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arXiv · 1509.09271

Optimal quantum algorithm for polynomial interpolation

Andrew M. Childs·Wim van Dam·Shih-Han Hung·Igor E. Shparlinski
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Abstract

We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over GF(q). A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2+1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2+1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithm's success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log q) with negligible decrease in the success probability. We end with a conjecture about the quantum query complexity of multivariate polynomial interpolation.

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Andrew M. Childs, Wim van Dam, Shih-Han Hung, Igor E. Shparlinski. 2015-09-30. Optimal quantum algorithm for polynomial interpolation. https://doi.org/10.4230/lipics.icalp.2016.16

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