April 20, 2006
Physics 681481; 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 bitflip errors, but
one wanted to take into account not only single bitflips (
X
i
) in the code words but also
double bitflips (
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 twodimensional subspaces for the uncor
rupted code words and all code words suffering either single or double bitflip 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 bitflip 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
bitflip errors.
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 Spring '08
 Ginsparg
 Cryptography, Leftwing politics, Code word, Political spectrum, Hilbert space, Konrad Zuse

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