I was reading a paper on Quantum Neural Networks where the authors discussed a new back propagation algorithm. They shared a schematic of the circuit. However, I am unable to understand how the proposed circuit can be derived from the equations derived. Can anybody shed some light?
The paper can be found here: https://arxiv.org/pdf/1905.10912.pdf
The following equations:
$$ \vert\psi(x)\rangle = \prod_{j=1}^{2^N} x_j \vert j\rangle $$
The above is the encoding.
$$ \begin{split} \hat{\mathcal{N}} \vert \psi\rangle = \, &e^{-i\hat{H}\delta t}\vert \psi \rangle \\ = &e^{-i\hat{\sum\limits_{j=1}^{2^N} w_j \Xi_j}\delta t}\vert\psi\rangle\\ =\, &\prod_{j=1}^{2^N} e^{ - iw_j \Xi_j \delta t} \vert \psi\rangle\\ =\, &e^{ - iw_1 \Xi_1 \delta t} * e^{ - iw_2 \Xi_2 \delta t} \dots e^{ - iw_k \Xi_k \delta t} \dots e^{ - iw_{2^N} \Xi_{2^N} \delta t}\vert \psi\rangle. \\ \end{split} $$
The above is the architecture where $w_{j}$ represents the $i\,th$ weight and the funny horizontal symbol is actually one of $\{I, \sigma_{x}, \sigma_{y}, \sigma_{z}\}$, the set of Identity and Pauli matrices, $\delta t$ is a small time interval under which the system is evolved, and $\hat{H}$ is the Hamiltonian of the system.
The authors go on to state each product term can be written as (example if the horizontal symbol was actually $\sigma_{x}$):
$$ \begin{split} e^{ - i w \sigma_x \delta t} &= cos(w \delta t)I - i sin(w \delta t) \sigma_x \\ &= \left[\begin{smallmatrix} cos(w \delta t) &0 \\ 0 & cos(w \delta t) \end{smallmatrix}\right] + \left[\begin{smallmatrix} 0 &-i sin(w \delta t) \\ -i sin(w \delta t) & 0 \end{smallmatrix}\right] \\ &= \left[\begin{smallmatrix} cos(w \delta t) &-i sin(w \delta t) \\ -i sin(w \delta t) & cos(w \delta t) \end{smallmatrix}\right] \end{split} $$
They defined the loss function as below:
$$ \begin{split} \mathcal{L} &= (\langle\psi\vert \hat{\mathcal{N}}^{\star} - \langle y\vert)*(\hat{\mathcal{N}}\vert\psi\rangle-\vert y\rangle)\\ &=\langle\psi\vert \hat{\mathcal{N}}^{\star} \hat{\mathcal{N}}\vert \psi\rangle - \langle y\vert\hat{\mathcal{N}}\vert\psi\rangle - \langle\psi\vert\hat{\mathcal{N}}^*\vert y\rangle+\langle y | y \rangle\\ &= 2 -\langle y\vert\hat{\mathcal{N}}\vert\psi\rangle - \langle\psi\vert\hat{\mathcal{N}}^*\vert y\rangle \end{split} $$
And the training involves the following equations:
$$ \begin{split} \frac{\partial \mathcal{L}}{\partial w_k} &= -\langle y\vert\frac{\partial \hat{\mathcal{N}}}{\partial w_k}\vert\psi\rangle - \langle\psi\vert\frac{\partial \hat{\mathcal{N}}^*}{\partial w_k}\vert y\rangle; \\ \frac{\partial \hat{\mathcal{N}}}{\partial w_k} &= e^{ - i\alpha_1 \Xi_1 \delta t} * e^{ - i\alpha_2 \Xi_2 \delta t} \dots (-i \Xi_k \delta t) \,e^{ - i\alpha_k \Xi_k \delta t} \dots e^{ - i\alpha_n \Xi_n \delta t};\\ \frac{\partial \hat{\mathcal{N}}^*}{\partial w_k} &= e^{ i\alpha_1 \Xi_1 \delta t} * e^{ i\alpha_2 \Xi_2 \delta t} \dots (i \Xi_k \delta t) \,e^{ i\alpha_k \Xi_k \delta t} \dots e^{ i\alpha_n \Xi_n \delta t}. \end{split} $$
Finally, they came up with this schematic:
Can anybody please explain how the circuit is derived from the equations?
