Toffoli gate

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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.

Contents

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, for all y. From common logic gates, NOT is reversible, as can be seen from its truth table below:

InputOutput
01
10

The common AND gate is not reversible, because 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]

More recent motivation comes from quantum computing. In quantum mechanics the quantum state can evolve in two ways: by Schrödinger's equation (unitary transformations), or by their collapse. Logic operations for quantum computers, of which the Toffoli gate is an example, are unitary transformations and therefore evolve reversibly. [2]

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 (hereafter CNOT), which XORs the first bit to the second bit and leaves the first bit unchanged.

Truth table Permutation matrix form
InputOutput
0000
0101
1011
1110

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. [3]

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 tablePermutation matrix form
InputOutput
000000
001001
010010
011011
100100
101101
110111
111110

It can be also described as mapping bits {a, b, c} to {a, b, c XOR (a AND b)}. This can also be understood as a modulo operation on bit c: {a, b, c} → {a, b, (c + ab) mod 2}, often written as {a, b, c} → {a, b, c ⨁ ab} [4]

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). A NOT gate, for example, can be constructed from a Toffoli gate by setting the three input bits to {a, 1, 1}, making the third output bit (1 XOR (a AND 1)) = NOT a; (a AND b) is the third output bit from {a, b, 0}. Essentially, this means that one can use Toffoli gates to build systems that will perform any desired Boolean function computation in a reversible manner.

The Toffoli gate can be constructed from single qubit T- and Hadamard-gates, and a minimum of six CNOTs. Qcircuit ToffolifromCNOT.svg
The Toffoli gate can be constructed from single qubit T- and Hadamard-gates, and a minimum of six CNOTs.

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 cannot 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. [11] A Toffoli gate based on quantum mechanics was successfully realized in January 2009 at the University of Innsbruck, Austria. [12] While the implementation of an n-qubit Toffoli with circuit model requires 2n CNOT gates, [13] the best known upper bound stands at 6n − 12 CNOT gates. [14] It has been suggested that trapped Ion Quantum computers may be able to implement an n-qubit Toffoli gate directly. [15] The application of many-body interaction could be used for direct operation of the gate in trapped ions, Rydberg atoms and superconducting circuit implementations. [16] [17] [18] [19] [20] [21] Following the dark-state manifold, Khazali-Mølmer Cn-NOT gate [17] operates with only three pulses, departing from the circuit model paradigm. The iToffoli gate was implemented in a single step using three superconducting qubits with pair-wise coupling. [22]

See also

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References

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  2. ↑ Colin P. Williams (2011). Explorations in Quantum Computing. Springer. pp. 25–29, 61. ISBN   978-1-84628-887-6.
  3. ↑ 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.
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  10. ↑ Kim, Y.; Morvan, A.; Nguyen, L.B.; Naik, R.K.; Jünger, C.; Chen, L.; Kreikebaum, J.M.; Santiago, D.I.; Siddiqi, I. (2 May 2022). "High-fidelity three-qubit iToffoli gate for fixed-frequency superconducting qubits". Nature Physics. 18 (5): 783–788. arXiv: 2108.10288 . Bibcode:2022NatPh..18..783K. doi: 10.1038/s41567-022-01590-3 .
  11. ↑ 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. doi:10.26421/QIC3.1-7.
  12. ↑ 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. 102 (4): 040501. arXiv: 0804.0082 . Bibcode:2009PhRvL.102d0501M. doi:10.1103/PhysRevLett.102.040501. PMID   19257408. S2CID   2051775.
  13. ↑ Shende, Vivek V.; Markov, Igor L. (2008-03-15). "On the CNOT-cost of TOFFOLI gates". arXiv: 0803.2316 [quant-ph].
  14. ↑ Maslov, Dmitri (2016). "Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization". Physical Review A. 93 (2): 022311. arXiv: 1508.03273 . Bibcode:2016PhRvA..93b2311M. doi:10.1103/PhysRevA.93.022311. S2CID   5226873.
  15. ↑ Katz, Or; Cetina, Marko; Monroe, Christopher (2022-02-08). " -Body Interactions between Trapped Ion Qubits via Spin-Dependent Squeezing". Physical Review Letters. 129 (6): 063603. arXiv: 2202.04230 . doi:10.1103/PhysRevLett.129.063603. PMID   36018637. S2CID   246679905.
  16. ↑ 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 . Bibcode:2021PhRvA.103e2437E. doi:10.1103/PhysRevA.103.052437. S2CID   223953418.
  17. 1 2 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. arXiv: 2006.07035 . Bibcode:2020PhRvX..10b1054K. doi: 10.1103/PhysRevX.10.021054 . ISSN   2160-3308.
  18. ↑ 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.
  19. ↑ 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.
  20. ↑ Nguyen, L.B.; Kim, Y.; Hashim, A.; Goss, N.; Marinelli, B.; Bhandari, B.; Das, D.; Naik, R.K.; Kreikebaum, J.M.; Jordan, A.; Santiago, D.I.; Siddiqi, I. (16 January 2024). "Programmable Heisenberg interactions between Floquet qubits". Nature Physics. 20 (1): 240–246. arXiv: 2211.10383 . Bibcode:2024NatPh..20..240N. doi: 10.1038/s41567-023-02326-7 .
  21. ↑ Nguyen, Long B.; Goss, Noah; Siva, Karthik; Kim, Yosep; Younis, Ed; Qing, Bingcheng; Hashim, Akel; Santiago, David I.; Siddiqi, Irfan (2023-12-29). "Empowering high-dimensional quantum computing by traversing the dual bosonic ladder". arXiv.org. Retrieved 2024-04-08.
  22. ↑ Kim, Y.; Morvan, A.; Nguyen, L.B.; Naik, R.K.; Jünger, C.; Chen, L.; Kreikebaum, J.M.; Santiago, D.I.; Siddiqi, I. (2 May 2022). "High-fidelity three-qubit iToffoli gate for fixed-frequency superconducting qubits". Nature Physics. 18 (5): 783–788. arXiv: 2108.10288 . Bibcode:2022NatPh..18..783K. doi: 10.1038/s41567-022-01590-3 .
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