How can I use an Arduino UNO as the controller in a two-stage switch?

Hello there! I'm having trouble trying to figure out the implementation of a digital two-stage space division switch. [SOLVED]

There are 3 inputs, 6 control signals and 3 outputs? Is that right? What voltage are the outputs and what do the inputs and control inputs require?

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Those logic gates look like they are wired so all three of the terminals in the middle will be the same (an OR of whichever of the three inputs are enabled by the control signals).

Did you want a 3-in/3-out crossbar? That's going to require two address bits for each output (6 total) but not wired the way you show.