Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the "sculpting problem." In this work, we give a full characterization of sculptable functions in the query complexity setting. We show that a total function f can be restricted to a promise P such that Q ( f P ) = O ( polylo g N ) and R ( f P ) = N (1) , if and only if f has a large number of inputs with large certificate complexity. The proof uses some interesting techniques: for one direction, we introduce new relationships between randomized and quantum query complexity in various settings, and for the other direction, we use a recent result from communication complexity due to Klartag and Regev. We also characterize sculpting for other query complexity measures, such as R ( f ) vs. R 0 ( f ) and R 0 ( f ) vs. D ( f ) .

Along the way, we prove some new relationships for quantum query complexity: for example, a nearly quadratic relationship between Q ( f ) and D ( f ) whenever the promise of f is small. This contrasts with the recent super-quadratic query complexity separations, showing that the maximum gap between classical and quantum query complexities is indeed quadratic in various settings - just not for total functions!

Lastly, we investigate sculpting in the Turing machine model. We show that if there is any BPP-bi-immune language in BQP, then every language outside BPP can be restricted to a promise which places it in PromiseBQP but not in PromiseBPP. Under a weaker assumption, that some problem in BQP is hard on average for P/poly, we show that every paddable language outside BPP is sculptable in this way.