### Quantum Computation

A well funded industry these days is research into the real world implementation of small sets of carefully chosen quantum operations, combinations of which are able to closely simulate any quantum computation. This is known as universal quantum computation, or UQC.

For example, in this paper by Kitaev and Bravyi, it is shown that the Clifford operations (which stabilise Pauli operators) along with certain magic states are sufficient for UQC. They assume that Clifford operations may be implemented ideally, and that the preparation of magic states is faulty. However, they first consider ideal magic states, such as

$|H \rangle = \textrm{cos} \frac{\pi}{8} |0 \rangle + \textrm{sin} \frac{\pi}{8} |1 \rangle$

$|T \rangle = \textrm{cos} \beta |0 \rangle + \omega \textrm{sin} \beta |1 \rangle$

where $\omega$ is a primitive eighth root of unity and $\textrm{cos} 2 \beta = \sqrt{3}^{-1}$. In particular, $| T \rangle$ may be used to implement a one qubit phase gate for the $12$-th root of unity. With the Clifford operations, this gate gives UQC.

The only Clifford generator that is not obviously a fun (field with one element) operation is the one qubit Hadamard gate (Fourier transform), but recall that (in MUB maths) this basis behaves like the zero of a finite field, or the standard choice of marked point for a fancy set! Now let's do UQC without complex numbers.

For example, in this paper by Kitaev and Bravyi, it is shown that the Clifford operations (which stabilise Pauli operators) along with certain magic states are sufficient for UQC. They assume that Clifford operations may be implemented ideally, and that the preparation of magic states is faulty. However, they first consider ideal magic states, such as

$|H \rangle = \textrm{cos} \frac{\pi}{8} |0 \rangle + \textrm{sin} \frac{\pi}{8} |1 \rangle$

$|T \rangle = \textrm{cos} \beta |0 \rangle + \omega \textrm{sin} \beta |1 \rangle$

where $\omega$ is a primitive eighth root of unity and $\textrm{cos} 2 \beta = \sqrt{3}^{-1}$. In particular, $| T \rangle$ may be used to implement a one qubit phase gate for the $12$-th root of unity. With the Clifford operations, this gate gives UQC.

The only Clifford generator that is not obviously a fun (field with one element) operation is the one qubit Hadamard gate (Fourier transform), but recall that (in MUB maths) this basis behaves like the zero of a finite field, or the standard choice of marked point for a fancy set! Now let's do UQC without complex numbers.

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