One of the main results of the theory of quantum error correction is that<\/p>\n\n\n\n
any general quantum error can be composed only of discrete errors represented<\/p>\n\n\n\n
by the Pauli operators X, Z, and Y = \u2212iXZ. Errors are induced on our qubit<\/p>\n\n\n\n
system due to e\ufb00ects of everything outside this system, which we will call<\/p>\n\n\n\n
the environment. The environment interacts weakly with the system to cause<\/p>\n\n\n\n
a change in the amplitudes of the basis states, a process called decoherence<\/p>\n\n\n\n
of the initial state. Initially, let\u2019s assume the system is created in a de\ufb01nite<\/p>\n\n\n\n
state |\u03c8i. The environment has been excluded experimentally, so that the<\/p>\n\n\n\n
combined environment-qubit system is in a product state: |ei|\u03c8i. Subsequent<\/p>\n\n\n\n
interaction between the two results in a change of this state. In order to<\/p>\n\n\n\n
model this evolution, we represent the transformation of the computational<\/p>\n\n\n\n
basis states by<\/p>\n\n\n\n
|ei|0i \u2212\u2192 |e<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i|0i + |e<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i|1i; (10.18a)<\/p>\n\n\n\n
|ei|1i \u2212\u2192 |e<\/p>\n\n\n\n
3<\/p>\n\n\n\n
i|0i + |e<\/p>\n\n\n\n
4<\/p>\n\n\n\n
i|1i. (10.18b)<\/p>\n\n\n\n
Here the kets |e<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i are (un-normalized) environment states that can be ex-<\/p>\n\n\n\n
pressed as fractions of |ei: |e<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i = a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|ei. For example, the bit-\ufb02ip error can<\/p>\n\n\n\n
be modelled with a<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= 0 = a<\/p>\n\n\n\n
4<\/p>\n\n\n\n
, a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1 = a<\/p>\n\n\n\n
3<\/p>\n\n\n\n
and the phase \ufb02ip by<\/p>\n\n\n\n
a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 0 = a<\/p>\n\n\n\n
3<\/p>\n\n\n\n
, a<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= 1 = \u2212a<\/p>\n\n\n\n
4<\/p>\n\n\n\n
. Now we want to be able to recognize the ef-<\/p>\n\n\n\n
fect of such an evolution on the superposition state |\u03c8i = \u03b1|0i + \u03b2|1i, as an<\/p>\n\n\n\n
operation on the qubit system alone. In order to separate the e\ufb00ects on |0i<\/p>\n\n\n\n
and |1i we\u2019ll now write a general error in the terms of the projectors<\/p>\n\n\n\n
P<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= |0ih0|, and P<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= |1ih1|. (10.19)<\/p>\n\n\n\n
So we can write Equations 10.18 as<\/p>\n\n\n\n
|ei|0i \u2212\u2192<\/p>\n\n\n\n
\ue000<\/p>\n\n\n\n
|e<\/p>\n\n\n\n
1<\/p>\n\n\n\n
iP<\/p>\n\n\n\n
0<\/p>\n\n\n\n
+ |e<\/p>\n\n\n\n
2<\/p>\n\n\n\n
iXP<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\ue001<\/p>\n\n\n\n
|0i (10.20a<\/p>\n\n\n\n
|ei|1i \u2212\u2192<\/p>\n\n\n\n
\ue000<\/p>\n\n\n\n
|e<\/p>\n\n\n\n
3<\/p>\n\n\n\n
iXP<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ |e<\/p>\n\n\n\n
4<\/p>\n\n\n\n
iP<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\ue001<\/p>\n\n\n\n
|1i. (10.20b)<\/p>\n\n\n\n
The error acting on |\u03c8i can be written as<\/p>\n\n\n\n
|ei|\u03c8i \u2212\u2192<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\ue000<\/p>\n\n\n\n
|e<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i + |e<\/p>\n\n\n\n
2<\/p>\n\n\n\n
iX<\/p>\n\n\n\n
\ue001<\/p>\n\n\n\n
P<\/p>\n\n\n\n
0<\/p>\n\n\n\n
+<\/p>\n\n\n\n
\ue000<\/p>\n\n\n\n
|e<\/p>\n\n\n\n
3<\/p>\n\n\n\n
iX + |e<\/p>\n\n\n\n
4<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\ue001<\/p>\n\n\n\n
P<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8i. (10.21)<\/p>\n\n\n\n
Now the projection operators can be written in terms of the Pauli matrices:<\/p>\n\n\n\n
Z = |0ih0| \u2212 |1ih1|; = |0ih0| + |1ih1|;<\/p>\n\n\n\n
=\u21d2 P<\/p>\n\n\n\n
0<\/p>\n\n\n\n
=<\/p>\n\n\n\n
+ Z<\/p>\n\n\n\n
2<\/p>\n\n\n\n
; P<\/p>\n\n\n\n
1<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u2212 Z<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. (10.22)<\/p>\n\n\n\n
Also, using XZ = iY , we get<\/p>\n\n\n\n
|ei|\u03c8i \u2212\u2192<\/p>\n\n\n\n
\ue000<\/p>\n\n\n\n
|E<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i + |E<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
X + |E<\/p>\n\n\n\n
3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y + |E<\/p>\n\n\n\n
4<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
\ue001<\/p>\n\n\n\n
|\u03c8i, (10.23)<\/p>\n\n\n\n
where we have appropriately regrouped the environment states |e<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i to obtain<\/p>\n\n\n\n
the new environment states |E<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i. (We do not care about the exact form of<\/p>\n\n\n\n
these states since we are not going to observe them.) We thus see that the<\/p>\n\n\n\n
generic error can be expressed as a linear combination of the discrete errors<\/p>\n\n\n\n
corresponding to the action of the Pauli matrices.<\/p>\n\n\n\n
If we encode using n qubits for error-correction, then a generic state would<\/p>\n\n\n\n
become<\/p>\n\n\n\n
|ei|<\/p>\n\n\n\n
\u02dc<\/p>\n\n\n\n
\u03c8i<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
|di +<\/p>\n\n\n\n
n<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i=1<\/p>\n\n\n\n
(|a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
+ |b<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y<\/p>\n\n\n\n
i<\/p>\n\n\n\n
+ |c<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
i<\/p>\n\n\n\n
)<\/p>\n\n\n\n
!<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u02dc<\/p>\n\n\n\n
\u03c8i<\/p>\n\n\n\n
n<\/p>\n\n\n\n
. (10.24)<\/p>\n\n\n\n
In order to diagnose the syndromes, the 2<\/p>\n\n\n\n
n<\/p>\n\n\n\n
-d Hilbert space must admit at<\/p>\n\n\n\n
least 1 + 3n 2-d subspaces:<\/p>\n\n\n\n
2<\/p>\n\n\n\n
n\u22121<\/p>\n\n\n\n
\u2265 1 + 3n,<\/p>\n\n\n\n
so n = 5, 7, 9…<\/p>\n\n\n\n
Thus the minimum codeword size is 5 qubits. We can see now that the<\/p>\n\n\n\n
9-qubit Shor code is not e\ufb03cient; we can make do with fewer qubits<\/p>\n","protected":false},"excerpt":{"rendered":"
One of the main results of the theory of quantum error correction is that any general quantum error can be composed only of discrete errors represented by the Pauli operators X, Z, and Y = \u2212iXZ. Errors are induced on our qubit system due to e\ufb00ects of everything outside this system, which we will call […]<\/p>\n","protected":false},"author":1,"featured_media":4044,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[499],"tags":[],"class_list":["post-4130","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-quantum-error-correction"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/error-state.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4130"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4130\/revisions"}],"predecessor-version":[{"id":4131,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4130\/revisions\/4131"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4044"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}