[Paper Review] A better lower bound for quantum algorithms searching an ordered list

Andris Ambainis|ArXiv.org|Feb 14, 1999

Quantum Computing Algorithms and Architecture5 references4 citations

TL;DR

This paper establishes a new lower bound of 1/12 log n − O(1) queries for quantum algorithms searching an ordered list of size n, demonstrating that quantum speedup for this problem is limited to a constant factor. Using a refined quantum adversary method, the author improves upon prior bounds and shows that quantum algorithms cannot achieve more than a constant-factor advantage over classical algorithms for ordered search.

ABSTRACT

We show that any quantum algorithm searching an ordered list of n elements needs to examine at least 1/12 log n-O(1) of them. Classically, log n queries are both necessary and sufficient. This shows that quantum algorithms can achieve only a constant speedup for this problem. Our result improves lower bounds of Buhrman and de Wolf(quant-ph/9811046) and Farhi, Goldstone, Gutmann and Sipser (quant-ph/9812057).

Motivation & Objective

• To establish a tighter lower bound on the number of queries required by quantum algorithms to search an ordered list.
• To address the fundamental question of whether quantum algorithms can achieve more than a constant-factor speedup over classical algorithms for ordered search.
• To improve upon existing lower bounds by Buhrman, de Wolf, and Farhi et al. using a refined quantum adversary technique.
• To clarify the limits of quantum advantage in the context of structured search problems.

Proposed method

• The author applies a refined version of the quantum adversary method to analyze the query complexity of searching an ordered list.
• The analysis focuses on the distinguishability of quantum states during the search process, tracking the evolution of amplitude distributions.
• A carefully constructed input pair is used to model the difficulty of distinguishing between two different list configurations.
• The method quantifies the minimal number of queries needed to achieve a significant success probability in identifying the target element.
• The bound is derived by analyzing the growth of the quantum state's variation over time, leveraging inner product inequalities.
• The final lower bound is expressed as 1/12 log n − O(1), improving upon previous results.

Experimental results

Research questions

• RQ1 What is the minimum number of queries required by any quantum algorithm to search an ordered list of size n?
• RQ2 Can quantum algorithms achieve more than a constant-factor speedup over classical algorithms for ordered search?
• RQ3 How does the quantum adversary method refine existing lower bounds in the context of ordered search?
• RQ4 What is the tightest possible lower bound on quantum query complexity for this problem?

Key findings

• The paper establishes a new lower bound of 1/12 log n − O(1) for quantum query complexity in ordered search.
• This bound improves upon the previous lower bounds by Buhrman and de Wolf and by Farhi et al.
• The result shows that quantum algorithms can achieve only a constant-factor speedup over classical algorithms for ordered search.
• The classical lower bound is log n, and the quantum bound is asymptotically 1/12 log n, indicating a limited quantum advantage.
• The analysis confirms that the quantum adversary method can yield tighter bounds when applied with careful input pair construction.
• The result implies that the ordered search problem does not exhibit a super-constant quantum speedup, limiting its potential for quantum advantage.