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Question

6. (This is an optional question for bonus credit)

Factorizing certain two-variable polynomials [8 points; 4 each].

Let m be an integer such that m≥ 2. Let a, b € Zm and define f: Zm x Zm → Zm as

f(x, y) = (x-a)(y - b) mod m,

(3)

for all x, y € Zm.

Suppose that you are given access to a black-box that, on input (x, y), produces f(x, y)

as output, but you do not know what the constants a and b are. Your goal is to determine

a and b exactly (meaning with success probability 1).

(a) Show that any classical algorithm for this problem requires at least three f-queries.

(b) Give a quantum algorithm that solves this problem with one f-query. The f-query

is a unitary operation that maps basis states x, y)2) to x, y) |z+f(x, y) mod m),

for all x, y, z € Zm. Explain why your algorithm works.

Note: If you submit a solution to this question then there is a size-limit of one page for

part (a) and one page for part (b). In fact, each part has a solution that can be clearly

explained in less than half a page.

Fig: 1