We explore the space "just above" BQP by defining a complexity class PDQP (Product Dynamical Quantum Polynomial time) which is larger than BQP but does not contain NP relative to an oracle. The class is defined by imagining that quantum computers can perform measurements that do not collapse the wavefunction. This (non-physical) model of computation can efficiently solve problems such as Graph Isomorphism and Approximate Shortest Vector which are believed to be intractable for quantum computers. Furthermore, it can search an unstructured N-element list in $\tilde O(N^{1/3})$ time, but no faster than $\Omega(N^{1/4})$, and hence cannot solve NP-hard problems in a black box manner. In short, this model of computation is more powerful than standard quantum computation, but only slightly so.

Our work is inspired by previous work of Aaronson on the power of sampling the histories of hidden variables. However Aaronson's work contains an error in its proof of the lower bound for search, and hence it is unclear whether or not his model allows for search in logarithmic time. Our work can be viewed as a conceptual simplification of Aaronson's approach, with a provable polynomial lower bound for search.