# The hadamard transform we will use a more general

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Unformatted text preview: ransform, we will use a more general form of type of Fourier transform). Fourier sampling is easy on a quantum computer, but appears to be diﬃcult to carry out on a classical 4.4. PHASE STATE 41 computer. In what follows, we will explore some of the power of Fourier sampling. 4.4 Phase State We will now see how to set up an interesting state for fourier sampling. Given a classical circuit for computing a boolean function f : {0, 1}n → {0, 1}, this procedure due to Deutsch and Jozsa, shows how to transform it into a quantum ￿ circuit that produces the quantum state |φ￿ = 1/2n/2 x (−1)f (x) |x￿. The quantum algorithm to carry out this task uses two quantum registers, the ﬁrst consisting of n qubits, and the second consisting of a single qubit. • Start with the registers in the state |0n ￿ |0￿ • Compute the Fourier transform on the ﬁrst register to get |0￿. • Compute f to get ￿ x | x￿ |f ( x) ￿ . • Apply a conditional phase based on f (x) to get • Uncompute f to get 4.5 ￿ x (−1) f ( x) | x ￿ ⊗ | 0￿ . ￿ ￿ x (−1) x∈ { 0 ,1 } n |x￿⊗ f ( x) | x ￿ | f ( x ) ￿ . Extracting n bits with 2 evaluations of Boolean Function Suppose we are given a black box (or an obfuscated classical circuit) that computes the function function fs : {0, 1}n → {1, −1}, where f (x) = s · x. s · x denotes the dot product s1 x1 + · · · + sn xn mod 2. The challenge is to use this black box to eﬃciently determine s. It is easy to see how to perform this task with n queries to the black box: simply input in turn the n inputs x of Hamming weight 1. The outputs of the black box are the bits of s. Since each query reveals only one bit of information, it is clear that n queries are necessary. Remarkably there is a quantum algorithm (the base case of the BernsteinVazirani algorithm) that requires only two (quantum) queries to the black box: • Use the black box to set up the phase state |φ￿ = 1/2n/2 ￿ x (−1) f (x) |x￿. 42 CHAPTER 4. FOURIER SAMPLING & SIMON’S ALGORITHM • Apply the Fourier transform H ⊗n and measure. The outcome of the measurement is s. To ￿ that the outcome of the measurement is s, recall that H ⊗n |s￿ = see s·x |x￿ = |φ￿. Since H ⊗n is its own inverse, it follows that x (−1) ⊗n | φ ￿ = | s ￿ . H More generally, the transformation H ⊗n maps the standard basis |s￿ to ￿ the fourier basis |φs ￿ = 1/2n/2 x (−1)s·x |x￿ and vice-versa. We have shown that a quantum algorithm can be more eﬃcient than any probabilistic algorithm in terms of the number of queries. One way to use this diﬀerence in the number of queries in order to demonstrate a gap between quantum and probabilistic algorithms is to make the queries very expensive. Then the quantum algorithm would be n/2 times faster than any probabilistic algorithm for the given task. But this does not help us in our goal, which is to show that quantum computers violate the extended Church-Turing thesi...
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## This document was uploaded on 09/22/2013.

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