# Toffoli gate

Circuit representation of Toffoli gate

In logic circuits, the Toffoli gate (also CCNOT gate), invented by Tommaso Toffoli, is a universal reversible logic gate, which means that any classical 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 both set to 1, it inverts the third bit, otherwise all bits stay the same.

## Background

An input-consuming logic gate L is reversible if it meets the following conditions: L(x) = y is a gate where for any output y, there is a unique input x. The gate L is reversible if there is a gate L′(y) = x which maps y to x. From common logic gates, NOT is reversible, as can be seen from its truth table 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).[1] If we think of a logic gate as consuming its input, information is 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 (Landauer's principle). 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.[citation needed]

More recent motivation comes from quantum computing. Quantum mechanics requires the transformations to be reversible, because quantum mechanics is unitary theory and unitary transformations are invertible. Moreover, quantum mechanics allows more general states of the computation than classical computers (superpositions).

## Universality and Toffoli gate

Any reversible gate that consumes its inputs and allows all input computations must have no more input bits than 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

${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{bmatrix}}}$

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. To compute an arbitrary function using reversible gates, another gate is needed. One possibility is the Toffoli gate, proposed in 1980 by Toffoli.[2]

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

${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\\\end{bmatrix}}}$

It can be also described as mapping bits {a, b, c} to {a, b, 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 that takes x1, x2, ..., xm and some extra bits set to 0 or 1 to 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

The Toffoli gate can be constructed from single qubit gates and a minimum of six CNOTs.
• The Fredkin gate is a universal 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,[3] but it can be shown that it is not possible using fewer than five.[4]
• A related quantum gate, the Deutsch gate, can be realized by five optical pulses with neutral atoms.[5] The Deutsch gate is a universal gate for quantum computing.[6]

## 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 some inherently quantum gate(s) in order to be universal for quantum computation. In fact, any single-qubit gate with real coefficients that can create a nontrivial quantum state suffices.[7] A Toffoli gate based on quantum mechanics was successfully realized in January 2009 at the University of Innsbruck, Austria.[8] While the implementation of an n-qubit Toffoli with circuit model requires 2n CNOT gates,[9] the best known upper bound stands at 6n-12 CNOT gates.[10] The application of many-body interaction could be used for direct operation of the gate in trapped ions, Rydberg atoms and superconducting circuit implementations.[11][12][13][14] Following the dark-state manifold, Khazali-Mølmer Cn-NOT gate[12] operates with only three pulses, departing from the circuit model paradigm.

## References

1. ^ Landauer, R. (July 1961). "Irreversibility and Heat Generation in the Computing Process". IBM Journal of Research and Development. 5 (3): 183–191. doi:10.1147/rd.53.0183. ISSN 0018-8646.
2. ^ Technical Report MIT/LCS/TM-151 (1980) and an adapted and condensed version: Toffoli, Tommaso (1980). J. W. de Bakker and J. van Leeuwen (ed.). Reversible computing (PDF). Automata, Languages and Programming, Seventh Colloquium. Noordwijkerhout, Netherlands: Springer Verlag. pp. 632–644. doi:10.1007/3-540-10003-2_104. ISBN 3-540-10003-2. Archived from the original (PDF) on 2010-04-15.
3. ^ Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (Nov 1995). "Elementary gates for quantum computation". Physical Review A. American Physical Society. 52 (5): 3457–3467. arXiv:quant-ph/9503016. Bibcode:1995PhRvA..52.3457B. doi:10.1103/PhysRevA.52.3457. PMID 9912645. S2CID 8764584.
4. ^ Yu, Nengkun; Duan, Runyao; Ying, Mingsheng (2013-07-30). "Five two-qubit gates are necessary for implementing the Toffoli gate". Physical Review A. 88 (1): 010304. arXiv:1301.3372. Bibcode:2013PhRvA..88a0304Y. doi:10.1103/physreva.88.010304. ISSN 1050-2947. S2CID 55486826.
5. ^ Shi, Xiao-Feng (May 2018). "Deutsch, Toffoli, and CNOT Gates via Rydberg Blockade of Neutral Atoms". Physical Review Applied. 9 (5): 051001. arXiv:1710.01859. Bibcode:2018PhRvP...9e1001S. doi:10.1103/PhysRevApplied.9.051001. S2CID 118909059.
6. ^ Deutsch, D. (1989). "Quantum Computational Networks". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 425 (1868): 73–90. Bibcode:1989RSPSA.425...73D. doi:10.1098/rspa.1989.0099. ISSN 0080-4630. JSTOR 2398494. S2CID 123073680.
7. ^ Shi, Yaoyun (Jan 2003). "Both Toffoli and Controlled-NOT need little help to do universal quantum computation". Quantum Information & Computation. 3 (1): 84–92. arXiv:quant-ph/0205115. Bibcode:2002quant.ph..5115S.
8. ^ Monz, T.; Kim, K.; Hänsel, W.; Riebe, M.; Villar, A. S.; Schindler, P.; Chwalla, M.; Hennrich, M.; Blatt, R. (Jan 2009). "Realization of the Quantum Toffoli Gate with Trapped Ions". Physical Review Letters. American Physical Society. 102 (4): 040501. arXiv:0804.0082. Bibcode:2009PhRvL.102d0501M. doi:10.1103/PhysRevLett.102.040501. PMID 19257408. S2CID 2051775.
9. ^ Shende, Vivek V.; Markov, Igor L. (2008-03-15). "On the CNOT-cost of TOFFOLI gates". arXiv:0803.2316 [quant-ph].
10. ^ Maslov, Dmitri (2015-08-13). "On the advantages of using relative phase Toffolis with an application to multiple control Toffoli optimization". arXiv:1508.03273 [quant-ph].
11. ^ Arias Espinoza, Juan Diego; Groenland, Koen; Mazzanti, Matteo; Schoutens, Kareljan; Gerritsma, Rene (2021-05-28). "High-fidelity method for a single-step $N$-bit Toffoli gate in trapped ions". Physical Review A. 103 (5): 052437. arXiv:2010.08490. doi:10.1103/PhysRevA.103.052437. Retrieved 2021-05-29.
12. ^ a b Khazali, Mohammadsadegh; Mølmer, Klaus (2020-06-11). "Fast Multiqubit Gates by Adiabatic Evolution in Interacting Excited-State Manifolds of Rydberg Atoms and Superconducting Circuits". Physical Review X. 10 (2): 021054. Bibcode:2020PhRvX..10b1054K. doi:10.1103/PhysRevX.10.021054. ISSN 2160-3308.
13. ^ Isenhower, L.; Saffman, M.; Mølmer, K. (2011-09-04). "Multibit CkNOT quantum gates via Rydberg blockade". Quantum Information Processing. 10 (6): 755. arXiv:1104.3916. doi:10.1007/s11128-011-0292-4. ISSN 1573-1332. S2CID 28732092.
14. ^ Rasmussen, S. E.; Groenland, K.; Gerritsma, R.; Schoutens, K.; Zinner, N. T. (2020-02-07). "Single-step implementation of high-fidelity n -bit Toffoli gates". Physical Review A. 101 (2): 022308. arXiv:1910.07548. Bibcode:2020PhRvA.101b2308R. doi:10.1103/physreva.101.022308. ISSN 2469-9926. S2CID 204744044.