more than one change at a time.

Note you can use circuit.diagram() to get a text diagram of the circuit, or circuit.diagram('timeline-svg') to get a picture diagram of the circuit, and if you add TICKs between layers then you can use circuit.diagram('detslice-with-ops-svg') to get the progression of the stabilizers corresponding to the detecting regions. That last one requires you to give decent qubit coordinates, so it doesn't just draw it on a line.

import stim
circuit_1=stim.Circuit('''
    QUBIT_COORDS(0, 0) 0
    QUBIT_COORDS(5, 0) 2
    QUBIT_COORDS(0, 5) 4
    QUBIT_COORDS(5, 5) 6
    QUBIT_COORDS(7, 7) 5
    QUBIT_COORDS(2, 7) 3
    QUBIT_COORDS(7, 2) 1
    QUBIT_COORDS(2.5, 2.5, 0) 7
    QUBIT_COORDS(6, 3.5, 0) 8
    QUBIT_COORDS(3.5, 6, 0) 9
    QUBIT_COORDS(2.5, 2.5, 1) 10
    QUBIT_COORDS(6, 3.5, 1) 11
    QUBIT_COORDS(3.5, 6, 1) 12

    #encode into code state
    MPP X3*X4*X5*X6
    MPP X1*X2*X5*X6
    MPP X0*X2*X4*X6
    TICK
    TICK
    
    #ancilla of X stabilizers
    RX 7 8 9 
    
    #ancilla of Z stabilizers
    R 10 11 12 
    TICK
    
    #CNOT of X stabilizers
    CX 7 6 8 5
    TICK
    CX 7 2 8 1
    TICK
    CX 7 4 8 6
    TICK
    CX 7 0 8 2 9 5
    TICK
    CX 9 6
    TICK
    CX 9 3
    TICK
    CX 9 4
    TICK
    
    #CNOT of Z stabilizers
    CX 6 12
    TICK
    CX 3 12
    TICK
    CX 4 12
    TICK
    CX 6 10 5 11
    TICK
    CX 2 10 1 11
    TICK
    CX 4 10 6 11    
    TICK
    CX 0 10 2 11 5 12
    TICK
    
    #measurement of stabilizers
    MRX 7 8 9
    MR 10 11 12
    DETECTOR rec[-3]
    DETECTOR rec[-2]
    DETECTOR rec[-1]
    DETECTOR rec[-4] rec[-9]
    DETECTOR rec[-5] rec[-8]
    DETECTOR rec[-6] rec[-7]
''')

circuit_1.diagram()

      /---------------------------------------------------------------\      /-\ /-\ /-\ /---\             /-\ /-\ /-\ /---\ /----------------------------------\
 q0: -QUBIT_COORDS(0,0)-----------------------------------MPP[X]:rec[2]------------------X-----------------------------@----------------DETECTOR:D5=rec[3]*rec[2]-
                                                          |                              |                             |
 q1: -QUBIT_COORDS(7,2)---------------------MPP[X]:rec[1]-|------------------------X-----|-----------------------@-----|----------------DETECTOR:D4=rec[4]*rec[1]-
                                            |             |                        |     |                       |     |
 q2: -QUBIT_COORDS(5,0)---------------------MPP[X]:rec[1]-MPP[X]:rec[2]----------X-|-----|-X-------------------@-|-----|-@----------------------------------------
                                            |             |                      | |     | |                   | |     | |
 q3: -QUBIT_COORDS(2,7)-------MPP[X]:rec[0]-|-------------|----------------------|-|-----|-|-----X-----@-------|-|-----|-|--------------DETECTOR:D3=rec[5]*rec[0]-
                              |             |             |                      | |     | |     |     |       | |     | |
 q4: -QUBIT_COORDS(0,5)-------MPP[X]:rec[0]-|-------------MPP[X]:rec[2]----------|-|-X---|-|-----|-X---|-@-----|-|-@---|-|----------------------------------------
                              |             |             |                      | | |   | |     | |   | |     | | |   | |
 q5: -QUBIT_COORDS(7,7)-------MPP[X]:rec[0]-MPP[X]:rec[1]-|--------------------X-|-|-|---|-|-X---|-|---|-|---@-|-|-|---|-|-@--------------------------------------
                              |             |             |                    | | | |   | | |   | |   | |   | | | |   | | |
 q6: -QUBIT_COORDS(5,5)-------MPP[X]:rec[0]-MPP[X]:rec[1]-MPP[X]:rec[2]------X-|-|-|-|-X-|-|-|-X-|-|-@-|-|-@-|-|-|-|-@-|-|-|--------------------------------------
                                                                             | | | | | | | | | | | | | | | | | | | | | | | |
 q7: -QUBIT_COORDS(2.5,2.5,0)---------------------------------------------RX-@-|-@-|-@-|-@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-MRX:rec[3]---------------------------
                                                                               |   |   |   | | | | | | | | | | | | | | | | |
 q8: -QUBIT_COORDS(6,3.5,0)-----------------------------------------------RX---@---@---@---@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-MRX:rec[4]---------------------------
                                                                                             | | | | | | | | | | | | | | | |
 q9: -QUBIT_COORDS(3.5,6,0)-----------------------------------------------RX-----------------@-@-@-@-|-|-|-|-|-|-|-|-|-|-|-|-MRX:rec[5]---------------------------
                                                                                                     | | | | | | | | | | | |
q10: -QUBIT_COORDS(2.5,2.5,1)---------------------------------------------R--------------------------|-|-|-X-|-X-|-X-|-X-|-|-MR:rec[6]--DETECTOR:D0=rec[6]--------
                                                                                                     | | |   |   |   |   | |
q11: -QUBIT_COORDS(6,3.5,1)-----------------------------------------------R--------------------------|-|-|---X---X---X---X-|-MR:rec[7]--DETECTOR:D1=rec[7]--------
                                                                                                     | | |                 |
q12: -QUBIT_COORDS(3.5,6,1)-----------------------------------------------R--------------------------X-X-X-----------------X-MR:rec[8]--DETECTOR:D2=rec[8]--------
      \---------------------------------------------------------------/      \-/ \-/ \-/ \---/             \-/ \-/ \-/ \---/ \----------------------------------/

circuit_1.diagram('timeline-svg')

circuit_1.diagram('detslice-with-ops-svg')