We proivde an implementation of the Variational Quantum Eigensolver for Heisenberg models following Bosse and Montanaro and Kattemölle and van Wezel.
Heisenberg problem#Creates a VQE problem instance for an isotropic Heisenberg model defined by a graph \(G=(V,E)\), the coupling constant \(J>0\) (antiferromagnetic), and the magnetic field strength \(B\). The model Hamiltonian is given by:
\[H = J\sum\limits_{(i,j)\in E}(X_iX_j+Y_iY_j+Z_iZ_j)+B\sum\limits_{i\in V}Z_i\]
Each Hamiltonian \(H_{i,j}=X_iX_j+Y_iY_j+Z_iZ_j\) has three eigenvectors for eigenvalue \(+1\) (triplet states):
\(\ket{00}\)
\(\ket{11}\)
\(\frac{1}{\sqrt{2}}\left(\ket{10}+\ket{01}\right)\)
and one eigenvector for eigenvalue \(-3\) (singlet state):
\(\frac{1}{\sqrt{2}}\left(\ket{10}-\ket{01}\right)\)
For the problem specific VQE ansatz, we choose a Hamiltonian Variational Ansatz as proposed here. This ansatz is inspired by the adiabatic theorem of quantum mechanics: A system is prepared in the ground state of an initial Hamiltonian \(H_0\) and then slowly evolved under a time-dependet Hamiltoniam \(H(t)\). Here, we set
\[H(t) = \left(1-\frac{t}{T}\right)H_0 + \frac{t}{T}H\]
where
\[H_0 = \sum\limits_{(i,j)\in M}(X_iX_j+Y_iY_j+Z_iZ_j)\]
for a maximal matching \(M\subset E\) of the graph \(G\).
For \(J>0\) the ground state of the initial Hamiltonian \(H_0\) is given by a tensor product of singlet states corresponding to the maximal matching \(M\).
The time evolution of \(H(t)\) is approximately implemented by trotterization, i.e., alternatingly applying \(e^{-iH_0\Delta t}\) and \(e^{-iH\Delta t}\), and if necessary trotterizing \(e^{-iH\Delta t}\).
In the scope of VQE, the short evolution times \(\Delta t\) are replaced by parameters \(\theta_i\) which are then optimized. This yields the following unitary ansatz with \(p\) layers:
\[U(\theta) = \prod_{l=1}^{p}e^{-i\theta_{l,0}H_0}e^{-i\theta_{l,1}H}\]
The unitary \(e^{-i\theta H}\) trotterized by:
\[U_H(\theta) = e^{-i\theta H_B}\prod\limits_{k=1}^{q}\prod_{(i,j)\in E_k}e^{-i\theta H_{ij}}\]
where \(E_1,\dotsc,E_q\) is an edge coloring of the graph \(G\), and \(H_B\) is the magnetic field Hamiltonian. Then all unitaries \(e^{-i\theta H_{ij}}\) for \((i,j)\in E_k\) commute. For implementing such unitaries, note that each two-qubit Heisenberg interaction unitary
\[\begin{split}\text{Heis}(\theta) \equiv e^{-i\theta/4}e^{-i\theta H_{ij}} = \begin{pmatrix} e^{-i\theta/2}&0&0&0\\ 0&\cos(\theta/2)&-i\sin(\theta/2)&0\\ 0&-i\sin(\theta/2)&\cos(\theta/2)&0\\ 0&0&0&e^{-i\theta/2} \end{pmatrix}\end{split}\]
becomes diagonal in Bell basis, i.e., \(e^{-i\theta/2}\text{diag}(1,1,1,e^{i\theta})\).
This ansatz can be further generalized by introducing parameters
per edge color (one parameter for each color)
per edge (one parameter for each edge)
in the unitary \(U_H(\theta)\).
The graph defining the lattice.
The positive coupling constant.
fhe magnetic field strength.
Specifies the Hamiltonian Variational Ansatz. Available are per hamiltonian, per edge color, per edge. The default is per hamiltonian.
VQE problem instance for a specific isotropic Heisenberg model.
Examples
import networkx as nx import matplotlib.pyplot as plt # Create a graph G = nx.Graph() G.add_edges_from([(0,1),(1,2),(2,3),(0,3)]) # Draw the graph with labels pos = nx.spring_layout(G) nx.draw(G, pos, with_labels=True, node_size=700, node_color='lightblue') nx.draw_networkx_labels(G, pos) plt.show()
from qrisp import QuantumVariable from qrisp.vqe.problems.heisenberg import * vqe = heisenberg_problem(G,1,1) vqe.set_callback() energy = vqe.run(QuantumVariable(G.number_of_nodes()),depth=2,max_iter=50) print(energy) # Yields -8.0
We visualize the optimization process:
>>> vqe.visualize_energy(exact=True)
This method creates the Hamiltonian for the Heisenberg model.
The graph defining the lattice.
The positive coupling constant.
The magnetic field strength.
The quantum Hamiltonian.
This method creates a function for applying one layer of the ansatz.
The graph defining the lattice.
The positive coupling constant.
The magnetic field strength.
A list of edges corresponding to a maximal matching of G.
An edge coloring of the graph G given by a list of lists of edges.
Specifies the Hamiltonian Variational Ansatz. Available are per hamiltonian, per edge color, per edge. The default is per hamiltonian.
A function that can be applied to a QuantumVariable and a list of parameters.
Creates the function that, when applied to a QuantumVariable, initializes a tensor product of singlet sates corresponding to a given matching.
A list of edges corresponding to a maximal matching of G.
A function that can be applied to a QuantumVariable.
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