having problem in deriving the upper bound mentioned in the paper proof of theorem 1

[kotoriii272]

Hi lockwo, it has beem very inspiring to watch your vedios on the noise induced barren plateau problem. While I'm learning this paper, it is quite hard to me to derive this equility in the Supplementary Note 2 - Proof of Theorem 1 :

, I have no idea about how the W and V are producted together here. I would be very grateful if you can spend some of your time helping me with this issue.

[lockwo]

W here is a quantum noise channel, so that is just the operation for the noise channel application

[kotoriii272]

W here is a quantum noise channel, so that is just the operation for the noise channel application

Thanks a lot lockwo. But I'm still a bit confused about what the a(L, j) stands for

[lockwo]

a is the Pauli coefficient that represents the expected value, see eq 45. L is just the layer and j is qubit index