To Move Fast, Quantum Maze Solvers Must Forget the Past

Imagine you visit a maze with some friends. You emerge from the exit shortly after going in, and wait around for hours before your friends emerge. Naturally, they ask about the path you took — surely you can retrace your steps and show them the way, right?

Last November, Coudron and two colleagues took a big step toward resolving that long-standing problem: They proved that no algorithm in a broad and natural class of fast quantum algorithms can find a path through that special maze, called a welded tree graph. The results show that any hypothetical path-finding algorithm that doesn’t blindly guess would have to temporarily lose track of the entrance to have any chance of succeeding. It seems that forgetting is inevitable.

“Classically, what you could do is just randomly try one and see if it’s good, and then try again and see if it’s good, and you keep on trying until you find a good element,” Apers said. This approach takes time proportional to the number of boxes. Multiply that number by 100, and the search will be 100 times slower.

Grover’s problem, for example, corresponds to searching a graph in which every node is connected to every other node . Different classical algorithms for a given search problem amount to different strategies for exploring the corresponding graph one node at a time, while quantum algorithms can move along multiple edges in superposition.In 2002, a team of computer scientists finally identified a classically intractable search problem that a quantum algorithm could solve easily.

Researchers wondered if there was some way to get the best of both worlds — a fast algorithm that identifies a path from entrance to exit. Then in 2019, Coudron encountered the welded tree graph in a different context: He and a colleague proved that all quantum walk algorithms that find the exit lack a property universal among algorithms that were known to yield exponential quantum speedups for other problems.

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