# Jordan-Wigner representation

Current quantum computers work with unitary gates. In order to represent operators relevant for quantum chemistry on a quantum computer, unitary operators are thus needed. The condition for a unitary operator is:

$UU^\dagger = I$

It can easily be checked if this condition is fulfilled for the annihilation and creation operators:

$\begin{eqnarray} aa^\dagger &=& \left(\begin{matrix} 0 & 1\\ 0 & 0 \end{matrix}\right)\left(\begin{matrix} 0 & 0\\ 1 & 0 \end{matrix}\right) = \left(\begin{matrix} 1 & 0\\ 0 & 0 \end{matrix}\right) \neq I \\ a^\dagger \left(a^\dagger\right)^\dagger &=& \left(\begin{matrix} 0 & 0\\ 1 & 0 \end{matrix}\right)\left(\begin{matrix} 0 & 1\\ 0 & 0 \end{matrix}\right) = \left(\begin{matrix} 0 & 0\\ 0 & 1 \end{matrix}\right) \neq I \end{eqnarray}$

It can thus be seen that the annihilation and creation operators are not unitary. Instead we can consider the Pauli operators:

$\begin{eqnarray} X &=& \left(\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix}\right) \\ Y &=& \left(\begin{matrix} 0 & -i\\ i & 0 \end{matrix}\right) \\ Z &=& \left(\begin{matrix} 1 & 0\\ 0 & -1 \end{matrix}\right) \end{eqnarray}$

It can easily be verified that these operators are indeed unitary:

$\begin{eqnarray} XX^\dagger &=& \left(\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix}\right)\left(\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix}\right) = \left(\begin{matrix} 1 & 0\\ 0 & 1 \end{matrix}\right) = I \\ YY^\dagger &=& \left(\begin{matrix} 0 & -i\\ i & 0 \end{matrix}\right)\left(\begin{matrix} 0 & -i\\ i & 0 \end{matrix}\right) = \left(\begin{matrix} 1 & 0\\ 0 & 1 \end{matrix}\right) = I \\ ZZ^\dagger &=& \left(\begin{matrix} 1 & 0\\ 0 & -1 \end{matrix}\right)\left(\begin{matrix} 1 & 0\\ 0 & -1 \end{matrix}\right) = \left(\begin{matrix} 1 & 0\\ 0 & 1 \end{matrix}\right) = I \end{eqnarray}$

Now let us consider the following construction:

$\frac{X+iY}{2} = \frac{\left(\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix}\right)+i\left(\begin{matrix} 0 & -i\\ i & 0 \end{matrix}\right)}{2} = \frac{\left(\begin{matrix} 0 & 1\\ 1 & 0 \end{matrix}\right)+\left(\begin{matrix} 0 & 1\\ -1 & 0 \end{matrix}\right)}{2} = \frac{\left(\begin{matrix} 0 & 2\\ 0 & 0 \end{matrix}\right)}{2} = \left(\begin{matrix} 0 & 1\\ 0 & 0 \end{matrix}\right) = a$

It can be seen that the annihilation operator can be expressed in terms of the Pauli operators. The same is true for the creation operator:

$a^\dagger=\frac{X-iY}{2}$

Now instead of using the phase-factor $$\Gamma$$ the sign change can be handled using $$Z$$ (I have not justified this). The two annihaltion operators working on the state $$\left|11\right>$$ can now be constructed as:

$a_1 = \left(\frac{X+iY}{2}\right)\otimes I$

and

$a_2 = Z\otimes \left(\frac{X+iY}{2}\right)$

Let us try to expand out $$a_2$$ to see it gives what is expected:

$a_2 = Z\otimes \left(\frac{X+iY}{2}\right) = \left(\begin{matrix} 1 & 0\\ 0 & -1 \end{matrix}\right)\otimes\left(\begin{matrix} 0 & 1\\ 0 & 0 \end{matrix}\right) = \left(\begin{matrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{matrix}\right)$

Using this $$a_2$$ on the state $$\left|11\right>$$ now gives:

$a_2\left|11\right> = \left(\begin{matrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 \end{matrix}\right)\left(\begin{matrix} 0\\ 0\\ 0\\ 1 \end{matrix}\right) = \left(\begin{matrix} 0\\ 0\\ -1\\ 0 \end{matrix}\right) = -\left|10\right>$

Which is the same as when using the phase-factor. The annihilation and creation operator has now been expressed in terms of unitary matrices.

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