Jordan-Wigner representation

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

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

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 unitrary matrices.

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