A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Quantum computation and quantum information
Quantum computation and quantum information
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Quantum lower bounds for the collision and the element distinctness problems
Journal of the ACM (JACM)
Quantum Algorithms for Element Distinctness
SIAM Journal on Computing
Quantum Walk Algorithm for Element Distinctness
SIAM Journal on Computing
Quantum Algorithms for the Triangle Problem
SIAM Journal on Computing
Span-program-based quantum algorithm for evaluating formulas
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Reflections for quantum query algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Span programs and quantum algorithms for st-connectivity and claw detection
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Time-Efficient quantum walks for 3-distinctness
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the element distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity O(n35/27) that is better than O(n13/10) of the best previously known algorithm by Magniez et al.