Quantum Fine-Grained Complexity
Subhasree Patro
Abstract:
There is considerable excitement around quantum computing because of so-called quantum speedups: quantum algorithms can solve many computational problems faster than their classical counterparts. However, the amount of speedup that is possible varies among different computational problems. It is expected that quantum computers will remain an expensive resource for a long time, and the extent to which a quantum speedup is possible, or not possible, may one day be a key factor in deciding whether or not to invest in the use of quantum computation, for example in an industrial setting. Therefore it is essential to understand how much quantum speedup is possible for a specific computational problem, and for this purpose we need to not only find classical and quantum algorithms but also understand their limits in the form of lower bounds.
Sadly, 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 Opn2 q time. Using a fine-grained reduction it can be shown that faster algorithms for computing 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. In addition to 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 does not help much for problems for which the best known algorithms take superlinear time, or for problems whose best known query algorithms aren’t time efficient. 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 thesis, we discuss some of these challenges and present results in which we circumvent these challenges. As a consequence we prove quantum time lower bounds for many problems in BQP conditioned on the conjectured quantum hardness of SAT (and its variants), 3SUM, and APSP problems.