fastdla.generators.z2lgt_hva.z2lgt_hva_generators

fastdla.generators.z2lgt_hva.z2lgt_hva_generators(num_fermions, gauge_op='X')

Construct the generators of the HVA for the 1+1-dimensional Z2 Lattice gauge theory model with periodic boundary condition.

The Hamiltonian of the 1+1d LGT with \(N_f\) Dirac fermions (\(N_s = 2 N_f\) lattice sites) and a periodic boundary condition is given by

\[H = f H_{\mathrm{g}} + m H_{\mathrm{m}} + \frac{J}{2} H_{\mathrm{h}}\]

where

\[\begin{split}H_{\mathrm{g}} = \sum_{n=0}^{N_s-1} X_{n,n+1}, \\ H_{\mathrm{m}} = \sum_{n=0}^{N_s-1} (-1)^n Z_n, \\ H_{\mathrm{h}} = \sum_{n=0}^{N_s-1} \frac{1}{2} (X_n Z_{n,n+1} X_{n+1} + Y_n Z_{n,n+1} Y_{n+1}).\end{split}\]

In the above expressions, \(P_n (P=X,Y,Z)\) are the Pauli operators acting on site \(n\), and \(P_{n,n+1} (P=X,Z)\) are those acting on the link between sites \(n\) and \(n+1\). By the boundary condition, we identify site \(N_s\) with site 0.

We then assume a register of \(4 N_f\) qubits in a ring topology, and map site \(n\) and link \(n,n+1\) to qubits \(2n\) and \(2n+1\), respectively. Under this mapping, we define the Hamiltonian variational ansatz (HVA) of this model as

\[U(\vec{\theta}) = \prod_{l=0}^{L-1} e^{-i \theta_{l,0} H_{\mathrm{g}}} e^{-i \theta_{l,1} H_{\mathrm{m}}^{\mathrm{(even)}}} e^{-i \theta_{l,2} H_{\mathrm{m}}^{\mathrm{(odd)}}} e^{-i \theta_{l,3} H_{\mathrm{h}}^{\mathrm{(even)}}} e^{-i \theta_{l,4} H_{\mathrm{h}}^{\mathrm{(odd)}}}\]

where \(L\) is the number of repeated circuit layers, and even (odd) superscripts indicate taking the sum over even (odd) \(n\) in the corresponding definition of the Hamiltonian term.

The generators of the HVA

\[\{ -iH_{\mathrm{g}}^{\mathrm{(even)}}, -iH_{\mathrm{g}}^{\mathrm{(odd)}}, -iH_{\mathrm{m}}^{\mathrm{(even)}}, -iH_{\mathrm{m}}^{\mathrm{(odd)}}}, -iH_{\mathrm{h}}^{\mathrm{(even)}}, -iH_{\mathrm{h}}^{\mathrm{(odd)}} \}\]

all commute with symmetry operators \(\{G_n\}_{n=0}^{N_s-1}\) (Gauss’s law), \(Q\) (total charge), and \(T_2\) (translation). The definitions of \(G_n\) and \(Q\) are given in terms of Pauli operators as

\[\begin{split}G_n = X_{n-1,n} Z_n X_{n,n+1}, \\ Q = \frac{1}{N_s} \sum_{n=0}^{N_s-1} Z_n.\end{split}\]

\(T_2\) is defined as an operation that shifts site index by 2: \(n \to n+2\) (qubit index by 4), and can be implemented with a series of qubit swap operations.

Parameters:

num_fermions (int) – Number of fermions \(N_f\).

Return type:

SparsePauliSumArray

Returns:

Five generators of the HVA.