Quantum computation and decision trees

TLDR

Many interesting computational problems can be reformulated in terms of decision trees. The authors devise a quantum algorithm that evolves a state from the root through the decision tree. They analyze a classical random‑walk strategy and introduce a quantum evolution that propagates a state from the root. They prove that if the classical random walk reaches level n in polynomial time, the quantum algorithm does too, and they exhibit trees where the quantum method is polynomial while the classical random walk is exponential, though other classical algorithms can also solve those trees in polynomial time.

Abstract

Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if the tree contains a node $n$ level from the root. We devise a quantum-mechanical algorithm that evolves a state, initially localized at the root, through the tree. We prove that if the classical strategy succeeds in reaching level $n$ in time polynomial in $n,$ then so does the quantum algorithm. Moreover, we find examples of trees for which the classical algorithm requires time exponential in $n,$ but for which the quantum algorithm succeeds in polynomial time. The examples we have so far, however, could also be solved in polynomial time by different classical algorithms.