The Gottesman-Knill theorem says that many circuits, including all Clifford curcuits can be simulated classically in polynomial time.
On the other hand it is believed that there is no polynomial time classical simulation of universal quantum computation.
Does there exist a (local?) gate-set which is non-universal and yet not known to be simulateable in polynomial time?
This is a commonly-asked question, whose answer is (I believe) still unknown even under standard complexity-separation conjectures like P$\ne$NP and BPP$\ne$BQP.
For example in "The Classification of Reversible Bit Operations" from 2015, Aaronson, Grier, and Schaffer (link to arXiv) list the key open problem of classifying all quantum circuits based on the gate-sets used.
This was even in Aaronson's 2005 posting on "The 10 Semi-Grand Challenges for Quantum Computing Theory" - with question 1 asking:
Also, is it true that every class of quantum gates is either universal for BQP, or else simulable in classical polynomial time? If so, what criterion separates the two? If not, do we get an interesting hierarchy of complexity classes between BPP and BQP?
Alternatively what we have learned is that there are certain restricted classes of quantum computing that might not be efficiently simulable classically - but might also not even capture all of what a classical computer can efficiently do! I'd refer to (1) IQP circuits, (2) DQC-1 circuits, and (3) BosonSampling linear optical circuits for examples of the above. These classes may contain problems outside of NP, but yet may not efficiently simulate every problem in P!
One thing that I've wondered about, but I haven't found a way to formalize, is that period-finding may not be in P but is in BQP and indeed is even in NP. For example, given an oracle $f$ with a promise that $f(x)=f(x+r)$ for some $r$, find $r$. This problem can be solved efficiently with a QFT, but also $r$ is an NP-certificate for a decision version of this problem. Thus oracularly this problem is in BQP and also in NP - which means, under the standard assumption that BQP$\not\subseteq$NP, it is somehow "sub-universal" for quantum computation.
Analogously people have studied a lot about "monotonic boolean functions" - circuits built solely out of AND and OR gates (without NOT gates). Satisfiability is efficiently solvable with such sets of gates.
I'm going to slightly reinterpret the question, because you cannot just talk about algorithms from gates. You have to talk about algorithms based on gates + initial states. For example, the simulation of stabilizer circuits in the Gottesman-Knill theorem is predicated on your starting state being a stabilizer state (such as the all 0s state).
So, is there a set of unitaries + initial states whose computational power is not know to be universal and not known to be efficiently simulatable? Yes. The model is called DQC-1. It is generally believed that it can solve some problems efficiently that classical computers cannot, and probably doesn't have the full power of a universal quantum computer (and may not even have the full power of a universal classical computer).
I think it might really depend on how you define universal, you can argue $CCX$ and $H$ are not universal as you cannot obtain any states with complex amplitudes. But at the same time any quantum circuit has an equivalent circuit with only $CCX$ and $H$ gates so that shouldn't be classically simulable!
