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ntwalibas committed Nov 14, 2018
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Expand Up @@ -149,10 +149,9 @@ To make our oracles reversible, we use the following scheme, dubbing it *XOR enc

.. admonition:: XOR encoding of boolean functions

Let :math:`f(x_1, x_2, \ldots, x_n):\mathbb{B}^n \to \mathbb{B}` be a boolean function.
| Define :math:`F(x_1, x_2, \ldots, x_n, z):\mathbb{B}^{n+1} \to \mathbb{B}` as
:math:`F(x_1, x_2, \ldots, x_n, z) = (x_1, x_2, \ldots, x_n, z \oplus f(x_1, x_2, \ldots, x_n))`.
| The function :math:`F:\mathbb{B}^{n+1} \to \mathbb{B}` is an encoding of :math:`f:\mathbb{B}^n \to \mathbb{B}`.
| Let :math:`f(x_1, x_2, \ldots, x_n):\mathbb{B}^n \to \mathbb{B}` be a boolean function.
| Define :math:`F(x_1, x_2, \ldots, x_n, z):\mathbb{B}^{n+1} \to \mathbb{B}` as :math:`F(x_1, x_2, \ldots, x_n, z) = (x_1, x_2, \ldots, x_n, z \oplus f(x_1, x_2, \ldots, x_n))`.
| The function :math:`F:\mathbb{B}^{n+1} \to \mathbb{B}` is an encoding of :math:`f:\mathbb{B}^n \to \mathbb{B}` and equivalent to it up to the ancilla :math:`z`.

So we have transformed our classical function into a new function that is equivalent to it but with additional properties:
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