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A quantum circuit operates on data structures which are quantum mechanical analogs of bitstrings of a given size ''n''. These data structures are sometimes referred to as ''n''-qubits, although Qubit is primarily used as a unit of quantum information. ''n''-qubits can be implemented using any system with an Observable quantity ''A'' which is ''conserved'' under Time Evolution and such that ''A'' has at least two discrete and sufficiently spaced consecutive Eigenvalues . An example of such an observable is spin. REVERSIBLE LOGIC GATES Ordinarily, the logic gates in a classical computer, other than the NOT gate are not Reversible . Thus for instance for an AND gate one cannot generally recover the two input bits from the output bit; the case ''both'' input bits are 1 is the exception. However, as a first step in describing a quantum computing device it is instructive to observe that reversible gates are theoretically possible; moreover, these are actually of practical interest, since they do not increase Entropy . A reversible gate is a reversible function on ''n''-bit data that returns ''n''-bit data, where an ''n''-bit datum is a String of bits ''x''1,''x''2, ...,''x''''n'' of length ''n''. The set of ''n''-bit data is the space {0,1}''n''.
In fact we are only interested in maps ''f'' which are different from the identity, and for reasons of practical engineering we are only interested in gates for small values of ''n'', e.g. ''n''=1, ''n''=2 or ''n''=3. These gates can be easily described by tables. Examples of these logic gates which have been studied are the controlled NOT gate (also called CNOT gate), the Toffoli Gate and the Fredkin Gate . To consider quantum gates, we first need to specify the quantum replacement of an ''n''-bit datum. The ''quantized version'' of classical ''n''-bit space {0,1}''n'' is Using Dirac Ket notation, if ''x''1,''x''2, ...,''x''''n'' is a classical bit string, then | ||
| :<math> W F( X 1, X 2, \cdots,x N Angle) | f(x_1, x_2, \cdots, x_n)
angle </math> |
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| :<math> U Heta 0 Angle | e^{i heta} 0
angle \quad U_ heta 1
angle = 1
angle </math> |
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| :<math> Ec{x},0 Angle | x_1, x_2, \cdots, x_n, \underbrace{0, \dots, 0}
angle </math> |
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| \big Ec{x},0 Ight Angle | \left\langle \operatorname{E}_{F(x)} U( ec{x},0
angle) \big U( ec{x},0
angle)
ight
angle \geq 1 - \epsilon</math> |
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