Well, I interpret the question as "what can computing be based on?". And the answer is: literally anything.
Vacuum tubes and miniaturizationminiaturisation
Tubes were around for a quite a time, there were some efforts to miniaturise them. Also, as the folk propaganda says, they are less affected by EMP, which is important in the Cuba crisis era.
There were real computersreal computers built with tubes. Efforts were made to miniaturise. Most prominent non-computing special purpose: radars and microwave ovens.
Why transistors?
Transistors can be manufactured relatively easily, they can be easily scaled down, creating a microchip, they can switch very fast, raising the tact frequency and hence the computational power.
However, at least in the very beginning, other technological bases were viable.
Pneumatics
Neil Stephenson made a fictional, but a very convincing example of an early computer based on compressed air and ventils, i.e. pneumatics.
Gears
Your favourite steampunk example is the Difference engineDifference engine (never built in reality at that time, modern replicas to some extent were built).
Relays
Follow the stepssteps of Zuse: why bother with transistors (not there yet) or tubes (meh), if you could use relays, that clacking switch things.
Biology
Use some cells capable of switching or accumulating signals, if you go very soft Sci-Fi to biopunk, maybe the neural cells as such.
Analogue computing
Span a resin sheet of given surface with given properties. Place some balls of given weight at finely calculated positions. Watch them move. Read up the final position. Voila, you have just solved a differential equation with an analogue computer.
Many more schemesMany more schemes were viable, from abacus to somewhat advanced physical simulations. They all died when digital computers became feasible enough.
Quantum computing
Even we (as in: humanity) have not really got there yet, but it does not mean, someone might not be faster or better. If you focus on reactors and jet engines instead of silicon, you might develop the physics for quantum computer faster than we did, even with some... deficits in computing power.
Basically, quantum computing is misusing the states of the matter to do you computing tasks, sort of similar to analogue computers, but on a wholly different level.