def get_hamiltonian_variables_with_edge_fields(
e_qs: EvolvingQubitSystem) -> Tuple[q.qarray, q.qarray, q.qarray]:
sx = q.pauli("X", sparse=True)
sz = q.pauli("Z", sparse=True)
qnum = (sz + q.identity(2, sparse=True)) / 2
dims = [2] * e_qs.N
# noinspection PyTypeChecker
time_independent_terms: q.qarray = 0
# noinspection PyTypeChecker
Omega_coeff_terms: q.qarray = 0
# noinspection PyTypeChecker
Delta_coeff_terms: q.qarray = 0
for i in range(e_qs.N):
Omega_coeff_terms += q.ikron(sx, dims=dims, inds=i, sparse=True) / 2
n_i = q.ikron(qnum, dims=dims, inds=i, sparse=True)
Delta_coeff_terms -= n_i
if e_qs.N <= 8:
if i == 0 or i == e_qs.N - 1:
time_independent_terms += n_i * 4.5e6 * 2 * np.pi
elif e_qs.N > 8:
if i == 0 or i == e_qs.N - 1:
time_independent_terms += n_i * 6e6 * 2 * np.pi
elif i == 3 or i == e_qs.N - 4:
time_independent_terms += n_i * 1.5e6 * 2 * np.pi
for j in range(i):
n_j = q.ikron(qnum, dims=dims, inds=j, sparse=True)
time_independent_terms += e_qs.V / e_qs.geometry.get_distance(
i, j)**6 * n_i * n_j
return (time_independent_terms, Omega_coeff_terms, Delta_coeff_terms)
def _get_hamiltonian_variables(
self) -> Tuple[q.qarray, q.qarray, q.qarray]:
sx = q.pauli("X", sparse=True)
sz = q.pauli("Z", sparse=True)
qnum = (sz + q.identity(2, sparse=True)) / 2
dims = [2] * self.N
# noinspection PyTypeChecker
time_independent_terms: q.qarray = 0
# noinspection PyTypeChecker
Omega_coeff_terms: q.qarray = 0
# noinspection PyTypeChecker
Delta_coeff_terms: q.qarray = 0
for i in range(self.N):
Omega_coeff_terms += q.ikron(sx, dims=dims, inds=i,
sparse=True) / 2
n_i = q.ikron(qnum, dims=dims, inds=i, sparse=True)
Delta_coeff_terms -= n_i
for j in range(i):
n_j = q.ikron(qnum, dims=dims, inds=j, sparse=True)
time_independent_terms += self.V / self.geometry.get_distance(
i, j)**6 * n_i * n_j
return (time_independent_terms, Omega_coeff_terms, Delta_coeff_terms)
def test_ndarrays(self):
a = qu.rand_matrix(2)
i = qu.eye(2)
b = qu.ikron([a], np.array([2, 2, 2]), [0, 2])
assert_allclose(b, a & i & a)
b = qu.ikron([a], [2, 2, 2], np.array([0, 2]))
assert_allclose(b, a & i & a)
def H_hop_boson(self, i, j):
'''Hopping term *without* anti-symmetrization
'''
c, cdag = annihil_op(), creation_op()
ccd = qu.ikron(ops=[c, cdag], dims=self._dims, inds=[i, j])
cdc = qu.ikron(ops=[c, cdag], dims=self._dims, inds=[j, i])
return ccd + cdc
def test_ikron_ownership(self, sparse, ri, rf):
dims = [7, 2, 4, 3]
X = qu.rand_matrix(2, sparse=sparse)
Y = qu.rand_matrix(3, sparse=sparse)
X1 = qu.ikron((X, Y), dims, (1, 3))[ri:rf, :]
X2 = qu.ikron((X, Y), dims, (1, 3), ownership=(ri, rf))
assert_allclose(X1.A, X2.A)
def test_sparse(self):
i = qu.eye(2, sparse=True)
a = qu.qu(qu.rand_matrix(2), sparse=True)
b = qu.ikron(a, [2, 2, 2], 1) # infer sparse
assert (qu.issparse(b))
assert_allclose(b.A, (i & a & i).A)
a = qu.rand_matrix(2)
b = qu.ikron(a, [2, 2, 2], 1, sparse=True) # explicit sparse
assert (qu.issparse(b))
assert_allclose(b.A, (i & a & i).A)
def test_overlap(self):
a = [qu.rand_matrix(4) for i in range(2)]
dims1 = [2, 2, 2, 2, 2, 2]
dims2 = [2, 4, 4, 2]
b = qu.ikron(a, dims1, [1, 2, 3, 4])
c = qu.ikron(a, dims2, [1, 2])
assert_allclose(c, b)
dims2 = [4, 2, 2, 4]
b = qu.ikron(a, dims1, [0, 1, 4, 5])
c = qu.ikron(a, dims2, [0, 3])
assert_allclose(c, b)
def test_basic(self):
a = qu.rand_matrix(2)
i = qu.eye(2)
dims = [2, 2, 2]
b = qu.ikron([a], dims, [0])
assert_allclose(b, a & i & i)
b = qu.ikron([a], dims, [1])
assert_allclose(b, i & a & i)
b = qu.ikron([a], dims, [2])
assert_allclose(b, i & i & a)
b = qu.ikron([a], dims, [0, 2])
assert_allclose(b, a & i & a)
b = qu.ikron([a], dims, [0, 1, 2])
assert_allclose(b, a & a & a)
def H_hop(self, i, j, f, dir, spin):
'''
Returns "hopping" term acting on sites i,j,f:
(XiXjOf + YiYjOf)/2
on vertex qdits i,j, face qdit f, spin sector `spin`,
where Of is {Xf, Yf, I} depending on edge.
dir: {'vertical', 'horizontal'}
'''
spin = {
0: 'u',
1: 'd',
'up': 'u',
'down': 'd',
'u': 'u',
'd': 'd'
}[spin]
X, Y, I = (qu.pauli(mu) for mu in ['x', 'y', 'i'])
#`spin` says which spin sector to act on
X_sigma = {'u': X & I, 'd': I & X}[spin]
Y_sigma = {'u': Y & I, 'd': I & Y}[spin]
Of = {'vertical': X_sigma, 'horizontal': Y_sigma}[dir]
#no associated face qbit
if f == None:
XXO = qu.ikron(ops=[X_sigma, X_sigma],
dims=self._sim_dims,
inds=[i, j])
YYO = qu.ikron(ops=[Y_sigma, Y_sigma],
dims=self._sim_dims,
inds=[i, j])
#otherwise, let Of act on index f
else:
XXO = qu.ikron(ops=[X_sigma, X_sigma, Of],
dims=self._sim_dims,
inds=[i, j, f])
YYO = qu.ikron(ops=[Y_sigma, Y_sigma, Of],
dims=self._sim_dims,
inds=[i, j, f])
return 0.5 * (XXO + YYO)
def test_2d_simple(self):
a = (qu.rand_matrix(2), qu.rand_matrix(2))
dims = ((2, 3), (3, 2))
inds = ((0, 0), (1, 1))
b = qu.ikron(a, dims, inds)
assert b.shape == (36, 36)
assert_allclose(b, a[0] & qu.eye(9) & a[1])
def stabilizer_at_face(self, fi, fj):
stab_data = self.loop_stabilizer_data(fi, fj)
oplist = [qu.pauli(Q) for Q in stab_data['opstring']]
return qu.ikron(ops=oplist,
dims=self._sim_dims,
inds=stab_data['inds'])
def test_mid_multi_reverse(self):
a = [qu.rand_matrix(2) for i in range(3)]
i = qu.eye(2)
dims = [2, 2, 2, 2, 2, 2]
inds = [5, 4, 1]
b = qu.ikron(a, dims, inds)
assert_allclose(b, i & a[2] & i & i & a[1] & a[0])
def test_sparse_format_outputs_with_dense(self, od1, stype, pos,
coo_build):
x = qu.ikron(od1, [3, 3, 3], pos, sparse=True,
stype=stype, coo_build=coo_build)
try:
default = "bsr" if (2 in pos and not coo_build) else "csr"
except TypeError:
default = "bsr" if (pos == 2 and not coo_build) else "csr"
assert x.format == default if stype is None else stype
def test_evo_at_times(self):
ham = qu.ham_heis(2, cyclic=False)
p0 = qu.up() & qu.down()
sim = qu.Evolution(p0, ham, method='solve')
ts = np.linspace(0, 10)
for t, pt in zip(ts, sim.at_times(ts)):
x = cos(t)
y = qu.expec(pt, qu.ikron(qu.pauli('z'), [2, 2], 0))
assert_allclose(x, y, atol=1e-15)
def test_graph_state_1d(self):
n = 5
p = graph_state_1d(n, cyclic=True)
for j in range(n):
k = ikron(
[pauli('x'), pauli('z'), pauli('z')],
dims=[2] * n,
inds=(j, (j - 1) % n, (j + 1) % n))
o = p.H @ k @ p
np.testing.assert_allclose(o, 1)
def state_occs(self, state):
Nk = [
qu.ikron(number_op(), self._dims, [site])
for site in self._V_ind.flatten()
]
n_ij = np.real(
np.array([qu.expec(Nk[k], state)
for k in range(self._N)])).reshape(self._shape)
return n_ij
def test_insert_operator(self):
p = MPS_rand_state(3, 7, tags='KET')
q = p.H.retag({'KET': 'BRA'})
qp = q & p
sz = qu.spin_operator('z').real
qp.insert_operator(sz, ('KET', 'I1'), ('BRA', 'I1'),
tags='SZ', inplace=True)
assert 'SZ' in qp.tags
assert len(qp.tensors) == 7
x1 = qp ^ all
x2 = qu.expec(p.to_dense(), qu.ikron(sz, [2, 2, 2], inds=1))
assert x1 == pytest.approx(x2)
def H_nn_int(self, i, j):
'''
Nearest neighbor repulsive
interaction, for vertex qbits
i, j.
Return (n_i)(n_j)
'''
return qu.ikron(ops=[self.number_op(),
self.number_op()],
dims=self._sim_dims,
inds=[i, j])
def H_onsite(self, i):
'''
Spin-spin repulsion at site i (should only
be called for vertex sites, not face sites)
(n^up_i)(n^down_i) --> (1-Vi^up)(1-Vi^down)/4
Projects onto [down (x) down] for qbits at index i
'''
return qu.ikron(ops=[self.q_number_op() & self.q_number_op()],
dims=self._sim_dims,
inds=[i])
def make_stabilizer(self):
'''
DEPREC -- DON'T USE
TODO:
* add a projector onto codespace (see method 2)
* generalize indices, rotation, reshape, etc
* always need to round? check always real
'''
X, Z = (qu.pauli(mu) for mu in ['x', 'z'])
## TODO: make general
# ops = [Z,Z,Z,Z,X]
# inds = [1,2,4,5,6]
# stabilizer = qu.ikron(ops=ops, dims=self._sim_dims, inds=inds)
_, Ux = qu.eigh(qu.pauli('x'))
stab_data = self.loop_stabilizer_data(0, 1)
oplist = [qu.pauli(Q) for Q in stab_data['opstring']]
stabilizer = qu.ikron(ops=oplist,
dims=self._sim_dims,
inds=stab_data['inds'])
#TODO: change to general rather than inds=6, Ux
U = qu.ikron(Ux.copy(), dims=self._sim_dims, inds=[6])
#TODO: can this be done without rounding()/real part?
Stilde = (U.H @ stabilizer @ U).real.round()
assert is_diagonal(Stilde)
U_plus = U[:, np.where(np.diag(Stilde) == 1.0)[0]]
HamCode = U_plus.H @ self.ham_sim() @ U_plus
self._stabilizer = stabilizer
self._Uplus = U_plus #+1 eigenstates written in full qubit basis
self._HamCode = HamCode # in +1 stabilizer eigenbasis
def test_variable_bond_ham(self):
import quimb as qu
HB = SpinHam(1 / 2)
HB[0, 1] += 0.6, 'Z', 'Z'
HB[1, 2] += 0.7, 'Z', 'Z'
HB[1, 2] += 0.8, 'X', 'X'
HB[2, 3] += 0.9, 'Y', 'Y'
H_mpo = HB.build_mpo(4)
H_sps = HB.build_sparse(4)
assert H_mpo.bond_sizes() == [3, 4, 3]
Sx, Sy, Sz = map(qu.spin_operator, 'xyz')
H_explicit = (qu.ikron(0.6 * Sz & Sz, [2, 2, 2, 2], [0, 1]) +
qu.ikron(0.7 * Sz & Sz, [2, 2, 2, 2], [1, 2]) +
qu.ikron(0.8 * Sx & Sx, [2, 2, 2, 2], [1, 2]) +
qu.ikron(0.9 * Sy & Sy, [2, 2, 2, 2], [2, 3]))
assert_allclose(H_explicit, H_mpo.to_dense())
assert_allclose(H_explicit, H_sps.A)
def jw_creation_op(self, k):
'''
Creation Jordan-Wigner qubit operator.
(Z_0)x(Z_1)x ...x(Z_k-1)x |1><0|_k
'''
Z = qu.pauli('z')
s_plus = qu.qu([[0, 0], [1, 0]]) #|1><0|
ind_list = [i for i in range(k + 1)]
op_list = [Z for i in range(k)] + [s_plus]
return qu.ikron(ops=op_list, dims=self._dims, inds=ind_list)
def jw_annihil_op(self, k):
'''
Annihilation Jordan-Wigner qubit operator.
(Z_0)x(Z_1)x ...x(Z_k-1)x |0><1|_k
'''
Z = qu.pauli('z')
s_minus = qu.qu([[0, 1], [0, 0]]) #|0><1|
ind_list = [i for i in range(k + 1)]
op_list = [Z for i in range(k)] + [s_minus]
return qu.ikron(ops=op_list, dims=self._dims, inds=ind_list)
def site_number_ops(self, sites, fermi=False):
'''
TODO: remove fermi?
Returns: list of qarrays, local number
operators acting on specified sites.
param `sites`: list of ints, indices of sites
param `fermi`: bool, indicates whether we are acting
in the large qubit space basis (False) or in the restricted
stabilizer eigenbasis (True)
'''
ds = {False: self._sim_dims, True: self._fermi_dims}[fermi]
return [qu.ikron(self.number_op(), ds, [site])
for site in sites] #.reshape(self._lat_shape)
def projected_ham_3(self):
'''
Should be equivalent to self.ham_code()
'''
#X because stabilizer acts with X on 7th qubit
_, Ux = qu.eigh(qu.pauli('x'))
U = qu.ikron(Ux.copy(), dims=self._sim_dims, inds=[6])
Stilde = (U.H @ self.stabilizer() @ U).real.round()
#Stilde should be diagonal!
U_plus = U[:, np.where(np.diag(Stilde) == 1.0)[0]]
HamProj = U_plus.H @ self.ham_sim() @ U_plus
return HamProj #in +1 stabilizer eigenbasis
def test_multisubsystem(self):
qu.seed_rand(42)
dims = [2, 2, 2]
IIX = qu.ikron(qu.rand_herm(2), dims, 2)
dcmp = qu.pauli_decomp(IIX, mode='c')
for p, x in dcmp.items():
if x == 0j:
assert (p[0] != 'I') or (p[1] != 'I')
else:
assert p[0] == p[1] == 'I'
K = qu.rand_iso(3 * 4, 4).reshape(3, 4, 4)
KIIXK = qu.kraus_op(IIX, K, dims=dims, where=[0, 2])
dcmp = qu.pauli_decomp(KIIXK, mode='c')
for p, x in dcmp.items():
if x == 0j:
assert p[1] != 'I'
else:
assert p[1] == 'I'
def state_local_occs(self, qstate=None, k=0, faces=False):
'''
TODO: fix face shapes?
Expectation values <k|local fermi occupation|k>
in the kth excited eigenstate, at vertex
sites (and faces if specified).
param qstate: vector in full qubit basis
param k: if `state` isn't specified, will compute for
the kth excited energy eigenstate (defaults to ground state)
param faces: if True, return occupations at face qubits as well
Returns: local occupations `nocc_v`, (`nocc_f`)
nocc_v (ndarray, shape (Lx,Ly))
nocc_f (ndarray, shape (Lx-1,Ly-1))
'''
if qstate is None:
qstate = self._eigstates[:, k]
#local number ops acting on each site j
# Nj = self.site_number_ops(sites=self.all_sites() , fermi=False)
Nj = [
qu.ikron(self.number_op(), self._sim_dims, [site])
for site in self.all_sites()
]
#expectation <N> for each vertex
nocc_v = np.array([
qu.expec(Nj[v], qstate) for v in self.vertex_sites()
]).reshape(self._lat_shape)
if not faces:
return nocc_v
#expectation <Nj> for each face
nocc_f = np.array([qu.expec(Nj[f], qstate) for f in self.face_sites()])
return nocc_v, nocc_f
def test_1d_vector_methods(self):
X = qu.spin_operator('X', sparse=True)
mera = qt.MERA.rand(16)
meraX = mera.gate(X.A, 7, inplace=False)
assert mera is not meraX
x1 = mera.H @ meraX
md = mera.to_dense()
mdX = qu.ikron(X, [2] * 16, 7) @ md
x2 = md.H @ mdX
# check against dense
assert x1 == pytest.approx(x2)
# check 'outside lightcone' unaffected
assert mera.select(3).H @ meraX.select(3) == pytest.approx(1.0)
# check only need 'lightcone' to compute local
assert mera.select(7).H @ meraX.select(7) == pytest.approx(x2)
def test_gate2split(self):
psi = MPS_rand_state(10, 3)
psi2 = psi.copy()
G = qu.eye(2) & qu.eye(2)
psi.gate2split(G, (2, 3), cutoff=0)
assert psi.bond_size(2, 3) == 6
assert psi.H @ psi2 == pytest.approx(1.0)
# check a unitary application
G = qu.rand_uni(2**2)
psi.gate2split(G, (7, 8))
psi.compress()
assert psi.bond_size(2, 3) == 3
assert psi.bond_size(7, 8) > 3
assert psi.H @ psi == pytest.approx(1.0)
assert abs(psi2.H @ psi) < 1.0
# check matches dense application of gate
psid = psi2.to_dense()
Gd = qu.ikron(G, [2] * 10, (7, 8))
assert psi.to_dense().H @ (Gd @ psid) == pytest.approx(1.0)
def jw_creation_op(self, k, spin):
'''
Jordan-Wigner transformed creation operator,
for site k and `spin` sector
(Z_0)x(Z_1)x ...x(Z_k-1)x |1><0|_k
'''
spin = {0: 0, 1: 1, 'up': 0, 'down': 1, 'u': 0, 'd': 1}[spin]
Z, I = qu.pauli('z'), qu.eye(2)
s_plus = qu.qu([[0, 0], [1, 0]])
N = self._Nverts
Z_sig = {0: Z & I, 1: I & Z}[spin]
s_plus_sig = {0: s_plus & I, 1: I & s_plus}[spin]
op_list = [Z_sig for i in range(k)] + [s_plus_sig]
ind_list = [i for i in range(k + 1)]
return qu.ikron(ops=op_list, dims=self._dims, inds=ind_list)