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assignment7 - Physics 681-481 CS 483 Assignment#7(please...

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April 20, 2006 Physics 681-481; CS 483: Assignment #7 (please hand in after the lecture, Thursday, May 4) This is the final assignment. 1. Suppose the only kinds of errors one had to worry about were bit-flip errors, but one wanted to take into account not only single bit-flips ( X i ) in the code words but also double bit-flips ( X i X j , i 6 = j ). (a) Show that there is an n for which the dimension of the n -Qbit state space is just large enough to accomodate mutually orthogonal two-dimensional subspaces for the uncor- rupted code words and all code words suffering either single or double bit-flip corruptions. What is that n ? (b) Show for the n you found in (a) that there is indeed a “perfect” n -Qbit code that corrects all single and double bit-flip errors, by writing down the states that encode | 0 i and | 1 i , and writing down a set of commuting hermitian operators whose squares are unity, that preserve both codewords, and have distinct patterns of commutations or anticommutations (show this explicitly) with each of the operators associated with all the single and double bit-flip errors.
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