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Topic: Toffoli gate

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	Rinton Press - Publisher in Science and Technology
		A simplified Toffoli gate coincides with the Toffoli gate except that the result is allowed to differ on one computational basis state by a phase factor.
		We prove that the simplified Toffoli gate implementation by Margolus is optimal, in the sense that it attains a lower bound of {\em three} controlled-not gates, and subject to that, a sharp lower bound of {\em four} single-qubit gates.
		We clarify the connections between the erasure scheme of probabilistic CNOT gate implementation recently proposed by Pittman, Jacobs and Franson [Phys.
	www.rintonpress.com /journals/qicabstracts/qicabstracts4-5.html (582 words)

	Toffoli gate - Wikipedia, the free encyclopedia
		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.
		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.
		Although it may never be implemented in an actual quantum computer, the Toffoli gate has already played an important role in theoretical research on quantum computing; for example in quantum error correction.
	en.wikipedia.org /wiki/Toffoli_gate (688 words)

	Quantum gate - Wikipedia, the free encyclopedia
		A quantum gate or quantum logic gate is a rudimentary quantum circuit operating on a small number of qubits.
		Some universal classical logic gates, such as the Toffoli gate, provide reversibility and can be directly mapped onto quantum logic gates.
		The controlled-U gate is a gate that operates on two qubits in such a way that the first qubit serves as a control.
	en.wikipedia.org /wiki/Quantum_gate (420 words)

	toffoli gate (Site not responding. Last check: 2007-10-29)
		The Fredkin gate is the reversible 3-bit gate that swaps the last two bits if the first bit is 1.
		It is almost universal and can be used in the same way as the Toffoli gate, except it can't change the total number of 0s or 1s.
		Therefore, Toffoli gate can be used for universal quantum computation.
	www.yourencyclopedia.net /Toffoli_gate.html (671 words)

	Fredkin Gate - Wikipedia, the free encyclopedia
		The Fredkin Gate is computational circuit suitable for reversible computing, invented by Ed Fredkin.
		A generalized n x n Fredkin Gate passes its first n-2 inputs to the corresponding outputs, and swaps its last two outputs if and only if the ﬁrst n-2 inputs are all 1.
		See also the Toffoli Gate, which is a controlled-controlled-NOT gate, often used in reversible computing.
	en.wikipedia.org /wiki/Fredkin_Gate (312 words)

	Reversible Benchmarks (Site not responding. Last check: 2007-10-29)
		In our calculations, quantum cost of the generalized Toffoli gates is taken from the following table, according to the known to us published results on their cost.
		The cost of a size k Fredkin gate is calculated by the formula: cost of size k Toffoli gate plus 2, as the size-k Fredkin gate can be efficiently simulated by a size k Toffoli gate and 2 CNOTs.
		The cost calculator finds each gate in these two tables and takes the minimal cost (if several gate implementations are known) with the condition that sum of values in columns Size and Garbage of this implementation does not exceed the number of lines in the analyzed circuit.
	www.cs.unb.ca /profs/gdueck/quantum/definitions.html (262 words)

	Reversible Computation/Gates
		Thus, one of the basic requirements for a reversible gate is that there are as many output variables as there are input variables.
		The Toffoli gate has one control bit and interchanges the value of the other two bits if it’s a 1.
		Using specific input values for the Fredkin gate, it is possible to produce reversible gates for operations that were previously deemed irreversible, such as AND.
	users.tpg.com.au /kataryna/thermodynamics_of_computation/information_pages/reversible_computation.htm (639 words)

	[No title]
		Toffoli also distinguished garbage lines, that is, lines whose value depends on the input data and thus cannot be used as source lines for a new computation; temporary storage consists instead of output lines with constant values, thus useful for further computations.
		Toffoli discussed methods for reducing the use of source and garbage lines, culminating in the following theorem: Using as a primitive the Toffoli gate, any function [phi] can be realized reversibly, possibly with temporary storage, but with no garbage.
		The latter is the Toffoli gate; the former is again the Toffoli gate, but with just one control bit (it is a 2-bit gate).
	www.research.ibm.com /journal/rd/456/zuliani.txt (5456 words)

	Simulating Quantum Circuits - Quantum Gates
		Gates are called universal gates if they can be used to create any logic circuit, such as the NAND gate in classical circuits.
		The square root of NOT gate is the first purely quantum gate that we shall examine and, from a classical point of view, is unusual in its behaviour.
		However two such gates linked sequentially produce an output that is the inverse of the input, and thus behave in the same way as the a classical NOT gate.
	www.themilkyway.com /quantum/FinalReport/QuantumGates.html (787 words)

	The controlled- NOT gate
		The fact that the Toffoli gate is universal for classical computation is made use of in the proof.
		Although this gate seems quite trivial from a logic point of view it is not easy to implement.
		The reason for this is that it is a two qubit gate and its implementation calls for a very precise control of couplings between qubits.
	beige.ucs.indiana.edu /M743-talk-2/node8.html (491 words)

	The Toffoli Gate
		But classical gates do not have to be reversible, and, as a matter of fact, most of them aren't.
		When you think of it, it is obvious that the number of reversible gates must be smaller, because reversible gates form a subset of an all-gates set.
		Now notice that by combining multiple n-bit Toffoli gates with appropriately adjusted controls a circuit can be built that transposes any two specific n-bit strings, since such transpositions can be built from the ones that transpose a single bit only.
	beige.ucs.indiana.edu /M743/node94.html (959 words)

	Quantum computation: theory and implementation (Site not responding. Last check: 2007-10-29)
		The Toffoli gate (a three-bit reversible gate, which flips the third bit if the first two are both 1) is universal for quantum computation, in the sense that appropriate applications of the Toffoli gate, combined with single-qubit rotations, suffice to implement any quantum computation (Deu89).
		The conditional relative phase shift is an very natural gate for implementing quantum computation on a linear ion trap (although it is less useful as a primitive in NMR quantum computation).
		For NMR quantum computation, the fundamental two-qubit computational gate is the J-coupling operator, which is the state-evolution operator corresponding to the scalar coupling of two spins for some period of time.
	www.stanford.edu /~eboyden3/mengthesis/thesis2.htm (3730 words)

	APS - 2005 APS March Meeting - Event - Quantum gates using a pulsed bias scheme (Site not responding. Last check: 2007-10-29)
		We show how a NOT gate (one qubit), a C-NOT gate (two qubits) and a Toffoli gate (three qubits) can be realized by clocking the bias on one of the qubits in the system.
		The key to the C-NOT gate is to maintain the control qubit in a memory state while forcing the target qubit into a transitional state.
		The reduced Hamiltonian approach is then extended further to a Toffoli gate which has two control qubits and one target qubit.
	meetings.aps.org /Meeting/MAR05/Event/22257 (247 words)

	Citations: A Universal Two--Bit Gate for Quantum Computation - Barenco (ResearchIndex)
		BMP 99] have recently shown that the set consisting of a Hadamard gate, a c NOT gate, and a phase rotation gate of angle =4 is universal.
		Deutsch defined the extended quantum XOR 3 qubit gate (known as the Deutsch Toffoli gate) and proved this gate, in combination with certain one qubit gates, is universal.
		Deutsch defined the extended quantum XOR 3 qubit gate (known as the DeutschToffoli gate) and proved this gate, in combination with certain one qubit gates, is universal.
	citeseer.ist.psu.edu /context/31470/0 (1998 words)

	Quantum Computing: An Introduction (Site not responding. Last check: 2007-10-29)
		The possibility of reversible logic gates was considered by Fredkin and Toffoli nearly 20 years ago [10].
		This is called a 'Controlled NOT' or CN gate, since the NOT operation on the lower input line is only operative when there is a '1' on the upper input: a '0' on the upper input means that the lower bit passes through unchanged.
		Well for one thing it is possible that use of such gates may one day be needed to reduce power consumption of microprocessors implemented in CMOS silicon technology.
	www.qtc.ecs.soton.ac.uk /lecture1/lecture1b.html (706 words)

	News from libquantum
		Release of libquantum 0.2.4: This release fixes a bug in the density operator formalism that caused a compiler error when using certain gates.
		Release of libquantum 0.2.1: The new release offers increased performance for many quantum gates, ports to GNU/Hurd and FreeBSD and some minor additions like an unbounded Toffoli gate.
		Note that this release breaks backward compatibility, so your programs will need recompilation.
	www.enyo.de /libquantum/news.html (267 words)

	Results on Two-Bit Gate Design for Quantum Computers - DiVincenzo, Smolin (ResearchIndex)
		We show that the Toffoli gate, which is a universal gate for all classical reversible computation, can be implemented using a particular sequence of exactly five two-bit gates.
		An arbitrary three-bit unitary gate, which can be used to build up any arbitrary quantum computation, can be implemented exactly with six two-bit gates.
		Five Two-Bit Quantum Gates are Sufficient to Implement the..
	citeseer.ist.psu.edu /divincenzo98results.html (513 words)

	Logic gates for quantum bits:
		Combinations of these are sufficient to construct a Toffoli gate for quantum bits or indeed any unitary operation on a finite number of bits.
		The quantum circuit for an XOR gate is illustrated in Fig.
		Fortunately, the Toffoli gate may be constructed by two-particle scattering processes alone [15,17,18,19,20].
	www-users.cs.york.ac.uk /~schmuel/comp/node7.html (562 words)

	Reference.com/Encyclopedia/Quantum gate
		This means that as matrices, quantum gates can be described by 2 x 2 or 4 x 4 matrices with orthonormal rows.
		The investigation of quantum logic gates is unrelated to quantum logic, which is a foundational formalism for quantum mechanics based on a modification of some of the rules of propositional logic.
		A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate,
	www.reference.com /browse/wiki/Quantum_gate (573 words)

	ipedia.com: Logic gate Article (Site not responding. Last check: 2007-10-29)
		A logic gate is an arrangement of electronically -controlled switches used to calculate operations in Boolean algebra.
		A logic gate is an arrangement of electronically-controlled switches used to calculate operations in Boolean algebra.
		In practice, the output of one gate can only drive a finite number of inputs to other gates, a number called the 'fanout limit', but this limit is rarely reached in real circuits.
	www.ipedia.com /logic_gate_1.html (1674 words)

	Re: Chen-Diao Exponential Quantum Search Speedup and Monte Carlo Integr
		It is a three-qubit gate, rather than a two-qubit gate, but it can be expressed as the product of five nondeterministic two-qubit gates.
		The effect of the Toffoli gate is to change a b 0> to the output ket a b aandb>, where aandb denotes the "and" of "a" and "b".
		A reversible OR gate can be constructed in the following fashion: (a) negate the input qubits a and b, (b) apply a Toffoli gate to produce an output qubit, and then (c) negate all qubits.
	www.lns.cornell.edu /spr/2001-06/msg0033723.html (873 words)

	Citations: Quantum block coding - Wootters (ResearchIndex)
		It differs from the classical logic gate in that the number of output bits must equal the number of input bits; quantum processes are reversible, and must involve no destruction of information.
		....gate into a single global phase factor (this amounts to considering SU(4) rather than U(4) matrices) Simple arithmetic shows that N 2 must be at least 5 in order that the objective function have at least 64 free parameters.
		All classical reversible logic can be generated by the Toffoli gate[7] U T = 0 B B B B B 1 1 1 1 1 1 0 1 1 0 1 C C C C C A ; 4) so we have explored the minimal implementation of this U(8) element.
	citeseer.ist.psu.edu /context/264682/0 (283 words)

	Quantum circuits with unbounded fan-out (Site not responding. Last check: 2007-10-29)
		Using this method, the Quantum Fourier Transform, and a shallow circuit for the increment operator, we construct approximate circuits for the Counting and linear Threshold gate.
		We first define a linear Rank gate (testing a linear equation on source qubits) and construct an exact circuit of size O(n log n) for it.
		Using this gate, we construct circuits for the Counting and linear Threshold gate of size O(n^2 p(n) log n).
	eccc.hpi-web.de /eccc-local/ECCC-Theses/spalek2.html (206 words)

	Efficient Decoherence Control Algorithms (Site not responding. Last check: 2007-10-29)
		Unitary operations are expressed in the quantum circuit model as a finite sequence of elementary gates, such as controlled-not gates and single qubit gates.
		We prove that the simplified Toffoli gate by Margolus, which coincides with the Toffoli gate up to a single change of sign, cannot be realized with less than three controlled-not gates.
		The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing.
	faculty.cs.tamu.edu /klappi/dec.html (1369 words)

	Thapliyal: "A Beginning in the Reversible Logic Synthesis of Sequential Circuits" (Site not responding. Last check: 2007-10-29)
		In the past, several reversible gates have been proposed, the important one being Feynman gate, the Toffoli gate, Fredkin gate and New gate.
		Outputs from one gate are used as inputs to the next gate without fan-out of more than one.
		The important reversible gates used for reversible logic synthesis are Feynman Gate, Toffoli Gate, and New Gate and Fredkin gate.
	klabs.org /mapld05/abstracts/1012_thapliyal_2_a.html (397 words)

	Logical reversibility
		Another model of logical reversibility, the Fredkin gate [5], is a 3-bit logic gate defined by the function FG:
		Toffoli's work on reversible computing [12] considered the problem of realizing in a reversible way any function
		Toffoli discussed methods for reducing the use of source and garbage lines, culminating in the following theorem: Using as a primitive the Toffoli gate, any function
	www.research.ibm.com /journal/rd/456/zuliani.html (4790 words)

	SWAP, Toffoli, Fredkin gates (Site not responding. Last check: 2007-10-29)
		Toffoli gate with the control bits a and b:
		Toffoli gate with the control bits b and c:
		Fredkin gate with the SWAPped bits a and b:
	www.qc.ee.es.osaka-u.ac.jp /~saitoh/silqcs/1.1.2/manual/node24.html (46 words)

	Generalised Quantum Gates Text - Physics Forums Library
		gate Phase gate and pi/8 gate or any other useful gates.
		There is no unique generalization of these gates and the one that you
		One possibility for a generalized controlled unitary gate which is
	www.physicsforums.com /archive/index.php/t-17380.html (283 words)

	Computation without ERASE:
		One solution, known as the Toffoli gate, is shown in Fig.
		A combination of many such gates could then be used for any computation and would still be reversible.
		This gate is clearly reversible since a second application of it retrieves the original input.
	www.weizmann.ac.il /chemphys/schmuel/comp/node5.html (361 words)

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