{"id":4122,"date":"2024-09-22T14:48:38","date_gmt":"2024-09-22T14:48:38","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4122"},"modified":"2024-09-22T14:48:38","modified_gmt":"2024-09-22T14:48:38","slug":"details-stabilizers","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/details-stabilizers\/","title":{"rendered":"Details: stabilizers"},"content":{"rendered":"\n

Why does this scheme work? It is possible to distinguish the syndromes,<\/p>\n\n\n\n

which are orthogonal states, if we measure a suitable observable of which<\/p>\n\n\n\n

they are eigenstates. It turns out that the bit-\ufb02ip syndrome states |\u03c8<\/p>\n\n\n\n

i<\/p>\n\n\n\n

i are<\/p>\n\n\n\n

eigenstates of the operators Z<\/p>\n\n\n\n

1<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

2<\/p>\n\n\n\n

and Z<\/p>\n\n\n\n

2<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

3<\/p>\n\n\n\n

with distinct sets of eigenvalues.<\/p>\n\n\n\n

In other words, for<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O<\/p>\n\n\n\n

I<\/p>\n\n\n\n

= \u2297 Z \u2297 Z and<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O<\/p>\n\n\n\n

II<\/p>\n\n\n\n

= Z \u2297 Z \u2297 ,<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O|\u03c8<\/p>\n\n\n\n

i<\/p>\n\n\n\n

i = \u00b1|\u03c8<\/p>\n\n\n\n

i<\/p>\n\n\n\n

i. (10.5)<\/p>\n\n\n\n

You can easily check that each |\u03c8<\/p>\n\n\n\n

i<\/p>\n\n\n\n

i has a di\ufb00erent set of eigenvalues for<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O<\/p>\n\n\n\n

I<\/p>\n\n\n\n

and<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O<\/p>\n\n\n\n

II<\/p>\n\n\n\n

(Table 10.2). These operators when acting on the full Hilbert space of<\/p>\n\n\n\n

\"\"<\/p>\n\n\n\n

Quantum Error Correction 201<\/p>\n\n\n\n

3-qubit states, do not change the subspaces containing the syndrome states.<\/p>\n\n\n\n

This subspace is said to be invariant under the action of these operators,<\/p>\n\n\n\n

which are therefore known as stabilizers. It is possible to understand why the<\/p>\n\n\n\n

TABLE 10.2: Eigenvalues of stabilizers.<\/p>\n\n\n\n

Error Syndrome 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

|\u03c8<\/p>\n\n\n\n

0<\/p>\n\n\n\n

i +1 +1<\/p>\n\n\n\n

X<\/p>\n\n\n\n

1<\/p>\n\n\n\n

|\u03c8<\/p>\n\n\n\n

1<\/p>\n\n\n\n

i +1 \u22121<\/p>\n\n\n\n

X<\/p>\n\n\n\n

2<\/p>\n\n\n\n

|\u03c8<\/p>\n\n\n\n

2<\/p>\n\n\n\n

i \u22121 \u22121<\/p>\n\n\n\n

X<\/p>\n\n\n\n

3<\/p>\n\n\n\n

|\u03c8<\/p>\n\n\n\n

3<\/p>\n\n\n\n

i \u22121 +1<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

Os of Equation 10.5 are the stabilizers and how they distinguish between the<\/p>\n\n\n\n

syndromes for single qubit-\ufb02ip errors. The uncorrupted codeword state is un-<\/p>\n\n\n\n

changed by the action of<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O. Each corrupted state is obtained by |\u03c8<\/p>\n\n\n\n

i<\/p>\n\n\n\n

i = X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

|<\/p>\n\n\n\n

\u02dc<\/p>\n\n\n\n

\u03c8i,<\/p>\n\n\n\n

and the operators Z<\/p>\n\n\n\n

j<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

k<\/p>\n\n\n\n

either commute or anti-commute with X<\/p>\n\n\n\n

i<\/p>\n\n\n\n

, depend-<\/p>\n\n\n\n

ing on whether i = j or k or not. It is a well-known concept in quantum<\/p>\n\n\n\n

mechanics that operators that commute or anti-commute with a transforma-<\/p>\n\n\n\n

tion operator are symmetries of the system: the states are left unchanged by<\/p>\n\n\n\n

them. Therefore, the corrupted states can be distinguished by measuring<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O<\/p>\n\n\n\n

I<\/p>\n\n\n\n

and<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O<\/p>\n\n\n\n

I<\/p>\n\n\n\n

I, without disturbing the states. Measuring Z<\/p>\n\n\n\n

i<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

j<\/p>\n\n\n\n

is like comparing the<\/p>\n\n\n\n

values of the i<\/p>\n\n\n\n

th<\/p>\n\n\n\n

and j<\/p>\n\n\n\n

th<\/p>\n\n\n\n

qubits, giving +1 if they match and \u22121 if not. Recall<\/p>\n\n\n\n

|ui<\/p>\n\n\n\n

O<\/p>\n\n\n\n

|0i<\/p>\n\n\n\n

H<\/p>\n\n\n\n

\u2022<\/p>\n\n\n\n

H<\/p>\n\n\n\n

FIGURE 10.6: Circuit for measuring an operator<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O.<\/p>\n\n\n\n

that measuring a unitary operator<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O having eigenvalues \u00b11 is achieved by the<\/p>\n\n\n\n

circuit in Figure 10.6, with |ui an eigenstate of<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

O.<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

\u2022 \u2022<\/p>\n\n\n\n

\u2261 \u2261<\/p>\n\n\n\n

H<\/p>\n\n\n\n

\u2022<\/p>\n\n\n\n

H H Z H<\/p>\n\n\n\n

FIGURE 10.7: Circuit equivalences for measuring<\/p>\n\n\n\n

\u02c6<\/p>\n\n\n\n

Z.<\/p>\n\n\n\n

If we need to measure Z<\/p>\n\n\n\n

1<\/p>\n\n\n\n

Z<\/p>\n\n\n\n

2<\/p>\n\n\n\n

, since the C-Z gate is symmetric, we can<\/p>\n\n\n\n

interchange the control and target qubits, and using X = HZH (see Fig-<\/p>\n\n\n\n

ure 10.7) and H<\/p>\n\n\n\n

2<\/p>\n\n\n\n

= 1, we can get the syndrome measurement circuit given<\/p>\n\n\n\n

by Figute 10.8. Check for yourself that each measurement in this process is<\/p>\n\n\n\n

identical to that in Figure 10.4<\/p>\n","protected":false},"excerpt":{"rendered":"

Why does this scheme work? It is possible to distinguish the syndromes, which are orthogonal states, if we measure a suitable observable of which they are eigenstates. It turns out that the bit-\ufb02ip syndrome states |\u03c8 i i are eigenstates of the operators Z 1 Z 2 and Z 2 Z 3 with distinct sets […]<\/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-4122","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\/4122","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=4122"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4122\/revisions"}],"predecessor-version":[{"id":4123,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4122\/revisions\/4123"}],"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=4122"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4122"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4122"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}