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3. Finding a hidden slope. Let p be a prime number. Suppose you are given a black-box function f: {0, 1,. ,p 1} x

3. Finding a hidden slope. Let p be a prime number. Suppose you are given a black-box function ..p  1}  {0,(b) Now suppose we apply U and measure the state of the third register in the computational basis (i.e., the

3. Finding a hidden slope. Let p be a prime number. Suppose you are given a black-box function f: {0, 1,. ,p 1} x {0, 1, .. ..p 1} {0, 1,...,p - 1} such that f(x,y) f(x', y') if and only if y' y = m(x x) mod p for some unknown integer m. In other words, the function is constant on lines of slope m, and distinct on different parallel lines of that slope. Your goal is to determine m mod p using as few queries as possible to f, which is given by a unitary operation Uf satisfying Uf|x)|y)|z) = |x)|y)|z + f(x, y) mod p) for all x, y, z {0, 1,...,p 1}. (Note that each of the three registers stores an integer modulo p, which we do not need to explicitly represent using qubits.) - (a) Let Fp denote the Fourier transform modulo p, the unitary operator p-1 2nixy/P\x) (y\. Fp = P = x,y=0 Suppose we begin with three registers in the state |0)|0)|0). If we apply Fp Fp I, what is the resulting state? (b) Now suppose we apply U and measure the state of the third register in the computational basis (i.e., the basis {|0), 1),..., p - 1)}). What are the probabilities of the different possible measurement outcomes, and what are the resulting post-measurement states of the first two registers? 1 (c) Show that by applying F F to the post-measurement state of the first two registers and then measuring in the computational basis, one can learn m mod p with probability 1-1/p. 3. Finding a hidden slope. Let p be a prime number. Suppose you are given a black-box function ..p 1} {0, 1,. .. ,p - 1} such that f(x, y) f(x', y') if and f: {0, 1,. ,p - 1} x {0, 1, .. only if y' y = m(x' x) mod p for some unknown integer m. In other words, the function is constant on lines of slope m, and distinct on different parallel lines of that slope. Your goal is to determine m mod p using as few queries as possible to f, which is given by a unitary operation Uf satisfying Uf|x)|y)|z) = |x)|y)]z + f(x, y) mod p) for all x, y, z {0, 1, ..., p 1}. (Note that each of the three registers stores an integer modulo p, which we do not need to explicitly represent using qubits.) - (a) Let Fp denote the Fourier transform modulo p, the unitary operator p-1 2nixy/P\x) (y\. Fp = P = x,y=0 Suppose we begin with three registers in the state |0)|0)|0). If we apply Fp Fp I, what is the resulting state? (b) Now suppose we apply U and measure the state of the third register in the computational basis (i.e., the basis {|0), 1),..., p - 1)}). What are the probabilities of the different possible measurement outcomes, and what are the resulting post-measurement states of the first two registers? 1 (c) Show that by applying F0 F to the post-measurement state of the first two registers and then measuring in the computational basis, one can learn m mod p with probability 1-1/p.

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