def fidelity(nn_state, target_psi, space, **kwargs):
r"""Calculates the square of the overlap (fidelity) between the reconstructed
wavefunction and the true wavefunction (both in the computational basis).
.. math:: F = \vert \langle \psi_{RBM} \vert \psi_{target} \rangle \vert ^2
:param nn_state: The neural network state (i.e. complex wavefunction or
positive wavefunction).
:type nn_state: qucumber.nn_states.WaveFunctionBase
:param target_psi: The true wavefunction of the system.
:type target_psi: torch.Tensor
:param space: The basis elements of the Hilbert space of the system :math:`\mathcal{H}`.
The ordering of the basis elements must match with the ordering of the
coefficients given in `target_psi`.
:type space: torch.Tensor
:param \**kwargs: Extra keyword arguments that may be passed. Will be ignored.
:returns: The fidelity.
:rtype: float
"""
Z = nn_state.compute_normalization(space)
target_psi = target_psi.to(nn_state.device)
psi = nn_state.psi(space) / Z.sqrt()
F = cplx.inner_prod(target_psi, psi)
return cplx.absolute_value(F).pow_(2).item()
def fidelity(nn_state, target, space=None, **kwargs):
r"""Calculates the square of the overlap (fidelity) between the reconstructed
state and the true state (both in the computational basis).
.. math::
F = \vert \langle \psi_{RBM} \vert \psi_{target} \rangle \vert ^2
= \left( \tr \lbrack \sqrt{ \sqrt{\rho_{RBM}} \rho_{target} \sqrt{\rho_{RBM}} } \rbrack \right) ^ 2
:param nn_state: The neural network state.
:type nn_state: qucumber.nn_states.NeuralStateBase
:param target: The true state of the system.
:type target: torch.Tensor
:param space: The basis elements of the Hilbert space of the system :math:`\mathcal{H}`.
The ordering of the basis elements must match with the ordering of the
coefficients given in `target`. If `None`, will generate them using
the provided `nn_state`.
:type space: torch.Tensor
:param \**kwargs: Extra keyword arguments that may be passed. Will be ignored.
:returns: The fidelity.
:rtype: float
"""
space = space if space is not None else nn_state.generate_hilbert_space()
Z = nn_state.normalization(space)
target = target.to(nn_state.device)
if isinstance(nn_state, WaveFunctionBase):
assert target.dim() == 2, "target must be a complex vector!"
psi = nn_state.psi(space) / Z.sqrt()
F = cplx.inner_prod(target, psi)
return cplx.absolute_value(F).pow_(2).item()
else:
assert target.dim() == 3, "target must be a complex matrix!"
rho = nn_state.rho(space, space) / Z
rho_rbm_ = cplx.numpy(rho)
target_ = cplx.numpy(target)
sqrt_rho_rbm = sqrtm(rho_rbm_)
prod = np.matmul(sqrt_rho_rbm, np.matmul(target_, sqrt_rho_rbm))
# Instead of sqrt'ing then taking the trace, we compute the eigenvals,
# sqrt those, and then sum them up. This is a bit more efficient.
eigvals = np.linalg.eigvals(
prod).real # imaginary parts should be zero
eigvals = np.abs(eigvals)
trace = np.sum(np.sqrt(eigvals))
return trace**2
def fidelity(nn_state, target_psi, space, **kwargs):
r"""Calculates the square of the overlap (fidelity) between the reconstructed
wavefunction and the true wavefunction (both in the computational basis).
:param nn_state: The neural network state (i.e. complex wavefunction or
positive wavefunction).
:type nn_state: WaveFunction
:param target_psi: The true wavefunction of the system.
:type target_psi: torch.Tensor
:param space: The hilbert space of the system.
:type space: torch.Tensor
:param \**kwargs: Extra keyword arguments that may be passed. Will be ignored.
:returns: The fidelity.
:rtype: float
"""
Z = nn_state.compute_normalization(space)
target_psi = target_psi.to(nn_state.device)
psi = nn_state.psi(space) / Z.sqrt()
F = cplx.inner_prod(target_psi, psi)
return cplx.absolute_value(F).pow_(2).item()
def test_vector_inner_prod(self):
vector = torch.tensor([[1, 2], [3, 4]], dtype=torch.double)
expect = torch.tensor([30, 0], dtype=torch.double)
self.assertTensorsEqual(
cplx.inner_prod(vector, vector), expect, msg="Vector inner product failed!"
)
def test_scalar_inner_prod(self):
scalar = torch.tensor([1, 2], dtype=torch.double)
expect = torch.tensor([5, 0], dtype=torch.double)
self.assertTensorsEqual(
cplx.inner_prod(scalar, scalar), expect, msg="Scalar inner product failed!"
)