def var_gate_exact(top_state, site, bottom_state):
'''
Goal:
to find argmax_{gate} <top_state | gate | down_state>
where gate is actting on (site, site+1)
Input:
top_state: (did not have conjugation yet!!!)
site: gate is applying on (site, site+1)
bottom_state
Return:
new_gate
'''
total_dim = top_state.size
L = int(np.log2(total_dim))
top_theta = np.reshape(top_state, [(2**site), 4, 2**(L - (site + 2))])
bottom_theta = np.reshape(bottom_state,
[(2**site), 4, 2**(L - (site + 2))])
M = np.tensordot(top_theta.conj(), bottom_theta, axes=([0, 2], [
0, 2
])) # [ ..., upper_p, ...], [..., lower_p, ...] --> upper_p, lower_p
## If the convention is lower_p, upper_p
## uncomment the following line.
# M = M.T # the convention is lower_p, upper_p
### For detailed explanation of the formula, see function var_gate
U, _, Vd = misc.svd(M, full_matrices=False)
new_gate = np.dot(U, Vd).conj()
# [TODO:remove] new_gate = new_gate.reshape([2, 2, 2, 2])
return new_gate
def get_renyi_n_entanglement(A_list, n):
'''
Goal:
Compute the renyi-n entanglement at each cut.
Input:
mps in left canonical form
Output:
list of bipartite entanglement [(0,1...), (01,2...), (012,...)]
'''
L = len(A_list)
copy_A_list = [A.copy() for A in A_list]
ent_list = [None] * (L - 1)
for i in range(L - 1, 0, -1):
d1, chi1, chi2 = copy_A_list[i].shape
X, Y, Z = misc.svd(np.reshape(np.transpose(copy_A_list[i], [1, 0, 2]),
[chi1, d1 * chi2]),
full_matrices=0)
chi1 = np.sum(Y > 1e-14)
arg_sorted_idx = (np.argsort(Y)[::-1])[:chi1]
Y = Y[arg_sorted_idx]
X = X[:, arg_sorted_idx]
Z = Z[arg_sorted_idx, :]
copy_A_list[i] = np.transpose(Z.reshape([chi1, d1, chi2]), [1, 0, 2])
R = np.dot(X, np.diag(Y))
new_A = np.tensordot(copy_A_list[i - 1], R,
axes=([2], [0])) #[p, 1l, (1r)] [(2l), 2r]
copy_A_list[i - 1] = new_A
bi_ent = np.log(np.sum(Y**(2 * n))) / (1 - n)
ent_list[i - 1] = bi_ent
return ent_list
def operator_2_MPO(op, L, chimax):
'''
Input:
op: the operator in matrix of size (2**L, 2**L)
L: system size
chimax: the maximum bond dimension
Return:
MPO in [p, l, r, q] form
'''
op_aR = np.reshape(op, (1, 2**L, 2**L))
Ms = []
for n in range(1, L + 1):
chi_n, dim_R1, dim_R2 = op_aR.shape
assert dim_R1 == 2**(L - (n - 1))
op_LR = np.reshape(op_aR, [chi_n * 2, dim_R1 // 2, 2, dim_R2 // 2])
op_LR = np.transpose(op_LR, [0, 2, 1, 3]).reshape(
[chi_n * 4, (dim_R1 // 2) * (dim_R2 // 2)])
M_n, lambda_n, op_tilde = misc.svd(op_LR, full_matrices=False)
if len(lambda_n) > chimax:
keep = np.argsort(lambda_n)[::-1][:chimax]
M_n = M_n[:, keep]
lambda_n = lambda_n[keep]
op_tilde = op_tilde[keep, :]
chi_np1 = len(lambda_n)
M_n = np.reshape(M_n, (chi_n, 2, 2, chi_np1))
Ms.append(M_n.transpose([1, 0, 3, 2]))
op_aR = lambda_n[:, np.newaxis] * op_tilde[:, :]
op_aR = op_aR.reshape([chi_np1, (dim_R1 // 2), (dim_R2 // 2)])
assert op_aR.shape == (1, 1, 1)
Ms[-1] = Ms[-1] * op_aR[0, 0, 0]
return Ms
def state_2_MPS(psi, L, chimax):
'''
Input:
psi: the state
L: the system size
chimax: the maximum bond dimension
'''
psi_aR = np.reshape(psi, (1, 2**L))
Ms = []
for n in range(1, L + 1):
chi_n, dim_R = psi_aR.shape
assert dim_R == 2**(L - (n - 1))
psi_LR = np.reshape(psi_aR, (chi_n * 2, dim_R // 2))
M_n, lambda_n, psi_tilde = misc.svd(psi_LR, full_matrices=False)
if len(lambda_n) > chimax:
keep = np.argsort(lambda_n)[::-1][:chimax]
M_n = M_n[:, keep]
lambda_n = lambda_n[keep]
psi_tilde = psi_tilde[keep, :]
chi_np1 = len(lambda_n)
M_n = np.reshape(M_n, (chi_n, 2, chi_np1))
Ms.append(M_n)
psi_aR = lambda_n[:, np.newaxis] * psi_tilde[:, :]
assert psi_aR.shape == (1, 1)
return lpr_2_plr(Ms)
def right_canonicalize(A_list, no_trunc=False, chi=None, normalized=True):
'''
Bring mps in right canonical form, assuming the input mps is in
left canonical form already.
modification in place
'''
L = len(A_list)
tot_trunc_err = 0.
for i in range(L - 1, 0, -1):
d1, chi1, chi2 = A_list[i].shape
X, Y, Z = misc.svd(np.reshape(np.transpose(A_list[i], [1, 0, 2]),
[chi1, d1 * chi2]),
full_matrices=0)
if no_trunc:
chi1 = np.size(Y)
else:
chi1 = np.sum(Y > 1e-14)
if chi is not None:
chi1 = np.amin([chi1, chi])
trunc_idx = (np.argsort(Y)[::-1])[chi1:]
trunc_error = np.sum(Y[trunc_idx]**2) / np.sum(Y**2)
tot_trunc_err = tot_trunc_err + trunc_error
arg_sorted_idx = (np.argsort(Y)[::-1])[:chi1]
Y = Y[arg_sorted_idx]
if normalized:
Y = Y / np.linalg.norm(Y)
X = X[:, arg_sorted_idx]
Z = Z[arg_sorted_idx, :]
A_list[i] = np.transpose(Z.reshape([chi1, d1, chi2]), [1, 0, 2])
R = np.dot(X, np.diag(Y))
new_A = np.tensordot(A_list[i - 1], R,
axes=([2], [0])) #[p, 1l, (1r)] [(2l), 2r]
A_list[i - 1] = new_A
if normalized:
A_list[0] = A_list[0] / np.linalg.norm(A_list[0])
return A_list, tot_trunc_err