Tfim simulation using 2 CNOT rather than 3 CNOT

[shangtai]

This pull request is to address issue 70 where we consider the special TFIM model:

$$H^{(a)}=X_{a}X_{a+1}+B_{a}Z_{a}$$

where we can express the time evolution as

$$e^{-itH^{(a)}}=CNOT(a,a+1)e^{-it(X_a+B_aZ_a)}CNOT(a,a+1).$$

[marekgluza]

Suggested change

	def circuit(self, a, t_duration, B_a, steps=None, order=None):
	def circuit(self, t_duration, steps=None, order=None):

This class must implement eo.circuit(0.1) without requiring other parameters.

What is a?

[shangtai]

I use the notation in the paper where a is an index and B_a is another parameter.

$$H^{(a)}=X_{a}X_{a+1}+B_{a}Z_{a}$$

[marekgluza]

This is commuting so it's equivalent to
circuit += self._time_evolution_step(a, t_duration, B_a)

Once you have for a in range(self.nqubits): (TFIM evolution oracle should be implemented for nqubits) then you need to do the CNOTs before every dt step here

[marekgluza]

Please provide test code demonstrating the functioning of this CNOT decomposition:

• as the number of steps increase the comparison with H_TFIM_symbolic.exp(t) should be improving
• in XXZ we used 2nd order, here we should do that too

[marekgluza]

Suggested change

	dt = t_duration / steps # Divide the time duration for Trotterization if needed
	dt = t_duration / steps # Divide the time duration for Trotterization if needed
	for a in range(nqubits:
	circuit.add(gates.CNOT(a, a + 1))
	circuit += self._time_evolution_step(a, dt, B_a)
	circuit.add(gates.CNOT(a, a + 1))

and then loop this over _ in range(steps)

[shangtai]

n_qubits = 3
h_coeff = 1
hamiltonian = SymbolicHamiltonian(nqubits=n_qubits)

oracle = TFIM_EvolutionOracle(h=hamiltonian, evolution_oracle_type="trotter", steps=1, B_a=0, order=2)

circuit = oracle.circuit(t_duration=1.0)

unitary = circuit.unitary()


from qibo import hamiltonians
from numpy.linalg import norm

def our_TFIM(nqubits, h: float = 0.0, dense: bool = True, backend=None):
        def multikron(matrix_list):
        """Calculates Kronecker product of a list of matrices."""
        return reduce(np.kron, matrix_list)

    from qibo.backends import matrices

    matrix = (
        - multikron([matrices.X, matrices.X]) - h * multikron([matrices.Z, matrices.I])
    )
    terms = [hamiltonians.terms.HamiltonianTerm(matrix, i, i + 1) for i in range(nqubits - 1)]
    terms.append(hamiltonians.terms.HamiltonianTerm(matrix, nqubits - 1, 0))
    ham = SymbolicHamiltonian(backend=backend)
    ham.terms = terms
    return ham

ham = our_TFIM(nqubits=n_qubits, h=h_coeff, dense=False)
truth = ham.exp(1)
verification_norm = []
for step in range(1, 21):
    oracle = TFIM_EvolutionOracle(h=hamiltonian, evolution_oracle_type="trotter", steps=step, B_a=h_coeff, order=2)
    circuit = oracle.circuit(t_duration=1.0)
    unitary = circuit.unitary()
    #print(norm(truth-unitary))
    verification_norm.append(norm(truth-unitary))

import matplotlib.pyplot as plt
x = np.array([i for i in range(1, 21)])
plt.plot(x, verification_norm, 'o')
plt.title("verification of TFIM 2 CNOT implementation")
plt.xlabel("steps")
plt.ylabel("norm of difference")

plt.show()

produces the following graph: