Invariant Quantum Algorithms for Insertion into an Ordered List

TL;DR

This paper introduces invariant quantum algorithms for inserting an item into an ordered list, demonstrating they can outperform classical methods and potentially solve the problem with fewer than logarithmic queries.

Contribution

The paper develops invariant quantum algorithms for list insertion, showing they can surpass classical query complexity and achieve sub-logarithmic query performance.

Findings

Invariant quantum algorithms outperform classical algorithms for list insertion.

Some invariant algorithms succeed with fewer queries than classical lower bounds.

Iterating certain algorithms reduces query complexity below 0.53 log N.

Abstract

We consider the problem of inserting one item into a list of N-1 ordered items. We previously showed that no quantum algorithm could solve this problem in fewer than log N/(2 log log N) queries, for N large. We transform the problem into a "translationally invariant" problem and restrict attention to invariant algorithms. We construct the "greedy" invariant algorithm and show numerically that it outperforms the best classical algorithm for various N. We also find invariant algorithms that succeed exactly in fewer queries than is classically possible, and iterating one of them shows that the insertion problem can be solved in fewer than 0.53 log N quantum queries for large N (where log N is the classical lower bound). We don't know whether a o(log N) algorithm exists.