Quantum Property Testing Algorithm for the Concatenation of Two Palindromes Language

TL;DR

The paper investigates quantum property testing for the context-free language $L_{REV}=\{uu^rvv^r: u,v\in\Sigma^*\}$, showing a quantum query algorithm with $O\left(\frac{1}{\varepsilon}n^{1/3}\log n\right)$ queries that surpasses the classical $Θ^*(\sqrt{n})$ bound in the property-testing setting. In the general decision problem, the authors obtain a quantum upper bound of $O\left(\sqrt{n}(\log n)^2\right)$ with a matching $\Omega(\sqrt{n})$ lower bound, implying an almost quadratic speed-up over the classical $Θ(n)$. The work leverages meet-in-the-middle techniques and Grover search, together with a trie-based data structure, to reduce the search space and achieve sublinear query complexity. Open questions include determining a quantum lower bound in the property-testing setting and extending the approach to other context-free languages. Overall, the results establish a measurable quantum advantage for recognizing a nontrivial context-free language under property testing and related settings.

Abstract

In this paper, we present a quantum property testing algorithm for recognizing a context-free language that is a concatenation of two palindromes $L_{REV}$. The query complexity of our algorithm is $O(\frac{1}{\varepsilon}n^{1/3}\log n)$, where $n$ is the length of an input. It is better than the classical complexity that is $Θ^*(\sqrt{n})$. At the same time, in the general setting, the picture is different a little. Classical query complexity is $Θ(n)$, and quantum query complexity is $Θ^*(\sqrt{n})$. So, we obtain polynomial speed-up for both cases (general and property testing).

Figures (4)

• Figure 1: The string $x^r$ is a substring of $y(x)$ and it starts from position $i$. We can see that $(x_{i+1},\dots,x_{n-1})$ is a palindrome.
• Figure 2: The string $x^r$ is a substring of $y(x)$ and it starts from position $i$. We can see that $(x_{0},\dots,x_{i})$ is a palindrome.
• Figure 3: Indexes $i$ and $j$ are symmetric with respect to the middle of the palindrome $uu^r$.
• Figure 4: Indexes $i$ and $j$ are symmetric with respect to the middle of the palindrome $vv^r$.

Theorems & Definitions (12)

• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof