Solutions to Assignment 7

Solutions to Assignment 7 - May 4 2006 Physics 681-481 CS...

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May 4, 2006 Physics 681-481; CS 483: Discussion of #7 1. (a) If only bit-flip errors are allowed but they can act on one or two qubits, then the general corruption of an n -qubit code word is ( | d i 1 + n X i =1 | a i i X i + X 1 i<j n | b ij i X i X j ) | Ψ i . (1) there are now 1+ n + 1 2 n ( n - 1) = 1 + 1 2 n ( n + 1) types of errors, so the condition for an n -qubit code becomes 2 n 2 + n ( n + 1) . (2) The smallest n for which this holds is n = 5, for which it again holds as an equality. (b) One simple realization of such a five-qubit code is to take the codewords to be | 0 i = | 00000 i , | 1 i = | 11111 i . (3) The four operators Z 1 Z 2 , Z 2 Z 3 , Z 3 Z 4 , and Z 4 Z 5 (4) commute with each other and act as the identity on both code words (3). Their patterns of commutation (+) or anticommutation ( - ) with each of the 16 operators appearing in (1) are given by the following table: Z 1 Z 2 Z 2 Z 3 Z 3 Z 4 Z 4 Z 5 1 + + + + 0 X 1 - + + + 8 X 2 - - + + 12 X 3 + - - + 6 X 4 + + - - 3 X 5 + + + - 1 X 1 X 2 + - + + 4 X 1 X 3 - - - + 14 X 1 X 4
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