hw07soln - May 5 2005 Physics 681-481 CS 483 Discussion...

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Unformatted text preview: May 5, 2005 Physics 681-481; CS 483: Discussion of #7 1. (a) If only one-qubit bit-flip errors are allowed, then the general corruption of an n-qubit code word | Ψ i is of the form ( | d i 1 + n X i =1 | a i i X i ) | Ψ i . (1) We need 2( n +1) dimensions to accomodate each of the n +1 terms in (1) in orthogonal two- dimensional subspaces, and an n-qubit code word provides 2 n dimensions. We therefore require 2 n ≥ 2 n + 2 . (2) The smallest n satisfying this condition is n = 3, for which it holds as an equality. (b) Take the two three-qubit codewords to be | i = | 000 i , | 1 i = | 111 i . (3) The two operators Z 1 Z 2 and Z 2 Z 3 (4) commute with each other and act as the identity on both code words (3). X 1 commutes with the second and anticommutes with the first; X 3 commutes with the first and anticommutes with the second; and X 2 anticommutes with both. So measuring both of the operators (4) projects the corrupted state (1) onto one of the four terms with a pattern of eigenvalues...
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hw07soln - May 5 2005 Physics 681-481 CS 483 Discussion...

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