Super-Polynomial Quantum Speed-ups for Boolean Evaluation Trees with Hidden Structure

Abstract

We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth $n$ tree using $O(n^{2+\log \omega })$ queries, where $\omega$ is independent of $n$ and depends only on the type of subformulas within the tree. We also prove a classical lower bound of $n^{\Omega (\log \log n)}$ queries, thus showing a (small) super-polynomial speed-up.

30 pages, 2 figures, v3: clarified exposition, matches journal reference

References in corpus (5)

• A Quantum Algorithm for the Hamiltonian NAND Tree
• Span-program-based quantum algorithm for evaluating formulas
• Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer
• Superpolynomial speedups based on almost any quantum circuit
• Quantum Lower Bound for Recursive Fourier Sampling

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• Taming Quantum Time Complexity
• Exponential improvements for quantum-accessible reinforcement learning
• NAND-Trees, Average Choice Complexity, and Effective Resistance
• Quantum Algorithms for Learning Symmetric Juntas via the Adversary Bound
• Quantum Algorithms for Graph Connectivity and Formula Evaluation
• Quantum Algorithm for Monotonicity Testing on the Hypercube
• Speed from Repetition