Schnirelmann density

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Circuit representation of Toffoli gate

In computer science, the Toffoli gate (also CCNOT gate), invented by Tommaso Toffoli, is a universal reversible logic gate, which means that any reversible circuit can be constructed from Toffoli gates. It is also known as the "controlled-controlled-not" gate, which describes its action. It has 3-bit inputs and outputs; if the first two bits are set, it inverts the third bit, otherwise all bits stay the same.

Background

A logic gate L is reversible if, for any output y, there is a unique input x such that applying L(x) = y. If a gate L is reversible, there is an inverse gate L′ which maps y to x for which L′(y) = x. From common logic gates, NOT is reversible, as can be seen from its truthtable below.

INPUT OUTPUT
0 1
1 0

The common AND gate is not reversible however. The inputs 00, 01 and 10 are all mapped to the output 0.

Reversible gates have been studied since the 1960s. The original motivation was that reversible gates dissipate less heat (or, in principle, no heat). In a normal gate, input states are lost, since less information is present in the output than was present at the input. This loss of information loses energy to the surrounding area as heat, because of thermodynamic entropy. Another way to understand this is that charges on a circuit are grounded and thus flow away, taking a small quantity of energy with them when they change state. A reversible gate only moves the states around, and since no information is lost, energy is conserved.

More recent motivation comes from quantum computing. Quantum mechanics requires the transformations to be reversible but allows more general states of the computation (superpositions). Thus, the reversible gates form a subset of gates allowed by quantum mechanics and, if we can compute something reversibly, we can also compute it on a quantum computer.

Universality and Toffoli gate

Any reversible gate must have the same number of input and output bits, by the pigeonhole principle. For one input bit, there are two possible reversible gates. One of them is NOT. The other is the identity gate which maps its input to the output unchanged. For two input bits, the only non-trivial gate is the controlled NOT gate which XORs the first bit to the second bit and leaves the first bit unchanged.

Truth table Permutation matrix form
INPUT OUTPUT
 0   0   0   0 
0 1 0 1
1 0 1 1
1 1 1 0

Unfortunately, there are reversible functions that cannot be computed using just those gates. In other words, the set consisting of NOT and XOR gates is not universal. If we want to compute an arbitrary function using reversible gates, we need another gate. One possibility is the Toffoli gate, proposed in 1980 by Toffoli.[1]

This gate has 3-bit inputs and outputs. If the first two bits are set, it flips the third bit. The following is a table of the input and output bits:

Truth table Permutation matrix form
INPUT OUTPUT
 0   0   0   0   0   0 
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 1
1 1 1 1 1 0

It can be also described as mapping bits a, b and c to a, b and c XOR (a AND b).

The Toffoli gate is universal; this means that for any Boolean function f(x1, x2, ..., xm), there is a circuit consisting of Toffoli gates which takes x1, x2, ..., xm and some extra bits set to 0 or 1 and outputs x1, x2, ..., xm, f(x1, x2, ..., xm), and some extra bits (called garbage). Essentially, this means that one can use Toffoli gates to build systems that will perform any desired Boolean function computation in a reversible manner.

Related logic gates

Fredkin & Toffoli billiard ball model for gates
  • The Fredkin gate is a reversible 3-bit gate that swaps the last two bits if the first bit is 1; a controlled-swap operation.
  • The n-bit Toffoli gate is a generalization of Toffoli gate. It takes n bits x1, x2, ..., xn as inputs and outputs n bits. The first n−1 output bits are just x1, ..., xn−1. The last output bit is (x1 AND ... AND xn−1) XOR xn.
  • The Toffoli gate can be realized by five two-qubit quantum gates.[2]
  • This gate is one of the reversible-gate cases that can be modeled with billiard balls (see Billiard-ball computer). The billiard ball modeling was introduced by Fredkin and Toffoli.[3] An example of how the collisions are used to model an electronic gate is shown in the figure.

Relation to quantum computing

Any reversible gate can be implemented on a quantum computer, and hence the Toffoli gate is also a quantum operator. However, the Toffoli gate can not be used for universal quantum computation, though it does mean that a quantum computer can implement all possible classical computations. The Toffoli gate has to be implemented along with single qubit gates to be used for universal quantum computation. The Toffoli gate has been successfully realized in January 2009 at the University of Innsbruck, Austria.[4]

See also

References

  1. Technical Report MIT/LCS/TM-151 (1980) and an adapted and condensed version: 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

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    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang