Abstract
Span program is a linear-algebraic model of computation which can be used to design quantum algorithms. For any Boolean function there exists a span program that leads to a quantum algorithm with optimal quantum query complexity. In general, finding such span programs is not an easy task.
In this work, given a query access to the adjacency matrix of a simple graph G with n vertices, we provide two new span-program-based quantum algorithms:
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an algorithm for testing if the graph is bipartite that uses \(O(n\sqrt{n})\) quantum queries;
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an algorithm for testing if the graph is connected that uses \(O(n\sqrt{n})\) quantum queries.
This work has been supported by the ERC Advanced Grant Methods for Quantum Computing.
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Acknowledgements
I am grateful to Andris Ambainis for the suggestion to solve the graph problems with span programs, and for many useful comments during the development of the paper.
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Āriņš, A. (2016). Span-Program-Based Quantum Algorithms for Graph Bipartiteness and Connectivity. In: Kofroň, J., Vojnar, T. (eds) Mathematical and Engineering Methods in Computer Science. MEMICS 2015. Lecture Notes in Computer Science(), vol 9548. Springer, Cham. https://doi.org/10.1007/978-3-319-29817-7_4
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