Quantum Algorithms for Finding Constant-Sized Sub-hypergraphs

Quantum query complexity is a model of quantum computation, in which the cost of computing a function is measured by the number of queries that are made to the input given as a black-box. In this model, it was exhibited in the early stage of quantum computing research that there exist quantum algorithms superior to the classical counterparts, such as Deutsch and Jozsa’s algorithm, Simon and Shor’s period finding algorithms, and Grover’s search algorithm. Extensive studies following them have invented a lot of powerful upper bound (i.e., algorithmic) techniques such as variations/generalizations of Grover’s search algorithm or quantum walks. Although these techniques give tight bounds for many problems, there are still quite a few cases for which no tight bounds are known. Intensively studied problems among them are the k-distinctness problem [1, 3, 4] and the triangle finding problem [2, 6, 8, 12, 14, 10]. A recent breakthrough is the concept of learning graph introduced by Belovs [2]. This concept enables one to easily find a special form of feasible solutions to the minimization form (i.e., the dual form) of the general adversary bound [7, 15], and makes possible to detour the need of solving a semidefinite program of exponential size to find a non-trivial upper bound. Indeed, Belovs [2] improved the long-standing Õ(n13/10) upper bound [14] (which was slightly improved to O(n13/10) [13]) of the triangle finding problem to O(n35/27). His idea was generalized by Lee, Magniez and Santha [11] and Zhu [23] to obtain a quantum algorithm that finds a constant-sized subgraph with complexity o(n2−2/k), improving the previous best boundO(n2−2/k) [14], where k is the size of the subgraph. Subsequently, Lee, Magniez and Santha [12] constructed a triangle finding algorithm with quantum query complexity O(n9/7). This bound was later shown by Belovs and Rosmanis [5] to be the best possible bound attained by the family of quantum algorithms whose complexities depend only on the index set of 1-certificates (very recently, Le Gall [10] broke this n9/7-barrier via combinatorial arguments to obtain the current best quantum upper bound of Õ(n5/4)). Ref. [12] also gave a framework of quantum algorithms for finding a constant-sized subgraph, based on which they showed that associativity testing (testing if a binary operator over a domain of size n is associative) has quantum query complexity O(n10/7). Jeffery, Kothari and Magniez [8] cast the idea of the above triangle finding algorithms into the framework of quantum walks (called nested quantum walks) by recursively performing the quantum walk algorithm given by Magniez, Nayak, Roland and Santha [13] (which extended two seminal works for quantum walk algorithms by Szegedy [18] and Ambainis [1]). Indeed, they presented two quantum-walk-based triangle finding algorithms of complexities Õ(n35/27) and Õ(n9/7), respectively. The nested quantum walk framework was further employed in [4] (but in a different way from [8]) to obtain Õ(n5/7) complexity for the 3-distinctness problem. This achieves the best known upper bound (up to poly-logarithmic factors), which was first obtained with the learning-graph-based approach [3]. The triangle finding problem also plays a central role in several areas beside query complexity, and it has been recently discovered that faster algorithms for (weighted versions of) triangle finding would lead to faster algorithms for matrix multiplication [9, 20], the 3SUM problem [19], and for Max-2SAT [21, 22]. In particular, Max-2SAT over n variables is reducible to finding a triangle with maximum weight over O(2n/3) vertices; in this context, although the final goal is a time-efficient classical or quantum algorithm that finds a triangle with maximum weight, studying triangle finding in the query complexity model is a first step toward this goal.