qit.utils.fermion_ladder

qit.utils.fermion_ladder(*args)

Fermionic ladder operators.

Parameters

n (int) – number of fermionic modes

Returns

fermionic annihilation operators $(f_1,f_2,\dots ,f_n)$

Return type

array[object]

Returns the fermionic annihilation operators for a system of n fermionic modes in the second quantization.

The annihilation operators are built using the Jordan-Wigner transformation for a chain of n qubits, where the state of each qubit denotes the occupation number of the corresponding mode. First define annihilation and number operators for a lone fermion mode:

$$\begin{array}{rl}{\sigma }^{-}& :=({\sigma }_x+i{\sigma }_y)/2,\\ n& :={\sigma }^{+}{\sigma }^{-}=(\mathrm{I}-{\sigma }_z)/2,\\ & {\sigma }^{-}|0⟩=0,{\sigma }^{-}|1⟩=|0⟩,n|k⟩=k|k⟩.\end{array}$$

Then define a phase operator to keep track of sign changes when permuting the order of the operators: ${\varphi }_k:=\sum _{j=1}^{k-1}{n}_j$. Now, the fermionic annihilation operators for the n-mode system are given by

$$f_k:=(-1{)}^{{\varphi }_k}{\sigma }_k^{-}=\left(\underset{j=1}{\overset{k-1}{⨂}}{{\sigma }_z}_j\right){\sigma }_k^{-}.$$

These operators fulfill the required anticommutation relations:

$$\begin{array}{rl}\{f_j,f_k\}& =0,\\ \{f_j,f_k^{†}\}& =\mathrm{I}{\delta }_{jk},\\ f_k^{†}{f}_k& =n_k.\end{array}$$