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Unformatted text preview: 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
n |d 1 +
i=1 |a i Xi +
1i<jn |bij Xi Xj | . (1) there are now 1+n + 1 n(n - 1) = 1 + 1 n(n + 1) types of errors, so the condition for an 2 2 n-qubit code becomes 2n 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 = |00000 , The four operators Z1 Z2 , Z2 Z3 , Z3 Z4 , and Z4 Z5 (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: Z1 Z2 + - - + + + + - - - - - - + + + Z2 Z3 + + - - + + - - + + + - - - - + 1 Z3 Z4 + + + - - + + - - + - - + + - - Z4 Z5 + + + + - - + + - - + - - - - + |1 = |11111 . (3) 1 X1 X2 X3 X4 X5 X1 X 2 X1 X 3 X1 X 4 X1 X 5 X2 X 3 X2 X 4 X2 X 5 X3 X 4 X3 X 5 X4 X 5 0 8 12 6 3 1 4 14 11 9 10 15 13 5 7 2 The uniqueness of the patterns is easily checked by noting that if the rows are read as binary numbers (with - as 1 and + as 0) and those binary numbers are expressed as decimal numbers (final column on the right) then every number from 0 to 15 occurs on the list of 16 numbers. So measuring the four operators (4) projects the corrupted state (1) onto one of the sixteen terms with a pattern of eigenvalues that tells which of the sixteen it is. 2. A full discussion of the circuit for 5-Qbit codewords has been added to the lecture notes for Chapter 5, on page 24 and in Figures 18-22. 2 ...
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