# Quantum circuit

In quantum information theory, a quantum circuit is a model for quantum computation in which a computation is a sequence of quantum gates, which are reversible transformations on a quantum mechanical analog of an n-bit register. This analogous structure is referred to as an n-qubit register. The graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation.

Reversible classical logic gates

The elementary logic gates of a classical computer, other than the NOT gate, are not reversible. Thus, for instance, for an AND gate one cannot always recover the two input bits from the output bit; for example, if the output bit is 0, we cannot tell from this whether the input bits are 0,1 or 1,0 or 0,0.

However, reversible gates in classical computers are easily constructed for bit strings of any length; moreover, these are actually of practical interest, since irreversible gates must always increase physical entropy. A reversible gate is a reversible function on n-bit data that returns n-bit data, where an n-bit data is a string of bits x1,x2, …,xn of length n. The set of n-bit data is the space {0,1}n, which consists of 2n strings of 0’s and 1’s.

More precisely: an n-bit reversible gate is a bijective mapping f from the set {0,1}n of n-bit data onto itself. An example of such a reversible gate f is a mapping that applies a fixed permutation to its inputs. For reasons of practical engineering, one typically studies gates only for small values of n, e.g. n=1, n=2 or n=3. These gates can be easily described by tables.

References

Biham, Eli; Brassard, Gilles; Kenigsberg, Dan; Mor, Tal (2004), “Quantum computing without entanglement”, Theoretical Computer Science, 320 (1): 15–33, arXiv:quant-ph/0306182, doi:10.1016/j.tcs.2004.03.041, MR 2060181.

Freedman, Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan (2003), “Topological quantum computation”, Bulletin of the American Mathematical Society, 40 (1): 31–38, arXiv:quant-ph/0101025, doi:10.1090/S0273-0979-02-00964-3, MR 1943131.

Hirvensalo, Mika (2001), Quantum Computing, Natural Computing Series, Berlin: Springer-Verlag, ISBN 3-540-66783-0, MR 1931238.

Kitaev, A. Yu. (1997), “Quantum computations: algorithms and error correction”, Uspekhi Mat. Nauk (in Russian), 52 (6(318)): 53–112, Bibcode:1997RuMaS..52.1191K, doi:10.1070/RM1997v052n06ABEH002155, MR 1611329.

Nielsen, Michael A.; Chuang, Isaac L. (2000), Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, ISBN 0-521-63235-8, MR 1796805.

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