Optimal Quantum Algorithm for Polynomial Interpolation
We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over F_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.
Quantum algorithms
query complexity
polynomial interpolation
finite fields
16:1-16:13
Regular Paper
Andrew M.
Childs
Andrew M. Childs
Wim
van Dam
Wim van Dam
Shih-Han
Hung
Shih-Han Hung
Igor E.
Shparlinski
Igor E. Shparlinski
10.4230/LIPIcs.ICALP.2016.16
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