Models of computation strictly between classical and quantum in terms of query complexity It is well known quantum computers are strictly more powerful than their classical counterparts in terms of query complexity. 


  Are there other models (natural or artificial) that are strictly between the quantum and classical in terms of query complexity? 


The seperation can be on


specific problems: model X computes function $f$ with strictly more queries than quantum, but fewer queries than the lower bound on classic, or
different problems: model X computes function $f_1$ with strictly more queries than quantum, but computes function $f_2$ with fewer queries than classical. 


In both cases, we want for every function $f$ to have $Q_2(f) \leq X(f) \leq R_2(f)$ to avoid examples that are hard to compare to quantum (like the certificate complexity of non-deterministic queries). Here $Q_2(f)$ (and $R_2(f)$) is the two-sided $1/3$-error quantum (and classical randomized) query complexity and the inequalities are within constant factors.
One easy way to come up with such a model is to first create a restricted model of quantum computation that can still do something non-classical, and then just give it classical computation for free.

An examples of this strategy is the one clean qubit model (along with a BPP machine). Some references: On the Power of One Bit of Quantum Information, Computation with Unitaries and One Pure Qubit and Estimating Jones polynomials is a complete problem for one clean qubit.

Another example would be to have a log-depth (or polylog depth) quantum circuit with access to a classical computer. This will yield something like $BPP^{BQNC}$.