Classical algorithms for Forrelation

We study the forrelation problem: given a pair of $n$-bit Boolean functions $f$ and $g$, estimate the correlation between $f$ and the Fourier transform of $g$. This problem is known to provide the largest possible quantum speedup in terms of its query complexity and achieves the landmark oracle separation between the complexity class BQP and the Polynomial Hierarchy. Our first result is a classical algorithm for the forrelation problem which has runtime $O(n2^{n/2})$. This is a nearly quadratic improvement over the best previously known algorithm. Secondly, we show that quantum query algorithm that makes $t$ queries to an $n$-bit oracle can be simulated by classical query algorithm making only $O(2^{n(1-1/2t)})$ queries. This fixes a gap in the literature arising from a recently discovered critical error in a previous proof; it matches recently established lower bounds (up to $poly(n,t))$ factors) and thus characterizes the maximal separation in query complexity between quantum and classical algorithms. Finally, we introduce a graph-based forrelation problem where $n$ binary variables live at vertices of some fixed graph and the functions $f,g$ are products of terms describing interactions between nearest-neighbor variables. We show that the graph-based forrelation problem can be solved on a classical computer in time $O(n)$ for any bipartite graph, any planar graph, or, more generally, any graph which can be partitioned into two subgraphs of constant treewidth. The graph-based forrelation is simply related to the variational energy achieved by the Quantum Approximate Optimization Algorithm (QAOA) with two entangling layers and Ising-type cost functions. By exploiting the connection between QAOA and the graph-based forrelation we were able to simulate the recently proposed Recursive QAOA with two entangling layers and $225$ qubits on a laptop computer.