nLab quantum logic gate

Idea

Analogously to how a classical logic gate is a function between (finite) sets of tuples of bits (truth values), so a quantum logic gate is a (unitary) linear operator on (finite-dimensional) Hilbert spaces of tensor products of qbits:

Specifically, one calls such a linear map a quantum gate if it is thought of as potentially implemented as a basic operation performed by a quantum computing machine.

As such, typical quantum logic gates operate on a small number of qbits, with more complicated such linear maps obtained by composing a given set of quantum logic gates into quantum logic circuits. Such compilation is hence one model of quantum computation.

Examples

The first examples are linearizations of classical logic gates, or rather of their reversible versions:

NOT or X:

XOR and CNOT:

AND:

The following examples have no classical analog:

Hadamard gate:

Pauli gate

(…)

(…)

Related concepts

• logic gate

• controlled quantum gate

• unitary quantum channel

• quantum circuit

• quantum logic

• qbit, qtrit, qdit

References

The notion of quantum logic gates and quantum circuits originates with

• Richard Feynman, Quantum mechanical computers, Foundations of Physics 16 (1986) 507–531 [doi:10.1007/BF01886518]

• David E. Deutsch, Quantum computational networks, Proceedings of the Royal Society A, 425 1868 (1989) 73-90 [doi:10.1098/rspa.1989.0099, jstor:2398494]

• Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, Harald Weinfurter, Elementary gates for quantum computation, Phys. Rev. A52 (1995) 3457 [arXiv:quant-ph/9503016, doi:10.1103/PhysRevA.52.3457]

See also:

• Wikipedia, Quantum logic gate

Implementation of quantum logic gates on qbits realized via nucleon-spin, via pulse protocols in nuclear magnetic resonance-technology:

• Price, Somaroo, Tseng, Gore, Fahmy,, Havel, Cory: Construction and Implementation of NMR Quantum Logic Gates for Two Spin Systems, Journal of Magnetic Resonance 140 2 (1999) 371-378 [doi;10.1006/jmre.1999.1851]

and analogously on electron-spin:

• M. Yu. Volkov and K. M. Salikhov, Pulse Protocols for Quantum Computing with Electron Spins as Qubits, Appl Magn Reson 41 (2011) 145–154 [doi:10.1007/s00723-011-0297-2]