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Unformatted text preview: C/CS/Phy191 Problem Set 3 Solutions Out: September 30, 2007 1. Show that if U and V are unitary operators acting on separate Hilbert spaces H 1 and H 2 respectively, then U ⊗ V acting on the product space H 1 ⊗ H 2 is also a unitary operator. If U and V are both unitary, then each can be written in some basis as a diagonal matrix with only complex phases e i φ n along the diagonal. Since operators in a tensor product act on different Hilbert spaces, we can diagonalize both of them independently. Let e i φ n be the diagonal elements of U , and let e i φ m be the diagonal elements of V . Then we use the rules for tensor products of matrices to see that U ⊗ V is a diagonal matrix with diagonal elements e i φ n e i φ m . Therefore it is also a diagonal matrix with only complex phases along the diagonal. So it is unitary. 2. Consider the unitary operation U resulting from applying the Hadamard gate to each of n qubits. Describe U by giving a formula for its ( i , j ) th entry (where i and j are the row and column indices of the matrix). Answer: U = H ⊗ H ⊗ ... ⊗ H , with n Hadamard operators in the product. This is often written U = H ⊗ n . Then we want to find y H ⊗ n x . To do this, let’s first find H ⊗ n x . Now, the state x contains n qubits, each of which is in the state 0 or 1 . Define x i to be the state of the i th qubit in x . H ⊗ n x = n O i = 1 ( + 1 ) / √ 2 if x i = (- 1 ) / √ 2 if x i = 1 = ∑ z 1 2 n / 2 n O i = 1 z i if x i = 0 or z i =- z i if x i = 1 and z i = 1 ! = ∑ z 1 2 n / 2 ( n ∏ i = 1 (- 1 ) x i z i ) z = 1 2 n / 2 ∑ z (- 1 ) x · z z where z ranges over all n-qubit computational basis states, and x · z ≡ ∑ n i = 1 x i z i ....
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- Fall '07
- mechanics, Alice, Hilbert space, unitary transformation, Unitary operator