Let us estimate the probability for the above technique to yield an uncor-<\/p>\n\n\n\n
rupted state, considering a channel characterized by \ufb02ipping of a qubit with<\/p>\n\n\n\n
probability p < 1\/2. We list in Table 10.3 the probability of occurrence of<\/p>\n\n\n\n
various corrupted states in order of decreasing probability.<\/p>\n\n\n\n
The probability that our procedure corrects errors is therefore<\/p>\n\n\n\n
P(correct) = (1 \u2212 p)<\/p>\n\n\n\n
3<\/p>\n\n\n\n
+ 3p(1 \u2212 p)<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1 \u2212 3p<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ 2p<\/p>\n\n\n\n
3<\/p>\n\n\n\n
. (10.7)<\/p>\n\n\n\n
and the probability that we have an erroneous state is<\/p>\n\n\n\n
P(incorrect) = p<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(1 \u2212 p) + p<\/p>\n\n\n\n
3<\/p>\n\n\n\n
= 3p<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2212 2p<\/p>\n\n\n\n
3<\/p>\n\n\n\n
< P(correct). (10.8)<\/p>\n\n\n\n
<\/p>\n\n\n\n
Quantum Error Correction 203<\/p>\n\n\n\n
TABLE 10.3: Probability of occurrence of corrupted states in a bit-\ufb02ip chan-<\/p>\n\n\n\n
nel.<\/p>\n\n\n\n
Number of \ufb02ips states probability<\/p>\n\n\n\n
0 |\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i (1 \u2212 p)<\/p>\n\n\n\n
3<\/p>\n\n\n\n
1<\/p>\n\n\n\n
X<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
X<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
X<\/p>\n\n\n\n
3<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
3p(1 \u2212 p)<\/p>\n\n\n\n
2<\/p>\n\n\n\n
2<\/p>\n\n\n\n
X<\/p>\n\n\n\n
1<\/p>\n\n\n\n
X<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
X<\/p>\n\n\n\n
2<\/p>\n\n\n\n
X<\/p>\n\n\n\n
3<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
X<\/p>\n\n\n\n
1<\/p>\n\n\n\n
X<\/p>\n\n\n\n
3<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
3p<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(1 \u2212 p)<\/p>\n\n\n\n
3 X<\/p>\n\n\n\n
1<\/p>\n\n\n\n
X<\/p>\n\n\n\n
2<\/p>\n\n\n\n
X<\/p>\n\n\n\n
3<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i p<\/p>\n\n\n\n
3<\/p>\n\n\n\n
Box 10.1: Error Correction and Fidelity<\/p>\n\n\n\n
People in the error-correcting business are not satis\ufb01ed with this, and try<\/p>\n\n\n\n
to work out schemes that are better by comparing \ufb01delities. We will see in<\/p>\n\n\n\n
Section 11.3.2 that the \ufb01delity of two states is de\ufb01ned by their degree of<\/p>\n\n\n\n
overlap. If we start with a pure state |\u03c8i, errors cause it to become a mixed<\/p>\n\n\n\n
state with probability p of transforming by X. This is represented by the<\/p>\n\n\n\n
density matrix<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
bf<\/p>\n\n\n\n
= pX|\u03c8ih\u03c8|X + (1 \u2212 p)|\u03c8ih\u03c8|. (10.9)<\/p>\n\n\n\n
The \ufb01delity of state transmission without error correction is given by<\/p>\n\n\n\n
F =<\/p>\n\n\n\n
p<\/p>\n\n\n\n
h\u03c8|\u03c1<\/p>\n\n\n\n
bf<\/p>\n\n\n\n
|\u03c8i (10.10)<\/p>\n\n\n\n
=<\/p>\n\n\n\n
q<\/p>\n\n\n\n
ph\u03c8|X|\u03c8i<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ (1 \u2212 p) (10.11)<\/p>\n\n\n\n
This has a minimum value of<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
1 \u2212 p, when the \ufb01rst term is zero. If we make<\/p>\n\n\n\n
use of the above protocol for error correction then for the 3-qubit encoded<\/p>\n\n\n\n
state,<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
corrected<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
(1 \u2212 p)<\/p>\n\n\n\n
3<\/p>\n\n\n\n
+ 3p(1 \u2212 p)<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
|\u03c8ih\u03c8|<\/p>\n\n\n\n
+ (\u03c1 for 2 or more bit \ufb02ips) , (10.12)<\/p>\n\n\n\n
and the \ufb01delity, using only the \ufb01rst two terms, is<\/p>\n\n\n\n
F \u2265<\/p>\n\n\n\n
p<\/p>\n\n\n\n
(1 \u2212 p)<\/p>\n\n\n\n
3<\/p>\n\n\n\n
+ 3p(1 \u2212 p)<\/p>\n\n\n\n
2<\/p>\n\n\n\n
,<\/p>\n\n\n\n
the same as the above.<\/p>\n","protected":false},"excerpt":{"rendered":"
Let us estimate the probability for the above technique to yield an uncor- rupted state, considering a channel characterized by \ufb02ipping of a qubit with probability p < 1\/2. We list in Table 10.3 the probability of occurrence of various corrupted states in order of decreasing probability. The probability that our procedure corrects errors is […]<\/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-4124","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\/4124","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=4124"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4124\/revisions"}],"predecessor-version":[{"id":4589,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4124\/revisions\/4589"}],"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=4124"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4124"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}