Problem
Research questions and friction points this paper is trying to address.
Quantum query complexity
Smoothed analysis
Beyond worst-case
Quantum speedup
Boolean functions
Quantum Query Complexity Beyond the Worst Case
🤖 AI Summary
This study addresses the limitation of traditional quantum query complexity, which focuses solely on worst-case scenarios and fails to capture algorithmic performance on practical inputs. We systematically establish the first quantum smoothed analysis framework by integrating smoothed analysis techniques, complexity characterizations of symmetric Boolean functions, and approximation methods for collision counts in non-repeating strings. Our results demonstrate that this framework successfully unifies worst-case and average-case analyses while revealing exponential separations as well as polynomial-to-superpolynomial quantum speedups. This work confirms that smoothed analysis can expose quantum advantages beyond worst-case bounds, thereby opening a new pathway for evaluating the potential of quantum computing in real-world scenarios.
📝 Abstract
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 $\varepsilon$-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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Smoothed analysis
Quantum query complexity
Symmetric Boolean functions
Quantum speedup
String problems
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