def real_hermitian_ansatz(p, n=2):
U = np.eye(4)
for w in p:
local1 = np.kron(Ry(w), I)
entangling = swap() @ CRy(np.pi - w) @ swap() @ CRy(-w)
U = entangling @ local1 @ U
return U
def ϵ(x):
u_state, v_state, e_state = state_function(
x, 'u_purity'), state_function(x, 'v_purity'), state_function(
x, 'energy')
v_purity = np.real(v_state.conj().T @ np.kron(
np.eye(2), np.kron(swap(), np.eye(2))) @ v_state)
u_purity = np.real(u_state.conj().T @ np.kron(
np.eye(4), np.kron(swap(), np.eye(4))) @ u_state)
energy = np.real(e_state.conj().T @ H @ e_state)
return energy + u_purity + v_purity
def ϵ_op(x):
uv_state, u_state, v_state, e_state = state_function(
x, 'uv_purity'), state_function(x, 'u_purity'), state_function(
x, 'v_purity'), state_function(x, 'energy')
v_purity = np.real(v_state.conj().T @ np.kron(
np.eye(2), np.kron(swap(), np.eye(2))) @ v_state)
u_purity = np.real(u_state.conj().T @ np.kron(
np.eye(4), np.kron(swap(), np.eye(4))) @ u_state)
uv_purity = np.real(uv_state.conj().T @ np.kron(
np.kron(np.eye(2), swap()), np.eye(4)) @ uv_state)
energy = np.real(e_state.conj().T @ H @ e_state)
k = 10
return energy, +k * u_purity, +k * v_purity, -2 * k * uv_purity
def ϵ(x):
uv_state, u_state, v_state, e_state = (state(x, 'uv_purity'),
state(x, 'u_purity'),
state(x, 'v_purity'),
state(x, 'energy'))
v_purity = np.real(v_state.conj().T @ np.kron(
np.eye(2), np.kron(swap(), np.eye(2))) @ v_state)
u_purity = np.real(u_state.conj().T @ np.kron(
np.eye(4), np.kron(swap(), np.eye(4))) @ u_state)
uv_purity = np.real(uv_state.conj().T @ np.kron(
np.kron(np.eye(2), swap()), np.eye(4)) @ uv_state)
energy = np.real(e_state.conj().T @ H @ e_state)
k = 1
return sum(
[energy, +k * u_purity, +k * v_purity, -2 * k * uv_purity])
def fixindices(v, ϵ=4e-2):
# don't feed the parameters straight into the
# lie algebra but add a swap and perturb slightly (ϵ)
# necessary to embed D=2 result into D=4 &c.
# swap makes i, j have same tensor product structure
# perturbation gets away from singular points
N = int(np.sqrt(len(v) + 1))
U = SU(v + ϵ, N)
n_qubits = int(np.log2(N)) + 1
S12 = reduce(np.kron, [np.eye(2)] * (n_qubits - 3) + [swap()])
U = U @ S12
return extractv(U)
def swapper():
return -np.kron(np.kron(np.eye(2), swap()), np.eye(8))