![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bge9.png\")
<\/p>\n\n\n\nThe number of syndromes in a 5-qubit scheme would be 5 \u00d7 3 + 1 = 16.<\/p>\n\n\n\n
We\u2019d thus need 4 stabilizer operators since 2<\/p>\n\n\n\n
4<\/p>\n\n\n\n
= 16. We\u2019ll simply give the<\/p>\n\n\n\n
operators (see Mermin [48] or La\ufb02amme et al. [44]):<\/p>\n\n\n\n
M<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= Z<\/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
Z<\/p>\n\n\n\n
4<\/p>\n\n\n\n
, M<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= Z<\/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
0<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
1<\/p>\n\n\n\n
,<\/p>\n\n\n\n
M<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= Z<\/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
Z<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, M<\/p>\n\n\n\n
3<\/p>\n\n\n\n
= Z<\/p>\n\n\n\n
4<\/p>\n\n\n\n
X<\/p>\n\n\n\n
0<\/p>\n\n\n\n
X<\/p>\n\n\n\n
1<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. (10.25)<\/p>\n\n\n\n
<\/p>\n\n\n\n
208 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
These operators satisfy<\/p>\n\n\n\n
M<\/p>\n\n\n\n
0<\/p>\n\n\n\n
M<\/p>\n\n\n\n
1<\/p>\n\n\n\n
M<\/p>\n\n\n\n
2<\/p>\n\n\n\n
M<\/p>\n\n\n\n
3<\/p>\n\n\n\n
= . (10.26)<\/p>\n\n\n\n
We can see that each operator \ufb02ips 2 qubits, and the encoding is more usefully<\/p>\n\n\n\n
de\ufb01ned in terms of these:<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
( + M<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)( + M<\/p>\n\n\n\n
1<\/p>\n\n\n\n
)( + M<\/p>\n\n\n\n
2<\/p>\n\n\n\n
)( + M<\/p>\n\n\n\n
3<\/p>\n\n\n\n
)|00000i, (10.27a)<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
( + M<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)( + M<\/p>\n\n\n\n
1<\/p>\n\n\n\n
)( + M<\/p>\n\n\n\n
2<\/p>\n\n\n\n
)( + M<\/p>\n\n\n\n
3<\/p>\n\n\n\n
)|11111i. (10.27b)<\/p>\n\n\n\n
One thing to notice is that |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i is composed of 16 basis states, each with an even<\/p>\n\n\n\n
number of 1\u2019s, while |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i is composed of states with an even number of 0\u2019s, so<\/p>\n\n\n\n
that the states are mutually orthogonal. Each M<\/p>\n\n\n\n
i<\/p>\n\n\n\n
commutes or anti-commutes<\/p>\n\n\n\n
with the X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
, Y<\/p>\n\n\n\n
i<\/p>\n\n\n\n
, and Z<\/p>\n\n\n\n
i<\/p>\n\n\n\n
error operators, so that the \ufb01fteen syndromes and the<\/p>\n\n\n\n
uncorrupted state are distinguished by di\ufb00erent sets of \u00b11 eigenvalues of the<\/p>\n\n\n\n
M\u2019s. Measuring them would therefore diagnose the syndromes.<\/p>\n\n\n\n
Exercise 10.3. Compute the 5-qubit codewords.<\/p>\n\n\n\n
Exercise 10.4. Verify that the circuit of Figure 10.11 performs the 5-qubit en-<\/p>\n\n\n\n
coding.<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
ZHZ<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022 \u2022<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
H<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/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
\u2022<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
\u03c8i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
|0i<\/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
\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
\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 10.11: The encoding circuit for the 5-qubit code<\/p>\n\n\n\n
As you are probably feeling, this code is harder to analyze and less trans-<\/p>\n\n\n\n
parent than the Shor code. For practical purposes, the 7-qubit code due to<\/p>\n\n\n\n
Steane is more popular.<\/p>\n\n\n\n
10.6 The 7-Qubit Code<\/p>\n\n\n\n
We again give the stabilizers, codewords for the logical bit states and the<\/p>\n\n\n\n
encoding circuit, for completeness. You can refer to the text by Mermin [48]<\/p>\n\n\n\n
![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bgea.png\")
<\/p>\n\n\n\n
Quantum Error Correction 209<\/p>\n\n\n\n
for a full discussion on how the scheme works to correct errors. The 7-qubit<\/p>\n\n\n\n
code is stabilized by 6 operators that distinguish the syndromes due to X, Y ,<\/p>\n\n\n\n
or Z acting on any one qubit. These are the Steane operators:<\/p>\n\n\n\n
N<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= X<\/p>\n\n\n\n
0<\/p>\n\n\n\n
X<\/p>\n\n\n\n
3<\/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
; N<\/p>\n\n\n\n
3<\/p>\n\n\n\n
= Z<\/p>\n\n\n\n
0<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
3<\/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
N<\/p>\n\n\n\n
1<\/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
X<\/p>\n\n\n\n
5<\/p>\n\n\n\n
X<\/p>\n\n\n\n
6<\/p>\n\n\n\n
; N<\/p>\n\n\n\n
4<\/p>\n\n\n\n
= Z<\/p>\n\n\n\n
1<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
3<\/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
N<\/p>\n\n\n\n
2<\/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
5<\/p>\n\n\n\n
X<\/p>\n\n\n\n
6<\/p>\n\n\n\n
; N<\/p>\n\n\n\n
5<\/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
5<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
6<\/p>\n\n\n\n
. (10.28)<\/p>\n\n\n\n
Observe that they mutually commute, and N<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= . The 7-qubit encoding is<\/p>\n\n\n\n
de\ufb01ned by the operations<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
8<\/p>\n\n\n\n
( + N<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)( + N<\/p>\n\n\n\n
1<\/p>\n\n\n\n
)( + N<\/p>\n\n\n\n
2<\/p>\n\n\n\n
)|0i<\/p>\n\n\n\n
7<\/p>\n\n\n\n
(10.29a)<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
8<\/p>\n\n\n\n
( + N<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)( + N<\/p>\n\n\n\n
1<\/p>\n\n\n\n
)( + N<\/p>\n\n\n\n
2<\/p>\n\n\n\n
)|1i<\/p>\n\n\n\n
7<\/p>\n\n\n\n
. (10.29b)<\/p>\n\n\n\n
You can see that |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i is a state with an odd number of 0\u2019s while |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i has an even<\/p>\n\n\n\n
number. The usefulness of this code lies in the easy way in which many 1-qubit<\/p>\n\n\n\n
operations generalize to operations on the 7-qubit codewords. For instance,<\/p>\n\n\n\n
de\ufb01ning<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
X = X<\/p>\n\n\n\n
\u22977<\/p>\n\n\n\n
,<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
Z = Z<\/p>\n\n\n\n
\u22977<\/p>\n\n\n\n
,<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
H = H<\/p>\n\n\n\n
\u22977<\/p>\n\n\n\n
, (10.30)<\/p>\n\n\n\n
we \ufb01nd that<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
X|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i = |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i;<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
Z|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i = |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i;<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
H|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i + |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i); (10.31a)<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
X|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i = |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i;<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
Z|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i = \u2212|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i;<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
H|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
0i \u2212 |<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
1i). (10.31b)<\/p>\n\n\n\n
This makes it a lot more convenient to use this encoding in various circuits.<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u00af<\/p>\n\n\n\n
\u03c8i<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2022<\/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
\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
\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
\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
\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 10.12: Circuit for the 7-qubit encoding. The qubits are arranged<\/p>\n\n\n\n
according to signi\ufb01cance from highest to lowest, top to bottom.<\/p>\n\n\n\n
210 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
You can also show that the rather cute circuit of Figure 10.12, again due<\/p>\n\n\n\n
to Mermin [48], performs this encoding.<\/p>\n\n\n\n
Exercise 10.5. Draw a circuit to measure the syndromes for the 7-qubit code.<\/p>\n\n\n\n
While we have discussed the basic reasons for the success of quantum error-<\/p>\n\n\n\n
correcting codes, we have barely scratched the surface of this complex and<\/p>\n\n\n\n
intriguing \ufb01eld. In general, errors need not be restricted to single-qubit errors.<\/p>\n\n\n\n
Nor need they be unitary. The full theory of quantum error-correcting codes is<\/p>\n\n\n\n
beyond the scope of this book. That theory examines how the system can be<\/p>\n\n\n\n
embedded in a larger system with a number of entangled qubits. Measurement<\/p>\n\n\n\n
of some of the ancilla qubits can lead to error diagnosis and correction. A very<\/p>\n\n\n\n
readable account of this is given in the book by Rei\ufb00el and Polak [58].<\/p>\n\n\n\n
Another important subject we are not dealing with is fault tolerant com-<\/p>\n\n\n\n
putation. The assumption in all we have studied so far is that the gates and<\/p>\n\n\n\n
circuits we employ are potentially error-free in themselves. This can hardly<\/p>\n\n\n\n
be guaranteed in practice. However by special coding a circuit or a gate can<\/p>\n\n\n\n
be made tolerant to errors<\/p>\n","protected":false},"excerpt":{"rendered":"
The number of syndromes in a 5-qubit scheme would be 5 \u00d7 3 + 1 = 16. We\u2019d thus need 4 stabilizer operators since 2 4 = 16. We\u2019ll simply give the operators (see Mermin [48] or La\ufb02amme et al. [44]): M 0 = Z 1 X 2 X 3 Z 4 , M 2 […]<\/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-4132","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\/4132","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=4132"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4132\/revisions"}],"predecessor-version":[{"id":4133,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4132\/revisions\/4133"}],"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=4132"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4132"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}