def _norbs_err(self, orbs, N, real_valued=True):
"""norbs - 'natural orbitals': eigenstates of the 1-body RDM"""
if real_valued:
psi = fermifab.FermiState(orbs,
N,
data=np.random.rand(int(binom(orbs, N))))
else:
psi = fermifab.FermiState(orbs,
N,
data=fermifab.crand(int(binom(orbs, N))))
# compute the one-body reduced density matrix
rho = fermifab.rdm(psi, 1)
# diagonalize rho and construct base change matrix operating on N-fold tensor product
_, U = fermifab.eig(rho)
UN = fermifab.tensor_op(U, N)
# apply inverse base change matrix to psi
psi = UN.H @ psi
G = fermifab.rdm(psi, 1)
# G should be a diagonal matrix
return np.linalg.norm(np.diag(np.diag(G.data)) - G.data)
def _err_calcQ(self, psi):
g1 = fermifab.rdm(psi, 1)
g2 = fermifab.rdm(psi, 2)
# calculate Q operator
Q = fermifab.calcQ(g2, g1)
# Q operator based solely on two-body RDM
Q2 = fermifab.calcQ_(g2, psi.N)
err = fermifab.norm(Q - Q2)
# TODO: implement test for QN
# generalized Q operator for p = 1 should be id - gamma^1(psi)
# TODO: implement test for QN using p = 1
return err
def _tensor_op_err(self, orbs, p, N):
err = 0
# random wavefunction "psi"
psi = fermifab.FermiState(orbs, N, data=fermifab.crand(int(binom(orbs, N))))
# compute one-body reduced density matrix
rho = fermifab.rdm(psi, 1)
U, _ = np.linalg.qr(rho.data)
U = fermifab.FermiOp(orbs, 1, 1, U)
Up = fermifab.tensor_op(U, p)
UN = fermifab.tensor_op(U, N)
err += np.linalg.norm((UN.H @ UN).data - np.eye(*UN.shape))
err += np.linalg.norm((Up.H @ fermifab.rdm(UN @ psi, p) @ Up - fermifab.rdm(psi, p)).data)
return err
def _err_calcT1(self, psi):
g1 = fermifab.rdm(psi, 1)
g2 = fermifab.rdm(psi, 2)
# calculate T1 operator
T = fermifab.calcT1(g2, g1)
# T1 operator based solely on two-body RDM
Td = fermifab.calcT1_(g2, psi.N)
err = fermifab.norm(T - Td)
# TODO: implement test for T1N
# generalized T1 operator for p = 2 should be gamma^2(psi) + Q
# TODO: implement test for T1 using p = 2
return err
def _p2N_err(self, orbs, p, N):
shape_data = int(binom(orbs, p)), int(binom(orbs, p))
h = fermifab.FermiOp(orbs, p, p, fermifab.crand(*shape_data))
# random wavefunction "psi"
psi = fermifab.FermiState(orbs,
N,
data=fermifab.crand(int(binom(orbs, N))))
return abs((psi.H @ fermifab.p2N(h, N) @ psi).data[0] -
fermifab.trace(h @ fermifab.rdm(psi, p)))