4. (6 points) (Classical and quantum algorithms for the OR problem (Part II)) (a) (3 points)...

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4. (6 points) (Classical and quantum algorithms for the OR problem (Part II)) (a) (3 points) Based on the results of Part I (Question 3), give a quantum algorithm using only unitary operations and measurements in the standard basis for the OR problem that makes a single query to f and: succeeds with probability 1 whenever f(0) v f(1) = 0; succeeds with probability 8/9 whenever f(0) V f(1) = 1. (b) (3 points) Optional for bonus credit for undergrad students: Note that the error probability of the algorithm from part (a) is one-sided in the sense that it is always correct in the case where f(0) V f(1) =0. Give a quantum algorithm for the OR problem that makes a single query to f and succeeds with probability 9/10. (Hint: take the output of the one-sided error algorithm from part (a) and do some classical post-processing on it, in order to turn it into a two-sided error algorithm with higher success probability.) 4. (6 points) (Classical and quantum algorithms for the OR problem (Part II)) (a) (3 points) Based on the results of Part I (Question 3), give a quantum algorithm using only unitary operations and measurements in the standard basis for the OR problem that makes a single query to f and: succeeds with probability 1 whenever f(0) v f(1) = 0; succeeds with probability 8/9 whenever f(0) V f(1) = 1. (b) (3 points) Optional for bonus credit for undergrad students: Note that the error probability of the algorithm from part (a) is one-sided in the sense that it is always correct in the case where f(0) V f(1) =0. Give a quantum algorithm for the OR problem that makes a single query to f and succeeds with probability 9/10. (Hint: take the output of the one-sided error algorithm from part (a) and do some classical post-processing on it, in order to turn it into a two-sided error algorithm with higher success probability.)

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