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<\/p>\n\n\n\nWe are studying classical gates to help us develop quantum gates. Quantum<\/p>\n\n\n\n
gates are unitary. This means they are reversible: they can be \u201crun backward\u201d.<\/p>\n\n\n\n
More practically, the meaning is that the inputs can be deduced from the<\/p>\n\n\n\n
outputs. Most classical gates however, are irreversible, and cannot as such<\/p>\n\n\n\n
be extended to quantum gates. For example, the AND gate, being 2 \u2192 1 is<\/p>\n\n\n\n
irreversible: it gives an output of 0 for more than one input set: (0, 0), (0, 1),<\/p>\n\n\n\n
and (1, 0). So given only the output, the input cannot be deduced. So is the<\/p>\n\n\n\n
OR gate and all the other famous 2-bit 2 \u2192 1 gates! For an n-bit gate to be<\/p>\n\n\n\n
reversible it must at least be a 1 \u2192 1 mapping. Further it must give distinct<\/p>\n\n\n\n
outputs for di\ufb00erent inputs. Thus the outputs are all simply permutations<\/p>\n\n\n\n
of the inputs. In terms of matrix representations, reversible gates must be<\/p>\n\n\n\n
invertible. The classical two-bit gates represented by non-square matrices can<\/p>\n\n\n\n
clearly not be inverted.<\/p>\n\n\n\n
The idea of reversibility in classical computation has been studied long<\/p>\n\n\n\n
before quantum gates were thought of (see for example Bennett [7]). It began<\/p>\n\n\n\n
with the ideas of Landauer [46], who argued that erasure of information is<\/p>\n\n\n\n
<\/p>\n\n\n\n
110 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
accompanied by a loss in energy. Irreversible gates essentially erase some bits<\/p>\n\n\n\n
of information in their functioning, and this should lead to intrinsic dissipation<\/p>\n\n\n\n
of energy. Thus if one wants the most energy-e\ufb03cient computing machine it<\/p>\n\n\n\n
should employ reversible gates.<\/p>\n\n\n\n
bit 0 bit 1<\/p>\n\n\n\n
FIGURE 6.3: A simple thermodynamic system encoding a bit of information.<\/p>\n\n\n\n
A simple way to understand how erasing information costs is in terms<\/p>\n\n\n\n
of the thermodynamic quantity known as entropy. We will see more of this<\/p>\n\n\n\n
concept when we study quantifying information. At present we want to see how<\/p>\n\n\n\n
Landauer argued that information erasure causes an increase in the entropy<\/p>\n\n\n\n
of the environment and therefore a decrease in the energy of the system. His<\/p>\n\n\n\n
main point was that information was not something abstract, but was in fact<\/p>\n\n\n\n
the physical system used to represent it. In an illustrative example due to<\/p>\n\n\n\n
Szilard [68], a bit of information can be encoded in terms of the location of a<\/p>\n\n\n\n
molecule in the left or right of a partition in a transparent box (Figure 6.3). If<\/p>\n\n\n\n
we look at the box and \ufb01nd the molecule in the left partition then the system<\/p>\n\n\n\n
encodes a logical 0, and if it is on the right side then it encodes a logical 1.<\/p>\n\n\n\n
We can write one bit of information in this system by putting the molecule in<\/p>\n\n\n\n
the appropriate half.<\/p>\n\n\n\n
One way to erase the information contained in the location of the molecule<\/p>\n\n\n\n
is to remove the partition and push the molecule to one end by compressing<\/p>\n\n\n\n
the \u201cgas\u201d with a piston. If we then replace the partition, the system reads 0<\/p>\n\n\n\n
irrespective of what was encoded in it initially (Figure 6.4).<\/p>\n\n\n\n
FIGURE 6.4: Erasing a bit of information.<\/p>\n\n\n\n
Thermodynamics tells us how to calculate the work done in this process.<\/p>\n\n\n\n
The entropy of a thermodynamic system is related to the logarithm of the<\/p>\n\n\n\n
number of microscopic states available to the system. Since the molecule could<\/p>\n\n\n\n
be in one of two locations, the entropy associated with the single bit encoded<\/p>\n","protected":false},"excerpt":{"rendered":"
We are studying classical gates to help us develop quantum gates. Quantum gates are unitary. This means they are reversible: they can be \u201crun backward\u201d. More practically, the meaning is that the inputs can be deduced from the outputs. Most classical gates however, are irreversible, and cannot as such be extended to quantum gates. For […]<\/p>\n","protected":false},"author":1,"featured_media":4040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[495],"tags":[],"class_list":["post-4054","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-computation-models-and-computational-complexity"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/download-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4054","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=4054"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4054\/revisions"}],"predecessor-version":[{"id":4055,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4054\/revisions\/4055"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4040"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4054"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4054"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}