As quantum computing progresses steadily from theory into practice, programmers will face

a common problem: How can they be sure that their code does what they intend it to do? This

paper presents encouraging results in the application of mechanized proof to the domain of quantum

programming in the context of the sqir development. It verifies the correctness of a range of a

quantum algorithms including Grover\’s algorithm and quantum phase estimation, a key component

of Shor\’s algorithm. In doing so, it aims to highlight both the successes and challenges of formal

verification in the quantum context and motivate the theorem proving community to target quantum

computing as an application domain.

We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

}, isbn = {978-3-95977-195-5}, doi = {10.4230/LIPIcs.ICALP.2021.55}, url = {https://drops.dagstuhl.de/opus/volltexte/2021/14124}, author = {Andrew M. Childs and Hung, Shih-Han and Li, Tongyang} }