What is the difference between a collection of Turing Machines and a family of Circuits?

Question

Given a Collection of Turing Machines $T_1, T_2, T_3,...T_n$ where $T_1$ denotes that the Turing machine can only take in an input of size 1. Is there any difference in computational power to a family of Circuits $C_1, C_2, C_3,...C_n$ ?

What if we assumed that each Turing machine, encoded special information for that specific input size to make each instance efficient?

If there is no difference in computational power, then maybe we could use this to define non-uniform algorithms instead of circuits?

Answer 1

What you describe is essentially Turing machines with advice, the advice for length $i$ being simply the description of $T_i$. It is a classic result that the two models are equivalent in the case of poly-time TMs and poly-sized circuits, that is, both produce the same class $\mathsf{P}/\mathrm{poly}$. If the description length of $T_i$ is allowed to be arbitrary, there is also no computational power to be gained since you can then express all Boolean functions for each length (just as with circuits).