Let\u2019s now consider a channel that can produce both bit \ufb02ips and phase<\/p>\n\n\n\n
\ufb02ips. A code that combines bit and phase \ufb02ip coding should protect against<\/p>\n\n\n\n
these errors. A simple way to do this is to \ufb01rst encode for phase \ufb02ips:<\/p>\n\n\n\n
|0i \u2192 |+ + +i; |1i \u2192 |\u2212 \u2212 \u2212i<\/p>\n\n\n\n
and then encode using the bit \ufb02ip code:<\/p>\n\n\n\n
|+i \u2192<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|000i + |111i) ; |\u2212i \u2192<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|000i \u2212 |111i) ,<\/p>\n\n\n\n
so that we have the \ufb01nal 9-qubit encoding<\/p>\n\n\n\n
|0i \u2192<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|000i + |111i)<\/p>\n\n\n\n
\u22973<\/p>\n\n\n\n
; |1i \u2192<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|000i \u2212 |111i)<\/p>\n\n\n\n
\u22973<\/p>\n\n\n\n
. (10.17)<\/p>\n\n\n\n
Such a code is called a concatenated code, and this particular 9-qubit code was<\/p>\n\n\n\n
\ufb01rst proposed by Peter Shor. The circuit to achieve this encoding is obtained<\/p>\n\n\n\n
by concatenating the circuits for the phase \ufb02ip and the bit \ufb02ip encoding, as<\/p>\n\n\n\n
shown in Figure 10.10. The syndrome generators are easy to construct: bit-<\/p>\n\n\n\n
\u03b1|0i + \u03b2|1i<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
FIGURE 10.10: Encoding circuit for the 9-qubit Shor code<\/p>\n\n\n\n
\ufb02ips in each block can be detected by measuring (Z<\/p>\n\n\n\n
1<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
2<\/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
), (Z<\/p>\n\n\n\n
4<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
5<\/p>\n\n\n\n
, Z<\/p>\n\n\n\n
5<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
6<\/p>\n\n\n\n
)<\/p>\n\n\n\n
and (Z<\/p>\n\n\n\n
7<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
8<\/p>\n\n\n\n
, Z<\/p>\n\n\n\n
8<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
9<\/p>\n\n\n\n
). Further, phase \ufb02ips between blocks can be distinguished<\/p>\n\n\n\n
by measuring 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
X<\/p>\n\n\n\n
4<\/p>\n\n\n\n
X<\/p>\n\n\n\n
5<\/p>\n\n\n\n
X<\/p>\n\n\n\n
6<\/p>\n\n\n\n
and X<\/p>\n\n\n\n
4<\/p>\n\n\n\n
X<\/p>\n\n\n\n
5<\/p>\n\n\n\n
X<\/p>\n\n\n\n
6<\/p>\n\n\n\n
X<\/p>\n\n\n\n
7<\/p>\n\n\n\n
X<\/p>\n\n\n\n
8<\/p>\n\n\n\n
X<\/p>\n\n\n\n
9<\/p>\n\n\n\n
.<\/p>\n\n\n\n
206 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
Exercise 10.2. Construct the circuit for error correction for this case.<\/p>\n\n\n\n
Note that with eight stabilizers, we have a possibility of correcting for 2<\/p>\n\n\n\n
8<\/p>\n\n\n\n
di\ufb00erent errors, but we have only tried to look at bit\/phase \ufb02ips of 9 qubits,<\/p>\n\n\n\n
which is 3 \u00d7 9 + 1 = 28! Thus this scheme is highly redundant. More e\ufb03cient<\/p>\n\n\n\n
schemes using fewer encoding qubits have been proposed, and Shor\u2019s 9-qubit<\/p>\n\n\n\n
code is of purely historical interest now. The reason it is important to study<\/p>\n\n\n\n
this code is that it shows that it is possible to simultaneously correct for both<\/p>\n\n\n\n
bit \ufb02ips and phase \ufb02ips. Now it turns out that this will actually correct for<\/p>\n\n\n\n
arbitrary single-qubit errors, since, as we are about to show, any such error<\/p>\n\n\n\n
can be thought of as a combination of just bit \ufb02ips and phase \ufb02ips<\/p>\n","protected":false},"excerpt":{"rendered":"
Let\u2019s now consider a channel that can produce both bit \ufb02ips and phase \ufb02ips. A code that combines bit and phase \ufb02ip coding should protect against these errors. A simple way to do this is to \ufb01rst encode for phase \ufb02ips: |0i \u2192 |+ + +i; |1i \u2192 |\u2212 \u2212 \u2212i and then encode using […]<\/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-4128","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\/4128","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=4128"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4128\/revisions"}],"predecessor-version":[{"id":4129,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4128\/revisions\/4129"}],"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=4128"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4128"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}