Quantum Fine-Grained Complexity (Subhasree Patro)

One of the major challenges in the field of complexity theory is the inability to prove unconditional time lower bounds, including for practical problems within the complexity class P. One way around this is the study of fine-grained complexity where we use special reductions to prove time lower bounds for many diverse problems in P based on the conjectured hardness of some key problems. For example, computing the edit distance between two strings, a problem that has a practical interest in the comparing of DNA strings, has an algorithm that takes O(n^2) time. Using a fine-grained reduction it can be shown that faster algorithms for edit distance also imply a faster algorithm for the Boolean Satisfiability (SAT) problem (that is believed to not exist) - strong evidence that it will be very hard to solve the edit distance problem faster. Other than SAT, 3SUM, and APSP are other such key problems that are very suitable to use as a basis for such reductions, since they are natural to describe and well-studied. The situation in the quantum regime is no better; almost all known lower bounds for quantum algorithms are defined in terms of query complexity, which doesn’t help much for problems for which the best-known algorithms take superlinear time. Therefore, employing fine-grained reductions in the quantum setting seems a natural way forward. However, translating the classical fine-grained reductions directly into the quantum regime is not always possible for various reasons. In this talk, I will present some recent results in which we circumvent these challenges and prove quantum time lower bounds for some problems in BQP conditioned on the conjectured quantum hardness of SAT (and its variants) and the 3SUM problem.