It is common knowledge that a universal Turing machine can simulate any Turing machine with logarithmic overhead. Is it possible to make this overhead constant by constructing an analogous Universal multi head Turing machine?
Universal Multi Head Turing Machine
turing machines
No. The logarithmic overhead comes from the fact that the machine you're simulating could have an arbitrarily large state set $Q$ and arbitrarily large tape alphabet $\Gamma$. To simulate any such machine using a fixed, constant number of states and a fixed, constant tape alphabet, you need to code $Q$ and $\Gamma$ as multiple symbols in the universal machine's tape alphabet: specifically, you need $\log |Q|$ symbols to write down the current state of the simulated machine, and each character of the simulated tape will take $\log |\Gamma|$ symbols on the universal machine's tape.  (The base of the logarithm will depend on how many symbols you use: for example, simulating in binary will give logs to base 2.)Because of this, the logarithmic slowdown is unavoidable, even if you use multiple tapes.  Indeed, if you only use a single tape, it'll be even worse than logarithmic.However, see Shreesh's answer for a more significant source of logarithmic overhead, which is logarithmic in the running time of the simulated machine, not just the length of its description.