Bit \ufb02ips alone are a very limited kind of error a qubit could undergo.<\/p>\n\n\n\n
Consider phase \ufb02ips, which have no classical equivalent. A phase-\ufb02ip quantum<\/p>\n\n\n\n
channel is de\ufb01ned as one that only allows single phase \ufb02ips, that is, with<\/p>\n\n\n\n
probability p, |1i \u2192 \u2212|1i. Under the action of this channel, a generic state<\/p>\n\n\n\n
transforms as<\/p>\n\n\n\n
|\u03c8i = \u03b1|0i + \u03b2|1i \u2192 \u03b1|0i \u2212 \u03b2|1i. (10.13)<\/p>\n\n\n\n
If we represent |\u03c8i in the X basis spanned by |\u00b1i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
[|0i \u00b1 |1i],<\/p>\n\n\n\n
|\u03c8i = \u03b1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|+i + \u03b2<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|\u2212i, (10.14)<\/p>\n\n\n\n
then phase \ufb02ip causes |+i \u2192 |\u2212i and |\u2212i \u2192 |+i. Thus, the phase-\ufb02ip case<\/p>\n\n\n\n
is unitarily equivalent to the bit-\ufb02ip case since we can change basis to the<\/p>\n\n\n\n
X-basis by applying a H transform. Error correction can be followed just as<\/p>\n\n\n\n
in the bit-\ufb02ip case, except that we now transform everything to the X basis<\/p>\n\n\n\n
by using the H gate at appropriate places. The 3-qubit encoding that will<\/p>\n\n\n\n
correct phase \ufb02ip errors should then be<\/p>\n\n\n\n
|0i \u2192 |+ + +i, |1i \u2192 |\u2212 \u2212 \u2212i, (10.15)<\/p>\n\n\n\n
which is achieved by the circuit in Figure 10.9.<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
H<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
\u03c8i = \u03b1|+ + +i + \u03b2|\u2212 \u2212 \u2212i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\uf8fc<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8fd<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8f4<\/p>\n\n\n\n
\uf8fe<\/p>\n\n\n\n
FIGURE 10.9: Encoding circuit for 3-qubit phase-\ufb02ip code.<\/p>\n\n\n\n
Syndrome measurement and recovery is now identical to the bit-\ufb02ip case,<\/p>\n\n\n\n
except that we work in the X basis by applying an H gate to each qubit. The<\/p>\n\n\n\n
stabilizers are the operators<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O<\/p>\n\n\n\n
0<\/p>\n\n\n\n
I<\/p>\n\n\n\n
= H<\/p>\n\n\n\n
\u22973<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
1<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
2<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u22973<\/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
,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O<\/p>\n\n\n\n
0<\/p>\n\n\n\n
II<\/p>\n\n\n\n
= H<\/p>\n\n\n\n
\u22973<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
2<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
3<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u22973<\/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
. (10.16)<\/p>\n\n\n\n
Measuring these operators distinguishes the syndromes. This is like com-<\/p>\n\n\n\n
paring the signs of the corresponding qubit values. Finally, recovery is per-<\/p>\n\n\n\n
formed by applying HXH = Z to the appropriate qubit.<\/p>\n\n\n\n
Exercise 10.1. Construct the circuit for detecting phase \ufb02ip syndromes and for<\/p>\n\n\n\n
correcting them<\/p>\n","protected":false},"excerpt":{"rendered":"
Bit \ufb02ips alone are a very limited kind of error a qubit could undergo. Consider phase \ufb02ips, which have no classical equivalent. A phase-\ufb02ip quantum channel is de\ufb01ned as one that only allows single phase \ufb02ips, that is, with probability p, |1i \u2192 \u2212|1i. Under the action of this channel, a generic state transforms as […]<\/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-4126","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\/4126","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=4126"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4126\/revisions"}],"predecessor-version":[{"id":4127,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4126\/revisions\/4127"}],"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=4126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}