Simple quantum algorithms for k-mismatch problem

Abstract

For many problems, quantum computation has significant advantage compared to classical computers. One of the most surprising example of this is the Grover Search Algorithm, which can search through any unstructured data structure of size n with O ( p n) queries. This algorithm gives quadratic speed-up to many searchbased problems. Another important example is the Shor’s factorization algorithm, which solves the factorizatio problem with ~O (n3) operations. No polynomial-time classical algorithm for factorization is known to this date. In particular, quantum computing has given significant speed-ups to many string-based problems. String processing is an active and interesting field, with many applications in fields such as bioinformatics. It is also of significant theoretical interest: a subquadratic classical solution (or lack thereof) to string problems such as can shed light to the existence (or lack) of solutions to quite a few problems. If SETH (Strong Exponential Time Hypothesis) is true, string problems such as LCS do not have strongly subquadratic time (O (n2􀀀 ) for some > 0) classical solutions. However, we can use quantum computing to get faster solutions to these problems: LCS has a quantum algorithm with ~O 􀀀 n2=3

[13] query complexity in contrast to the classical known best of ~O (n2). In this thesis, we work on the k-mismatch problem. Here, given a pattern and a text, we have to find if the text has a substring with a Hamming distance of at most k from the pattern. This is a “fault-tolerant” version of the well-known string search problem. This important problem has been extensively studied in classical setting but had not been studied through quantum computation until 2022 by Jin and Nogler [16]. In that paper, they provided an ~O (k p n)-time quantum algorithm and showed that the problem has a quantum query lower-bound of

p kn

. They posed the question of whether there is a quantum algorithm with better query complexity than ~O 􀀀 k3=4n1=2+o(1)

. In 2024, Kociumaka, Nogler, and Wellnitz [18] found an algorithm with optimal query complexity ~O ( p kn) and time complexity ~O p n=m( p km + k2)

. This thesis provides simple quantum algorithms for two variants of the k-mismatch problems. The first is when r = m􀀀k is small. For this case, we have a ~O ( p rmn)-time quantum algorithm. The second algorithm solves the problem in an approximate manner, given a parameter > 0. It returns an occurrence as a match only if it is a (1 + )k-mismatch. If it does not return any occurrence, then there is no k-mismatch. This algorithm has a time complexity of ~O ( 􀀀1 p mn=k).