On exponential polynomials and quantum computing

We calculate the zeros of an exponential polynomial of three variables by a classical algorithm and quantum algorithms which are based on the method of van Dam and Shparlinski, they treated the case of two variables, and compare with the time complexity of those cases. Further we compare the case of van Dam and Shparlinski with our case by considering the ratio (classical/quantum) of the time complexity. Then we can observe the ratio decreases.

💡 Research Summary

The paper investigates the problem of locating zeros of a three‑variable exponential polynomial, extending the two‑variable quantum approach originally developed by van Dam and Shparlinski. An exponential polynomial of the form
(f(x_1,x_2,x_3)=\sum_{i=1}^{k} a_i \exp!\big(b_{i1}x_1+b_{i2}x_2+b_{i3}x_3\big))
is considered over a finite field (\mathbb{F}_p) with integer variables constrained to a box (