Quantum Search with In-Place Queries

Quantum query complexity is typically characterized in terms of XOR queries |x,y> to |x,y+f(x)> or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is unitary: in-place queries |x> to |f(x)>. Some problems are known to require exponentially fewer in-place queries than XOR queries, but no separation has been shown in the opposite direction. A candidate for such a separation was the problem of inverting a permutation over N elements. This task, equivalent to unstructured search in the context of permutations, is solvable with $O(\sqrt{N})$ XOR queries but was conjectured to require $\Omega(N)$ in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using $O(\sqrt{N})$ in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer XOR queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 XOR query and $\Theta(\sqrt{N})$ in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem.