![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg93.png\")
<\/p>\n\n\n\nWe are now ready to see how computing with qubits can be done. In this<\/p>\n\n\n\n
book, we will mainly use the circuit model for computation which was \ufb01rst<\/p>\n\n\n\n
introduced by Deutsch [25]. We will represent by quantum \u201cwires,\u201d the qubits<\/p>\n\n\n\n
upon which manipulations. The length of the wire is to be interpreted as the<\/p>\n\n\n\n
time axis. Manipulations on qubits can be done using basic unitary operators<\/p>\n\n\n\n
that are the equivalents of logic \u201cgates.\u201d An algorithm, or a complete set<\/p>\n\n\n\n
of steps for achieving a processing task, is a combination of wires and gates<\/p>\n\n\n\n
representing a quantum circuit. This circuit must be thought of as a time<\/p>\n\n\n\n
sequence of events with every wire a way of representing qubit states, and<\/p>\n\n\n\n
with gates representing processing of those states.<\/p>\n\n\n\n
|i<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
G<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|o<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|i<\/p>\n\n\n\n
n\u22121<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
G<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|o<\/p>\n\n\n\n
n\u22121<\/p>\n\n\n\n
i<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
G<\/p>\n\n\n\n
3<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
|i<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
G<\/p>\n\n\n\n
4<\/p>\n\n\n\n
|o<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
FIGURE 7.1: Illustrating a quantum circuit with n qubits.<\/p>\n\n\n\n
This notation is based on one by Richard Feynman, with the convention<\/p>\n\n\n\n
that time \ufb02ows from left to right.<\/p>\n\n\n\n
Sometimes, an n-qubit state is represented by a wire with a \/<\/p>\n\n\n\n
n<\/p>\n\n\n\n
decoration<\/p>\n\n\n\n
on it, that is referred to as a register.<\/p>\n\n\n\n
|ii<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\/<\/p>\n\n\n\n
n<\/p>\n\n\n\n
Circuit<\/p>\n\n\n\n
\/<\/p>\n\n\n\n
n<\/p>\n\n\n\n
|oi<\/p>\n\n\n\n
n<\/p>\n\n\n\n
In practice the circuit is e\ufb00ectively a unitary operator acting on the input<\/p>\n\n\n\n
qubits. A few major di\ufb00erences between classical circuits and quantum ones<\/p>\n\n\n\n
are:<\/p>\n\n\n\n
\u2022 Quantum circuits never contain loops or feedbacks: they are acyclic<\/p>\n\n\n\n
\u2022 Quantum wires are never fanned out: since arbitrary quantum states<\/p>\n\n\n\n
cannot be cloned<\/p>\n\n\n\n
121<\/p>\n\n\n\n
<\/p>\n\n\n\n
122 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
\u2022 Though the action of a circuit can be analyzed using classical states, the<\/p>\n\n\n\n
e\ufb00ect on superpositions is what gives it true quantum power.<\/p>\n\n\n\n
The fact that quantum evolution is unitary results in quantum gates (and<\/p>\n\n\n\n
circuits) being reversible. This means that any manipulation of quantum infor-<\/p>\n\n\n\n
mation can be undone, unless an irreversible process such as measurement or<\/p>\n\n\n\n
decoherence happens on the system. This, and the peculiar features of qubits<\/p>\n\n\n\n
discussed in Chapter 4, makes for startling di\ufb00erences in the way we must<\/p>\n\n\n\n
think about quantum algorithms.<\/p>\n\n\n\n
Mathematically a gate can be represented as a matrix. Classical reversible<\/p>\n\n\n\n
gates can have only ones and zeros as elements: reversibility implies that they<\/p>\n\n\n\n
can only perform a permutation of the inputs. For example, a reversible XOR<\/p>\n\n\n\n
gate is given by y<\/p>\n\n\n\n
0<\/p>\n\n\n\n
output in the truth table of what has sometimes been called<\/p>\n\n\n\n
a Feynman gate:<\/p>\n\n\n\n
x y x<\/p>\n\n\n\n
0<\/p>\n\n\n\n
y<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0 0 0 0<\/p>\n\n\n\n
0 1 0 1<\/p>\n\n\n\n
1 0 1 1<\/p>\n\n\n\n
1 1 1 0<\/p>\n\n\n\n
=\u21d2<\/p>\n\n\n\n
00 01 10 11<\/p>\n\n\n\n
\uf8eb<\/p>\n\n\n\n
\uf8ec<\/p>\n\n\n\n
\uf8ec<\/p>\n\n\n\n
\uf8ed<\/p>\n\n\n\n
\uf8f6<\/p>\n\n\n\n
\uf8f7<\/p>\n\n\n\n
\uf8f7<\/p>\n\n\n\n
\uf8f8<\/p>\n\n\n\n
00 1 0 0 0<\/p>\n\n\n\n
01 0 1 0 0<\/p>\n\n\n\n
10 0 0 0 1<\/p>\n\n\n\n
11 0 0 1 0<\/p>\n\n\n\n
(7.1)<\/p>\n\n\n\n
Such a gate is also implementable as a quantum gate, but the most generic<\/p>\n\n\n\n
quantum gate is represented by a complex matrix<\/p>\n","protected":false},"excerpt":{"rendered":"
We are now ready to see how computing with qubits can be done. In this book, we will mainly use the circuit model for computation which was \ufb01rst introduced by Deutsch [25]. We will represent by quantum \u201cwires,\u201d the qubits upon which manipulations. The length of the wire is to be interpreted as the time […]<\/p>\n","protected":false},"author":1,"featured_media":4061,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[496],"tags":[],"class_list":["post-4060","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-quantum-gates-and-circuits"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/idea-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4060","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=4060"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4060\/revisions"}],"predecessor-version":[{"id":4062,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4060\/revisions\/4062"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4061"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4060"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4060"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4060"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}