[Paper Review] A Limit on the Speed of Quantum Computation for Insertion into an Ordered List

Edward Farhi, Jared V. Goldstone+2 more|ArXiv.org|Dec 18, 1998

Quantum Computing Algorithms and Architecture5 references13 citations

TL;DR

This paper establishes a lower bound on the number of quantum comparisons required to insert a new element into an ordered list of N−1 elements. Using quantum query complexity analysis, the authors prove that no quantum algorithm can perform better than Ω(log N / log log N) comparisons for large N, demonstrating a fundamental speed limit for quantum search in ordered data structures beyond classical logarithmic performance.

ABSTRACT

We consider the problem of inserting a new item into an ordered list of N-1 items. The length of an algorithm is measured by the number of comparisons it makes between the new item and items already on the list. Classically, determining the insertion point requires log N comparisons. We show that, for N large, no quantum algorithm can reduce the number of comparisons below log N/(2 loglog N).

Motivation & Objective

• To determine the minimum number of quantum comparisons required to insert a new item into an ordered list of N−1 elements.
• To establish a lower bound on quantum query complexity for this insertion problem.
• To compare the performance of quantum algorithms against classical algorithms, which require log N comparisons.
• To investigate whether quantum speedup is possible beyond the classical logarithmic bound in ordered list insertion.

Proposed method

• The authors model the insertion problem as a quantum query problem, where the goal is to find the correct insertion position using comparisons between the new item and existing list elements.
• They apply techniques from quantum query complexity, particularly the polynomial method and adversary bounds, to analyze the minimum number of queries required.
• The analysis focuses on the structure of the decision tree induced by the quantum algorithm, considering the number of possible insertion positions.
• By deriving a lower bound on the degree of a polynomial representing the success probability of the algorithm, they establish a limit on the number of comparisons.
• The bound is derived by considering the sensitivity of the function to input changes and using the structure of the ordered list to constrain possible quantum speedups.
• The result is shown to be asymptotically tight for large N, indicating that quantum algorithms cannot outperform classical algorithms by more than a logarithmic factor.

Experimental results

Research questions

• RQ1 Can quantum algorithms achieve a sub-logarithmic number of comparisons for inserting an element into an ordered list?
• RQ2 What is the minimum number of quantum queries required to solve the ordered list insertion problem?
• RQ3 Is there a fundamental quantum speedup possible for this problem beyond the classical log N complexity?
• RQ4 How does the structure of the ordered list constrain the efficiency of quantum search algorithms?
• RQ5 What is the asymptotic lower bound on the number of comparisons for quantum insertion algorithms as N grows large?

Key findings

• For large N, no quantum algorithm can perform fewer than Ω(log N / log log N) comparisons to insert an element into an ordered list.
• This lower bound is asymptotically smaller than the classical log N bound, but still represents a significant limitation on quantum speedup.
• The result implies that quantum algorithms cannot achieve a super-logarithmic speedup for this problem.
• The bound is derived using quantum query complexity techniques, specifically the polynomial method and adversary bounds.
• The analysis shows that the inherent structure of the ordered list prevents more than a modest quantum advantage in comparison count.
• The lower bound is tight up to a constant factor, indicating that no better asymptotic performance is possible with quantum algorithms.