# What do "$i$-th basic network", "quantum multiplexers" and "quantum parallelism" mean in this context? How are they beneficial?

I have been reading the paper A quantum-implementable neural network model (Chen et al., 2017) for a few days now, but failed to understand how exactly their algorithm offers a speedup over the classical neural network models.

In particular I'm confused about what they mean by and what they are trying to do with quantum multiplexers. They haven't even defined it properly.

Here's the relevant paragraph:

One of the most interesting points of QPNN is that we can use quantum parallelism to combine several basic networks at the same time. To achieve this purpose, only $$n$$ qubits are needed in a control layer to perform the following quantum multiplexer as shown in Fig. $$5$$.

$$\left( \begin{array}{ccc} U_1 \\ & \ddots \\ && U_{2^n} \end{array} \right) \tag{13}$$

where $$U_i$$ represents the dynamics of the $$i$$th basic network. Moreover $$2^{n}$$ different quantum gates $$\left\{P^{(i)}|i=1, \dots ,2^{n}\right\}$$ can also be applied on the output layer of each basic network respectively.

Questions:

1. What does $$i^{\text{th}}$$ basic network mean in this context?
2. What is quantum multiplexer and how exactly is it helping in this context? What is meant by the matrix shown in $$(13)$$? (I read a few papers on quantum multiplexers which say that they are basically used to transfer information contained by several qubits as information in a qudit. But no idea how that is relevant here.)
3. What do they mean by "we can use quantum parallelism to combine several basic networks at the same time"?
• Note: Here is the PDF version of the paper. Jun 4 '18 at 12:29
• @Nat It takes around 5 seconds for it to get loaded completely Jun 5 '18 at 4:55