On the Quantum Hardness of Matching Colored Triangles
Koen Leijnse
Abstract:
Classically, many problems can be shown to have computational lower bounds in their time complexity conditioned on the hardness of three popular problems; k-SAT, 3SUM and APSP. This is done through fine-grained reductions and research in this classical field has recently exploded, as can be seen in V. Williams’ overview. The reductions from all three popular problems to two intuitive graph triangle problems, ∆-Matching Triangles and Triangle Collection, makes for a notable result in the field of fine-grained computational complexity. By reducing all three problems to the two graph triangle problems Abboud, V. Williams and Yu prove that if we find an O(n3−ε ) time algorithm, with n the number of nodes and ε > 0, for either of the two graph triangle problems, all three hardness conjectures must be false. This result makes the two problems very worthwhile to work with.
Many computational problems allow for speed-ups on quantum computers and recent work has taken the concept of fine-grained reductions to quantum computers to prove many new conditional quantum lower bounds based on quantum hardness conjectures for k-SAT and 3SUM. Continuing this work, we formulate an n2.5−o(1) quantum hardness conjecture for APSP and provide arguments for why this hardness conjecture is likely to be valid in the quantum case. Based on this conjecture, we prove a series of quantum lower bounds for graph and matrix problems. Starting at APSP we follow the classical chain of reductions all the way through to ∆-Matching Triangles and Triangle Collection, first reviewing the classical reductions and then applying them in the quantum setting. We do the same for k-SAT, proving the quantum hardness of the two graph triangle problems based on the Quantum Strong Exponential Time Hypothesis, using classical reductions. Combined with quantum hardness results based on 3SUM, we show that an O(n1.5−ε ) algorithm for ∆-Matching Triangles, with ω(1) ≤ ∆ ≤ no(1) , or Triangle Collection would contradict all three quantum hardness conjectures, mirroring the classical result.
The thesis finishes by showing matching upper bounds for all problems that we reduce APSP to. For ∆-Matching Triangles and Triangle Collection we find matching upper bounds by using the Variable Time Grover Search algorithm as suggested by Ambainis. For all other problems, matching upper bounds are found through a simple application of Grover Search.