Getting started

Simple example

Here we demonstrate the standard workflow using a simple example solving a Heisenberg problem on a 5x5 grid.

To begin, we import the required libraries, and build our Hamiltonian as a QubitOperator:

import scipy.sparse.linalg as spla
from qiskit.transpiler import CouplingMap
import fulqrum as fq

# Build 16-qubit coupling map
cmap = CouplingMap.from_grid(4, 4)
num_qubits = cmap.size()

# Generate Hamiltonian
H = fq.QubitOperator(num_qubits, [])
touched_edges = set({})
coeffs = [-1/2, -1/2, -1]
for edge in cmap.get_edges():
    if edge[::-1] not in touched_edges:
        touched_edges.add(edge)
        H += fq.QubitOperator(num_qubits, [("XX", edge, coeffs[0]),
                                           ("YY", edge, coeffs[1]),
                                           ("ZZ", edge, coeffs[2])])

The subspace in which our Hamiltonian will be defined is derived from the counts sampled from a quantum computer and stored as a Subspace object (note that we do not use the counts data itself):

counts = []
for kk in range(2**num_qubits):
    counts.append(bin(kk)[2:].zfill(num_qubits))

S = fq.Subspace([counts])

Right now we have a Hamiltonian and a subspace, but they know nothing about each other. We can combine them into a SubspaceHamiltonian object, which is where all the magic happens, and is what we pass on to the eigensolver:

Hsub = fq.SubspaceHamiltonian(H, S)
evals, evecs = spla.eigsh(Hsub, k=1, which='SA')

In passing the SubspaceHamiltonian object directly we have solved the problem using a matrix-free method, but one can build a collection of different, more performant, matrix-representations from this object. The found eigenenergy is:

print("Eigenvalue:", evals[0])

Eigenvalue: -24.000000000000007

there is of course an associated eigenvector, but because of the way Fulqrum solves the problem, it is not immediately usable. Instead, we can use SubspaceHamiltonian.interpret_vector() to cast the solution as a dictionary the statevector amplitudes as values:

Hsub.interpret_vector(evecs)

{'0000000000000000': -0.9896849063220712,
 '1111111111111111': 0.14326125155907726}