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 elementary logic gates of a classical computer, other than the NOT gate, are not reversible. Thus, for instance, for an AND gate one cannot 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 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 data is a string of bits x₁,x₂, ...,x_n 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.

To define 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 the Hilbert space

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