The edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [21]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an O(n1.858) quantum algorithm that approximates the edit distance within a factor of 7. We further extend this result to an O(n1.781) quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of 3, with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.
The edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [21]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an O(n1.858) quantum algorithm that approximates the edit distance within a factor of 7. We further extend this result to an O(n1.781) quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of 3, with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.
Approximate Pattern Matching is among the most fundamental string-processing tasks. Given a text T of length n, a pattern P of length m, and a threshold k, the task is to identify the fragments of T that are at distance at most k to P. We consider the two most common distances: Hamming distance (the number of mismatches or character substitutions) in Pattern Matching with Mismatches and edit distance (the minimum number of character insertions, deletions, and substitutions) in Pattern Matching with Edits. We revisit the complexity of these two problems in the quantum setting.
Classically, the edit distance of two length-n strings can be computed in O(n2) time, whereas an O(n2−ε)-time procedure would falsify the Orthogonal Vectors Hypothesis. If the edit distance does not exceed k, the running time can be improved to O(n+k2), which is near-optimal (conditioned on OVH) as a function of n and k. Our first main contribution is a quantum O~(nk+k2)-time algorithm that uses O~(nk) queries, where O~(⋅) hides polylogarithmic factors. This query complexity is unconditionally optimal, and any significant improvement in the time complexity would resolve a long-standing open question of whether edit distance admits an O(n2−ε)-time quantum algorithm. Our divide-and-conquer quantum algorithm reduces the edit distance problem to a case where the strings have small Lempel-Ziv factorizations. Then, it combines a quantum LZ compression algorithm with a classical edit-distance subroutine for compressed strings. The LZ factorization problem can be classically solved in O(n) time, which is unconditionally optimal in the quantum setting. We can, however, hope for a quantum speedup if we parameterize the complexity in terms of the factorization size z. Already a generic oracle identification algorithm yields the optimal query complexity of O~(nz) at the price of exponential running time. Our second main contribution is a quantum algorithm that achieves the optimal time complexity of O~(nz). The key tool is a novel LZ-like factorization of size O(zlog2n) whose subsequent factors can be efficiently computed through a combination of classical and quantum techniques. We can then obtain the string's run-length encoded Burrows-Wheeler Transform (BWT), construct the r-index, and solve many fundamental string processing problems in time O~(nz).
Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor w, where w is the computer word size. For example, edit distance of two strings of length n can be solved in O(n2/w) time. In a reasonable classical model of computation, one can assume w=Θ(logn). There are conditional lower bounds for such problems stating that speed-ups with factor nε for any ε>0 would lead to breakthroughs in complexity theory. However, these conditional lower bounds do not cover quantum models of computing. Indeed, Boroujeni et al. (J. ACM, 2021) showed that edit distance can be approximated within a factor 3 in sub-quadratic time O(n1.81) using quantum computing. They also showed that, in their chosen model of quantum computing, the approximation factor cannot be improved using sub-quadractic time. To break through the aforementioned classical conditional lower bounds and this latest quantum lower bound, we enrich the model of computation with a quantum random access memory (QRAM), obtaining what we call the word QRAM model. Under this model, we show how to convert the bit-parallelism of quadratic time solvable problems into quantum algorithms that attain speed-ups with factor n. The technique we use is simple and general enough to apply to many bit-parallel algorithms that use Boolean logics and bit-shifts. To apply it to edit distance, we first show that the famous O(n2/w) time bit-parallel algorithm of Myers (J. ACM, 1999) can be adjusted to work without arithmetic + operations. As a direct consequence of applying our technique to this variant, we obtain linear time edit distance algorithm under the word QRAM model for constant alphabet. We give further results on a restricted variant of the word QRAM model to give more insights to the limits of the model.
Smoothed analysis is a central framework in classical algorithms for explaining the performance of algorithms beyond the worst case, often explaining why algorithms perform well in practice. We initiate a systematic study of its quantum counterpart and show the following results. (1) We show that there is a total function whose smoothed quantum query complexity is exponentially smaller than its classical query complexity. (2) We give near-tight characterizations of smoothed randomized and quantum query complexities for symmetric Boolean functions, unifying the worst-case complexity results of [Beals et al, FOCS'98] and average-case complexity results of [Ambainis and de Wolf, STACS'00]. (3) We study string problems such as pattern matching and edit distance and, in various regimes, give polynomial to superpolynomial quantum speedups. Our main technical ingredients include a near-tight quantum algorithm for ε-approximating the number of collisions between two non-repetitive strings, improving the result of Le Gall and Ng [QIC'22]. Together, our results show that smoothing can reveal larger quantum speedups than worst-case analysis suggests, opening a path towards quantum advantage on more realistic inputs.
Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor w, where w is the computer word size. A classic example is computing the edit distance of two strings of length n, which can be solved in O(n2/w) time. In a reasonable classical model of computation, one can assume w=Θ(logn), and obtaining significantly better speed-ups is unlikely in the light of conditional lower bounds obtained for such problems. In this paper, we study the connection of bit-parallelism to quantum computation, aiming to see if a bit-parallel algorithm could be converted to a quantum algorithm with better than logarithmic speed-up. We focus on string matching in labeled graphs, the problem of finding an exact occurrence of a string as the label of a path in a graph. This problem admits a quadratic conditional lower bound under a very restricted class of graphs (Equi et al. ICALP 2019), stating that no algorithm in the classical model of computation can solve the problem in time O(∣P∣∣E∣1−ε) or O(∣P∣1−ε∣E∣). We show that a simple bit-parallel algorithm on such restricted family of graphs (level DAGs) can indeed be converted into a realistic quantum algorithm that attains subquadratic time complexity O(∣E∣∣P∣).