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Unformatted text preview: February 3, 2005 Physics 681-481; CS 483: Discussion of #1 I. Manipulating simple operators. (a) An algebraic way (ridiculously complicated, of course, since the result is obvious from the non-algebraic definition) to check that the square of the SWAP operator, S ij = n i n j + n i n j + ( X i X j )( n i n j + n i n j ) , (1) is 1 , is to note first that X i X j commutes with both ( n i n j + n i n j ) and with ( n i n j + n i n j ), as a consequence of the fact that X and n operators associated with the same qubit satisfy nX = X n , nX = Xn , (2) while all operators associated with different qubits commute. Therefore bringing X i X j through either ( n i n j + n i n j ) or ( n i n j + n i n j ) simply interchanges each n and n . But both quantities are invariant under such an interchange. Consequently S 2 ij = ( n i n j + n i n j ) 2 + ( X i X j ) 2 ( n i n j + n i n j ) 2 + 2( X i X j ) ( n i n j + n i n j )( n i n j + n i n j ) . (3) Because number operators associated with the same qubit satisfy n 2 = n , n 2 = n , n n = nn = , (4) the first term in (3) simplifies to ( n...
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- Spring '05