A Quantum Algorithm for the Hamiltonian NAND Tree
E. Farhi, J. Goldstone, S. Gutmann
TL;DR
This paper addresses evaluating a binary NAND tree in the Hamiltonian oracle model by leveraging a continuous-time quantum walk on a graph that combines a bifurcating tree with a long runway. The input is encoded in the Hamiltonian oracle, and the root value is read out from the energy-0 transmission coefficient of a right-moving packet, yielding a runtime of $O(\sqrt{N})$. A matching $\Omega(\sqrt{N})$ lower bound is proven via a Hamiltonian parity reduction, establishing the algorithm's optimality in this model. The method connects scattering theory and energy-parameterized transmission to NAND-tree evaluation and demonstrates that quantum walks can achieve square-root speedups in Hamiltonian oracle settings.
Abstract
We give a quantum algorithm for the binary NAND tree problem in the Hamiltonian oracle model. The algorithm uses a continuous time quantum walk with a run time proportional to sqrt N. We also show a lower bound of sqrt N for the NAND tree problem in the Hamiltonian oracle model.