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This tutorial describes the FQE's capabilities for Hamiltonian time-evolution and expectation value estimation
Where possible, LiH will be used as an example molecule for the API.
try:
import fqe
except ImportError:
!pip install fqe --quiet
Print = True
from openfermion import FermionOperator, MolecularData
from openfermion.utils import hermitian_conjugated
import numpy
import fqe
numpy.set_printoptions(floatmode='fixed', precision=6, linewidth=80, suppress=True)
numpy.random.seed(seed=409)
curl -O https://raw.githubusercontent.com/quantumlib/OpenFermion-FQE/master/tests/unittest_data/build_lih_data.py
% Total % Received % Xferd Average Speed Time Time Time Current
Dload Upload Total Spent Left Speed
100 62086 100 62086 0 0 259k 0 --:--:-- --:--:-- --:--:-- 260k
import build_lih_data
h1e, h2e, wfn = build_lih_data.build_lih_data('energy')
lih_hamiltonian = fqe.get_restricted_hamiltonian(([h1e, h2e]))
lihwfn = fqe.Wavefunction([[4, 0, 6]])
lihwfn.set_wfn(strategy='from_data', raw_data={(4, 0): wfn})
if Print:
lihwfn.print_wfn()
Sector N = 4 : S_z = 0 a'000011'b'000011' (-0.9870890035778126+0j) a'000011'b'000101' (-0.0384854879340124+0j) a'000011'b'100001' (-0.0036477466809967+0j) a'000101'b'000011' (-0.0384854879340073+0j) a'000101'b'000101' (0.0344508540105218+0j) a'000101'b'100001' (-0.0592953695921639+0j) a'000101'b'100010' (-0.0010777697106179+0j) a'001001'b'001001' (0.0271656962509733+0j) a'001001'b'001010' (-0.0031209939957822+0j) a'010001'b'010001' (0.0271656962509733+0j) a'010001'b'010010' (-0.0031209939957822+0j) a'100001'b'000011' (-0.003647746680997+0j) a'100001'b'000101' (-0.0592953695921649+0j) a'100001'b'100001' (0.1135892009408387+0j) a'000110'b'000110' (0.0037623492033852+0j) a'001010'b'001001' (-0.0031209939957822+0j) a'001010'b'001010' (0.001794202149606+0j) a'010010'b'010001' (-0.0031209939957822+0j) a'010010'b'010010' (0.001794202149606+0j) a'100010'b'000101' (-0.001077769710618+0j) a'100010'b'100010' (0.0010515758855392+0j)
Application of one- and two-body fermionic gates
The API for time propagation can be invoked through the fqe namespace or the wavefunction object
# dummy geometry
from openfermion.chem.molecular_data import spinorb_from_spatial
from openfermion import jordan_wigner, get_sparse_operator, InteractionOperator, get_fermion_operator
h1s, h2s = spinorb_from_spatial(h1e, numpy.einsum("ijlk", -2 * h2e) * 0.5)
mol = InteractionOperator(0, h1s, h2s)
ham_fop = get_fermion_operator(mol)
ham_mat = get_sparse_operator(jordan_wigner(ham_fop)).toarray()
from scipy.linalg import expm
time = 0.01
evolved1 = lihwfn.time_evolve(time, lih_hamiltonian)
if Print:
evolved1.print_wfn()
evolved2 = fqe.time_evolve(lihwfn, time, lih_hamiltonian)
if Print:
evolved2.print_wfn()
assert numpy.isclose(fqe.vdot(evolved1, evolved2), 1)
cirq_wf = fqe.to_cirq(lihwfn)
evolve_cirq = expm(-1j * time * ham_mat) @ cirq_wf
test_evolve = fqe.from_cirq(evolve_cirq, thresh=1.0E-12)
assert numpy.isclose(fqe.vdot(test_evolve, evolved1), 1)
Sector N = 4 : S_z = 0 a'000011'b'000011' (-0.9832017407384288-0.08751592993809965j) a'000011'b'000101' (-0.038333927909412015-0.003412147794935153j) a'000011'b'100001' (-0.003633381455202132-0.00032341158146809005j) a'000101'b'000011' (-0.038333927909406935-0.0034121477949347074j) a'000101'b'000101' (0.03431518282571416+0.003054434433325243j) a'000101'b'100001' (-0.05906185803441181-0.005257164690403508j) a'000101'b'100010' (-0.001073525344578889-9.555564554363788e-05j) a'001001'b'001001' (0.02705871481472957+0.0024085276354275367j) a'001001'b'001010' (-0.003108703183821363-0.00027670930255657j) a'010001'b'010001' (0.02705871481472957+0.0024085276354275367j) a'010001'b'010010' (-0.003108703183821363-0.00027670930255657j) a'100001'b'000011' (-0.0036333814552024306-0.0003234115814681176j) a'100001'b'000101' (-0.0590618580344128-0.0052571646904036005j) a'100001'b'100001' (0.11314187441703756+0.010070889930469404j) a'000110'b'000110' (0.003747532670094563+0.0003335721898296519j) a'001010'b'001001' (-0.003108703183821363-0.0002767093025565706j) a'001010'b'001010' (0.0017871363875789756+0.00015907509845870593j) a'010010'b'010001' (-0.003108703183821363-0.0002767093025565706j) a'010010'b'010010' (0.0017871363875789756+0.00015907509845870593j) a'100010'b'000101' (-0.0010735253445789887-9.555564554364526e-05j) a'100010'b'100010' (0.0010474346607506372+9.323348063182962e-05j) Sector N = 4 : S_z = 0 a'000011'b'000011' (-0.9832017407384288-0.08751592993809965j) a'000011'b'000101' (-0.038333927909412015-0.003412147794935153j) a'000011'b'100001' (-0.003633381455202132-0.00032341158146809005j) a'000101'b'000011' (-0.038333927909406935-0.0034121477949347074j) a'000101'b'000101' (0.03431518282571416+0.003054434433325243j) a'000101'b'100001' (-0.05906185803441181-0.005257164690403508j) a'000101'b'100010' (-0.001073525344578889-9.555564554363788e-05j) a'001001'b'001001' (0.02705871481472957+0.0024085276354275367j) a'001001'b'001010' (-0.003108703183821363-0.00027670930255657j) a'010001'b'010001' (0.02705871481472957+0.0024085276354275367j) a'010001'b'010010' (-0.003108703183821363-0.00027670930255657j) a'100001'b'000011' (-0.0036333814552024306-0.0003234115814681176j) a'100001'b'000101' (-0.0590618580344128-0.0052571646904036005j) a'100001'b'100001' (0.11314187441703756+0.010070889930469404j) a'000110'b'000110' (0.003747532670094563+0.0003335721898296519j) a'001010'b'001001' (-0.003108703183821363-0.0002767093025565706j) a'001010'b'001010' (0.0017871363875789756+0.00015907509845870593j) a'010010'b'010001' (-0.003108703183821363-0.0002767093025565706j) a'010010'b'010010' (0.0017871363875789756+0.00015907509845870593j) a'100010'b'000101' (-0.0010735253445789887-9.555564554364526e-05j) a'100010'b'100010' (0.0010474346607506372+9.323348063182962e-05j)
Exact evolution implementation of quadratic Hamiltonians
Listed here are examples of evolving the special Hamiltonians.
Diagonal Hamiltonian evolution is supported.
wfn = fqe.Wavefunction([[4, 2, 4]])
wfn.set_wfn(strategy='random')
if Print:
wfn.print_wfn()
diagonal = FermionOperator('0^ 0', -2.0) + \
FermionOperator('1^ 1', -1.7) + \
FermionOperator('2^ 2', -0.7) + \
FermionOperator('3^ 3', -0.55) + \
FermionOperator('4^ 4', -0.1) + \
FermionOperator('5^ 5', -0.06) + \
FermionOperator('6^ 6', 0.5) + \
FermionOperator('7^ 7', 0.3)
if Print:
print(diagonal)
evolved = wfn.time_evolve(time, diagonal)
if Print:
evolved.print_wfn()
Sector N = 4 : S_z = 2 a'0111'b'0001' (0.06410384332680892+0.006457824553220213j) a'0111'b'0010' (-0.04750282105641742-0.004806746363757904j) a'0111'b'0100' (-0.13678251578172124-0.14669354294155937j) a'0111'b'1000' (-0.17590908446219902+0.18037032855388507j) a'1011'b'0001' (-0.024476509055323645-0.35002973813257837j) a'1011'b'0010' (-0.29306549119285774-0.24993699345042755j) a'1011'b'0100' (-0.24463823979331925+0.216230018507552j) a'1011'b'1000' (0.2630370061724331-0.3219176890144332j) a'1101'b'0001' (0.06881598452335648-0.11820307831930973j) a'1101'b'0010' (0.07453400242973146-0.06864240203689581j) a'1101'b'0100' (-0.10280462830183762-0.20883766019199357j) a'1101'b'1000' (0.18607595463278417-0.08025697543327445j) a'1110'b'0001' (0.11456775354687777+0.3103763043669054j) a'1110'b'0010' (-0.06508766236659613-0.19892441460633234j) a'1110'b'0100' (0.10203424947892725-0.20758467025020982j) a'1110'b'1000' (0.07609612213727357+0.04672521448420223j) -2.0 [0^ 0] + -1.7 [1^ 1] + -0.7 [2^ 2] + -0.55 [3^ 3] + -0.1 [4^ 4] + -0.06 [5^ 5] + 0.5 [6^ 6] + 0.3 [7^ 7] Sector N = 4 : S_z = 2 a'0111'b'0001' (0.0637484451008294+0.009334986580321275j) a'0111'b'0010' (-0.047315172642354755-0.006395096304800911j) a'0111'b'0100' (-0.132531714879653-0.15054499897167728j) a'0111'b'1000' (-0.18036290425047674+0.17591669673256966j) a'1011'b'0001' (-0.010810197548109522-0.35071791614011905j) a'1011'b'0010' (-0.2860822817529272-0.25790077721904764j) a'1011'b'0100' (-0.24946214917080445+0.21064644645479583j) a'1011'b'1000' (0.26910559751344887-0.31686218219200535j) a'1101'b'0001' (0.07267851126583812-0.11586820724539217j) a'1101'b'0010' (0.07599247436942777-0.06702418007556614j) a'1101'b'0100' (-0.09932391825824108-0.21051536565432893j) a'1101'b'1000' (0.1871035427295063-0.07783127453836906j) a'1110'b'0001' (0.108337728499401+0.31260543349216774j) a'1110'b'0010' (-0.06339447392546516-0.19947046697326787j) a'1110'b'0100' (0.1027808914964281-0.207216002598293j) a'1110'b'1000' (0.07609612213727357+0.04672521448420223j)
Exact evolution of dense quadratic hamiltonians is supported. Here is an evolution example using a spin restricted Hamiltonian on a number and spin conserving wavefunction
norb = 4
h1e = numpy.zeros((norb, norb), dtype=numpy.complex128)
for i in range(norb):
for j in range(norb):
h1e[i, j] += (i+j) * 0.02
h1e[i, i] += i * 2.0
hamil = fqe.get_restricted_hamiltonian((h1e,))
wfn = fqe.Wavefunction([[4, 0, norb]])
wfn.set_wfn(strategy='random')
initial_energy = wfn.expectationValue(hamil)
print('Initial Energy: {}'.format(initial_energy))
evolved = wfn.time_evolve(time, hamil)
final_energy = evolved.expectationValue(hamil)
print('Final Energy: {}'.format(final_energy))
Initial Energy: (12.222344460445168+5.551115123125783e-17j) Final Energy: (12.222344460445179+1.9168694409543718e-16j)
The GSO Hamiltonian is for evolution of quadratic hamiltonians that are spin broken and number conserving.
norb = 4
h1e = numpy.zeros((2*norb, 2*norb), dtype=numpy.complex128)
for i in range(2*norb):
for j in range(2*norb):
h1e[i, j] += (i+j) * 0.02
h1e[i, i] += i * 2.0
hamil = fqe.get_gso_hamiltonian((h1e,))
wfn = fqe.get_number_conserving_wavefunction(4, norb)
wfn.set_wfn(strategy='random')
initial_energy = wfn.expectationValue(hamil)
print('Initial Energy: {}'.format(initial_energy))
evolved = wfn.time_evolve(time, hamil)
final_energy = evolved.expectationValue(hamil)
print('Final Energy: {}'.format(final_energy))
Initial Energy: (28.720309503101298-2.393918396847994e-16j) Final Energy: (28.72030950310125+4.0939474033052647e-16j)
The BCS hamiltonian evovles spin conserved and number broken wavefunctions.
norb = 4
time = 0.001
wfn_spin = fqe.get_spin_conserving_wavefunction(2, norb)
hamil = FermionOperator('', 6.0)
for i in range(0, 2*norb, 2):
for j in range(0, 2*norb, 2):
opstring = str(i) + ' ' + str(j + 1)
hamil += FermionOperator(opstring, (i+1 + j*2)*0.1 - (i+1 + 2*(j + 1))*0.1j)
opstring = str(i) + '^ ' + str(j + 1) + '^ '
hamil += FermionOperator(opstring, (i+1 + j)*0.1 + (i+1 + j)*0.1j)
h_noncon = (hamil + hermitian_conjugated(hamil))/2.0
if Print:
print(h_noncon)
wfn_spin.set_wfn(strategy='random')
if Print:
wfn_spin.print_wfn()
spin_evolved = wfn_spin.time_evolve(time, h_noncon)
if Print:
spin_evolved.print_wfn()
6.0 [] + (0.05-0.15000000000000002j) [0 1] + (0.25-0.35000000000000003j) [0 3] + (0.45-0.55j) [0 5] + (0.65-0.75j) [0 7] + (0.05+0.05j) [0^ 1^] + (0.15000000000000002+0.15000000000000002j) [0^ 3^] + (0.25+0.25j) [0^ 5^] + (0.35000000000000003+0.35000000000000003j) [0^ 7^] + (0.05-0.05j) [1 0] + (0.15000000000000002-0.15000000000000002j) [1 2] + (0.25-0.25j) [1 4] + (0.35000000000000003-0.35000000000000003j) [1 6] + (0.05+0.15000000000000002j) [1^ 0^] + (0.15000000000000002+0.25j) [1^ 2^] + (0.25+0.35000000000000003j) [1^ 4^] + (0.35000000000000003+0.45j) [1^ 6^] + (0.15000000000000002-0.25j) [2 1] + (0.35000000000000003-0.45j) [2 3] + (0.55-0.65j) [2 5] + (0.75-0.8500000000000001j) [2 7] + (0.15000000000000002+0.15000000000000002j) [2^ 1^] + (0.25+0.25j) [2^ 3^] + (0.35000000000000003+0.35000000000000003j) [2^ 5^] + (0.45+0.45j) [2^ 7^] + (0.15000000000000002-0.15000000000000002j) [3 0] + (0.25-0.25j) [3 2] + (0.35000000000000003-0.35000000000000003j) [3 4] + (0.45-0.45j) [3 6] + (0.25+0.35000000000000003j) [3^ 0^] + (0.35000000000000003+0.45j) [3^ 2^] + (0.45+0.55j) [3^ 4^] + (0.55+0.65j) [3^ 6^] + (0.25-0.35000000000000003j) [4 1] + (0.45-0.55j) [4 3] + (0.65-0.75j) [4 5] + (0.8500000000000001-0.9500000000000001j) [4 7] + (0.25+0.25j) [4^ 1^] + (0.35000000000000003+0.35000000000000003j) [4^ 3^] + (0.45+0.45j) [4^ 5^] + (0.55+0.55j) [4^ 7^] + (0.25-0.25j) [5 0] + (0.35000000000000003-0.35000000000000003j) [5 2] + (0.45-0.45j) [5 4] + (0.55-0.55j) [5 6] + (0.45+0.55j) [5^ 0^] + (0.55+0.65j) [5^ 2^] + (0.65+0.75j) [5^ 4^] + (0.75+0.8500000000000001j) [5^ 6^] + (0.35000000000000003-0.45j) [6 1] + (0.55-0.65j) [6 3] + (0.75-0.8500000000000001j) [6 5] + (0.9500000000000001-1.05j) [6 7] + (0.35000000000000003+0.35000000000000003j) [6^ 1^] + (0.45+0.45j) [6^ 3^] + (0.55+0.55j) [6^ 5^] + (0.65+0.65j) [6^ 7^] + (0.35000000000000003-0.35000000000000003j) [7 0] + (0.45-0.45j) [7 2] + (0.55-0.55j) [7 4] + (0.65-0.65j) [7 6] + (0.65+0.75j) [7^ 0^] + (0.75+0.8500000000000001j) [7^ 2^] + (0.8500000000000001+0.9500000000000001j) [7^ 4^] + (0.9500000000000001+1.05j) [7^ 6^] Sector N = 2 : S_z = 2 a'0011'b'0000' (-0.03822050376543366+0.04722267594475418j) a'0101'b'0000' (0.11266351755026058-0.08045492107765523j) a'1001'b'0000' (0.20320077458016653-0.12569862804377432j) a'0110'b'0000' (0.05619850454412105+0.09496478341558393j) a'1010'b'0000' (0.10559885537015444+0.11842688052122244j) a'1100'b'0000' (0.0077637482095351895+0.31531590367980844j) Sector N = 4 : S_z = 2 a'0111'b'0001' (0.2824942893468279+0.03355206579818916j) a'0111'b'0010' (-0.05772107497136629-0.02636869741981419j) a'0111'b'0100' (0.023785796648682776-0.0020224394959580906j) a'0111'b'1000' (0.21640450174424197-0.04378508800670816j) a'1011'b'0001' (-0.05498328060358216+0.02734692187255261j) a'1011'b'0010' (-0.04928412842784853+0.14503876601698823j) a'1011'b'0100' (0.15795758958015643-0.4182365189352067j) a'1011'b'1000' (-0.17447882444539087-0.009486157624653761j) a'1101'b'0001' (-0.1445095969304484-0.18205850652200903j) a'1101'b'0010' (-0.03991499779909757-0.15433749648217981j) a'1101'b'0100' (0.06570900360151104+0.004102951665073998j) a'1101'b'1000' (0.059818195036244165-0.12107474028451727j) a'1110'b'0001' (-0.06872552283670944-0.03502572909303779j) a'1110'b'0010' (0.01187150936783494-0.08220106790262457j) a'1110'b'0100' (0.245935671612729+0.2003604108656353j) a'1110'b'1000' (-0.08678168538325304-0.15114749448953507j) Sector N = 6 : S_z = 2 a'1111'b'0011' (-0.07458559356076411-0.07500964684800192j) a'1111'b'0101' (0.00026534389687275984-0.07669551428142163j) a'1111'b'1001' (-0.09607654962925911-0.03066759719778399j) a'1111'b'0110' (0.09851785462192013+0.04062010213534433j) a'1111'b'1010' (-0.06147228452062744+0.19634181999024788j) a'1111'b'1100' (0.203465187540792+0.15201301478908172j) Sector N = 2 : S_z = 2 a'0011'b'0000' (-0.0380977431136074+0.04753994545449976j) a'0101'b'0000' (0.11218647680909519-0.08121989427531981j) a'1001'b'0000' (0.20259647757874874-0.12691391634375246j) a'0110'b'0000' (0.05681562481162151+0.09444679519610702j) a'1010'b'0000' (0.10609790338961551+0.11805904723618599j) a'1100'b'0000' (0.009711552915153255+0.31499411161394925j) Sector N = 4 : S_z = 2 a'0111'b'0001' (0.28273340818288784+0.031919134370123885j) a'0111'b'0010' (-0.057769577759326396-0.02609721113767848j) a'0111'b'0100' (0.023769115093997907-0.0021755817497583017j) a'0111'b'1000' (0.216055008347028-0.04524984142713354j) a'1011'b'0001' (-0.0548838771340961+0.027665253462434567j) a'1011'b'0010' (-0.048527689130426155+0.14528128840122673j) a'1011'b'0100' (0.15548424682043685-0.41896605472582654j) a'1011'b'1000' (-0.17451327690229243-0.008546537415431606j) a'1101'b'0001' (-0.1455551722473013-0.18111178278510337j) a'1101'b'0010' (-0.040747946468046796-0.15419679959996307j) a'1101'b'0100' (0.06575967845578273+0.0037369092000684705j) a'1101'b'1000' (0.058961384441977176-0.12164982883882651j) a'1110'b'0001' (-0.06899064502908153-0.034623881367538016j) a'1110'b'0010' (0.0112445552937786-0.08231180714509592j) a'1110'b'0100' (0.24720056742353771+0.1990727751786515j) a'1110'b'1000' (-0.08770534327870655-0.1507028840348224j) Sector N = 6 : S_z = 2 a'1111'b'0011' (-0.07500249390617938-0.07463966565297969j) a'1111'b'0101' (-0.0002759642307366578-0.07657867707171738j) a'1111'b'1001' (-0.09614340563343121-0.03022655476659553j) a'1111'b'0110' (0.09886734056718283+0.04001330102091265j) a'1111'b'1010' (-0.06049739731068088+0.1966835109908998j) a'1111'b'1100' (0.2044745383937896+0.1508623543631741j)
Exact Evolution Implementation of Diagonal Coulomb terms
norb = 4
wfn = fqe.Wavefunction([[5, 1, norb]])
vij = numpy.zeros((norb, norb, norb, norb), dtype=numpy.complex128)
for i in range(norb):
for j in range(norb):
vij[i, j] += 4*(i % norb + 1)*(j % norb + 1)*0.21
wfn.set_wfn(strategy='random')
if Print:
wfn.print_wfn()
hamil = fqe.get_diagonalcoulomb_hamiltonian(vij)
evolved = wfn.time_evolve(time, hamil)
if Print:
evolved.print_wfn()
Sector N = 5 : S_z = 1 a'0111'b'0011' (0.1767197633238888+0.07705027591386174j) a'0111'b'0101' (0.26511616271206534+0.008961114965964636j) a'0111'b'1001' (-0.21500788967807205+0.18883222357115242j) a'0111'b'0110' (0.05899888340530983+0.24718508330513655j) a'0111'b'1010' (0.026779941886877087+0.178826903379466j) a'0111'b'1100' (0.040105194632471065-0.13765308148589672j) a'1011'b'0011' (-0.07737195675705617-0.13361350504300393j) a'1011'b'0101' (-0.13687313057404554+0.29187074296331617j) a'1011'b'1001' (-0.06315193751752252-0.015963783432630548j) a'1011'b'0110' (0.1412070788641405+0.1547599681431957j) a'1011'b'1010' (0.19284938811527336+0.05815106367478927j) a'1011'b'1100' (0.0601422593196422+0.28502769589072846j) a'1101'b'0011' (-0.18150775903742453+0.016874070450244708j) a'1101'b'0101' (-0.0348624766019713-0.13176446392846256j) a'1101'b'1001' (-0.0025269874019722425+0.15813527253622475j) a'1101'b'0110' (0.2350325470053182+0.00014186481294887606j) a'1101'b'1010' (-0.05882009410190486-0.17131076761878405j) a'1101'b'1100' (-0.0805012226704995-0.22440941132365497j) a'1110'b'0011' (0.03604887823935152+0.03924039987192205j) a'1110'b'0101' (0.01791557236805855+0.049178708081171994j) a'1110'b'1001' (0.043394070527428906+0.04684086505377531j) a'1110'b'0110' (-0.06429592223560363-0.0008430408719242828j) a'1110'b'1010' (0.2544814342392375-0.29724856959461543j) a'1110'b'1100' (0.030682862125531125+0.04316975493381036j) Sector N = 5 : S_z = 1 a'0111'b'0011' (0.17246919202766037+0.08614521210711293j) a'0111'b'0101' (0.2640044105946365+0.02585638217504639j) a'0111'b'1001' (-0.22865588002522289+0.1720537410593985j) a'0111'b'0110' (0.039317599959618595+0.25106863600429263j) a'0111'b'1010' (0.01016559936487448+0.18053500283686325j) a'0111'b'1100' (0.05486817182308945-0.13246237654467485j) a'1011'b'0011' (-0.06892245945470493-0.1381618000932499j) a'1011'b'0101' (-0.15852659275216138+0.2806993121856342j) a'1011'b'1001' (-0.06148222691741648-0.021516165247931745j) a'1011'b'0110' (0.12632522987699016+0.16712876220911863j) a'1011'b'1010' (0.18549537660753673+0.0785104958599347j) a'1011'b'1100' (0.023846943350919615+0.2903260272067709j) a'1101'b'0011' (-0.18226377985147815+0.00311695341631818j) a'1101'b'0101' (-0.022781831587704093-0.13438100453072824j) a'1101'b'1001' (-0.019218653219683423+0.15698341776715583j) a'1101'b'0110' (0.2336171388678771+0.025755595598224452j) a'1101'b'1010' (-0.03682570987971141-0.1773444379284259j) a'1101'b'1100' (-0.04695986639822977-0.23374088579008845j) a'1110'b'0011' (0.03227444359792561+0.042399185073162045j) a'1110'b'0101' (0.012449211992330995+0.05083827477971962j) a'1110'b'1001' (0.037163719046413746+0.05192273088508236j) a'1110'b'0110' (-0.06364995585406888-0.00913013985728614j) a'1110'b'1010' (0.29549075882045067-0.25652041624443533j) a'1110'b'1100' (0.02295227609762028+0.047731109260085336j)
Exact evolution of individual n-body anti-Hermitian gnerators
norb = 3
nele = 4
ops = FermionOperator('5^ 1^ 2 0', 3.0 - 1.j)
ops += FermionOperator('0^ 2^ 1 5', 3.0 + 1.j)
wfn = fqe.get_number_conserving_wavefunction(nele, norb)
wfn.set_wfn(strategy='random')
wfn.normalize()
if Print:
wfn.print_wfn()
evolved = wfn.time_evolve(time, ops)
if Print:
evolved.print_wfn()
Sector N = 4 : S_z = -2 a'001'b'111' (0.15192470437087405+0.002030736551023001j) a'010'b'111' (-0.03463482137049871+0.27009718718635434j) a'100'b'111' (-0.227167326143701-0.042680013855770464j) Sector N = 4 : S_z = 0 a'011'b'011' (-0.3135681471723931+0.02343272674191714j) a'011'b'101' (-0.02499985483324474+0.08816424695516287j) a'011'b'110' (0.08135663091791555-0.2354257740286914j) a'101'b'011' (0.10687726150559686-0.2956005054158534j) a'101'b'101' (0.17170942273731685+0.06147474695535962j) a'101'b'110' (-0.06659232012041506-0.17366091158923022j) a'110'b'011' (0.10656876871906572+0.05034757634263565j) a'110'b'101' (-0.5439254382876493-0.12805668259580452j) a'110'b'110' (0.05259450329119359+0.030994001966578325j) Sector N = 4 : S_z = 2 a'111'b'001' (-0.27915567439519695-0.16997476921880741j) a'111'b'010' (0.21779510862047607-0.13520849680378372j) a'111'b'100' (-0.062444490395427814+0.08399365005639217j) Sector N = 4 : S_z = -2 a'001'b'111' (0.15192470437087405+0.002030736551023001j) a'010'b'111' (-0.03463482137049871+0.27009718718635434j) a'100'b'111' (-0.2277896098680148-0.04319797642130905j) Sector N = 4 : S_z = 0 a'011'b'011' (-0.3135681471723931+0.02343272674191714j) a'011'b'101' (-0.02499985483324474+0.08816424695516287j) a'011'b'110' (0.08135663091791555-0.2354257740286914j) a'101'b'011' (0.10687726150559686-0.2956005054158534j) a'101'b'101' (0.17170942273731685+0.06147474695535962j) a'101'b'110' (-0.06659232012041506-0.17366091158923022j) a'110'b'011' (0.10656876871906572+0.05034757634263565j) a'110'b'101' (-0.5439254382876493-0.12805668259580452j) a'110'b'110' (0.05259450329119359+0.030994001966578325j) Sector N = 4 : S_z = 2 a'111'b'001' (-0.27915567439519695-0.16997476921880741j) a'111'b'010' (0.2174388128701414-0.13456899986199047j) a'111'b'100' (-0.062444490395427814+0.08399365005639217j)
Approximate evolution of sums of n-body generators
Approximate evolution can be done for dense operators.
lih_evolved = lihwfn.apply_generated_unitary(time, 'taylor', lih_hamiltonian, accuracy=1.e-8)
if Print:
lih_evolved.print_wfn()
Sector N = 4 : S_z = 0 a'000011'b'000011' (-0.9870501056639676-0.008762984250848186j) a'000011'b'000101' (-0.0384839713479148-0.00034165891188627537j) a'000011'b'100001' (-0.003647602935297334-3.238325409440231e-05j) a'000101'b'000011' (-0.0384839713479097-0.00034165891188623075j) a'000101'b'000101' (0.03444949641617426+0.0003058410149547989j) a'000101'b'100001' (-0.059293032957665366-0.000526400752638063j) a'000101'b'100010' (-0.0010777272393492372-9.568002262034463e-06j) a'001001'b'001001' (0.02716462574072948+0.0002411662625463919j) a'001001'b'001010' (-0.003120871007714629-2.770694732178501e-05j) a'010001'b'010001' (0.02716462574072948+0.0002411662625463919j) a'010001'b'010010' (-0.003120871007714629-2.770694732178501e-05j) a'100001'b'000011' (-0.0036476029352976343-3.238325409440512e-05j) a'100001'b'000101' (-0.059293032957666365-0.0005264007526380719j) a'100001'b'100001' (0.11358472476587893+0.0010083998412408197j) a'000110'b'000110' (0.0037622009416752988+3.340063743457015e-05j) a'001010'b'001001' (-0.003120871007714629-2.7706947321785068e-05j) a'001010'b'001010' (0.0017941314460251166+1.5928215395129844e-05j) a'010010'b'010001' (-0.003120871007714629-2.7706947321785068e-05j) a'010010'b'010010' (0.0017941314460251166+1.5928215395129844e-05j) a'100010'b'000101' (-0.0010777272393493371-9.568002262035205e-06j) a'100010'b'100010' (0.0010515344463539172+9.335483532366947e-06j)
norb = 2
nalpha = 1
nbeta = 1
nele = nalpha + nbeta
time = 0.05
h1e = numpy.zeros((norb*2, norb*2), dtype=numpy.complex128)
for i in range(2*norb):
for j in range(2*norb):
h1e[i, j] += (i+j) * 0.02
h1e[i, i] += i * 2.0
hamil = fqe.get_general_hamiltonian((h1e,))
spec_lim = [-1.13199078e-03, 6.12720338e+00]
wfn = fqe.Wavefunction([[nele, nalpha - nbeta, norb]])
wfn.set_wfn(strategy='random')
if Print:
wfn.print_wfn()
evol_wfn = wfn.apply_generated_unitary(time, 'chebyshev', hamil, spec_lim=spec_lim)
if Print:
evol_wfn.print_wfn()
Sector N = 2 : S_z = 0 a'01'b'01' (0.3495794429313121+0.022158760002413482j) a'01'b'10' (0.5664453364412798-0.3945979322462498j) a'10'b'01' (-0.39816876837018494+0.14573910133744658j) a'10'b'10' (-0.4662060110250305-0.06007542464228622j) Sector N = 2 : S_z = 0 a'01'b'01' (0.344436410676463-0.05101455179951593j) a'01'b'10' (0.4210294240976466-0.548162567744529j) a'10'b'01' (-0.33530047561074994+0.26085960635871003j) a'10'b'10' (-0.4509597513637166+0.13105059079720655j)
API for determining desired expectation values
rdm1 = lihwfn.expectationValue('i^ j')
if Print:
print(rdm1)
val = lihwfn.expectationValue('5^ 3')
if Print:
print(2.*val)
trdm1 = fqe.expectationValue(lih_evolved, 'i j^', lihwfn)
if Print:
print(trdm1)
val = fqe.expectationValue(lih_evolved, '5 3^', lihwfn)
if Print:
print(2*val)
[[ 1.999908+0.000000j -0.000284+0.000000j 0.000441+0.000000j -0.000000+0.000000j -0.000000+0.000000j -0.001285+0.000000j] [-0.000284+0.000000j 1.951766+0.000000j 0.073757+0.000000j 0.000000+0.000000j 0.000000+0.000000j 0.010948+0.000000j] [ 0.000441+0.000000j 0.073757+0.000000j 0.012402+0.000000j 0.000000+0.000000j 0.000000+0.000000j -0.017277+0.000000j] [-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j 0.001525+0.000000j 0.000000+0.000000j 0.000000+0.000000j] [-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j 0.001525+0.000000j -0.000000+0.000000j] [-0.001285+0.000000j 0.010948+0.000000j -0.017277+0.000000j 0.000000+0.000000j -0.000000+0.000000j 0.032874+0.000000j]] (0.0737569627134323+0j) [[ 0.000170-0.017754j 0.000284+0.000003j -0.000441-0.000004j 0.000000+0.000000j 0.000000+0.000000j 0.001285+0.000011j] [ 0.000284+0.000003j 0.048311-0.017327j -0.073754-0.000655j -0.000000-0.000000j -0.000000-0.000000j -0.010948-0.000097j] [-0.000441-0.000004j -0.073754-0.000655j 1.987599-0.000110j -0.000000-0.000000j -0.000000-0.000000j 0.017276+0.000153j] [ 0.000000+0.000000j -0.000000-0.000000j -0.000000-0.000000j 1.998475-0.000014j -0.000000-0.000000j -0.000000-0.000000j] [ 0.000000+0.000000j -0.000000-0.000000j -0.000000-0.000000j -0.000000-0.000000j 1.998475-0.000014j 0.000000+0.000000j] [ 0.001285+0.000011j -0.010948-0.000097j 0.017276+0.000153j -0.000000-0.000000j 0.000000+0.000000j 1.967127-0.000292j]] (-0.07375405619488316-0.0006547850894203785j)
2.B.1 RDMs In addition to the above API higher order density matrices in addition to hole densities can be calculated.
rdm2 = lihwfn.expectationValue('i^ j k l^')
if Print:
print(rdm2)
rdm2 = fqe.expectationValue(lihwfn, 'i^ j^ k l', lihwfn)
if Print:
print(rdm2)
[[[[-0.000047+0.000000j -0.000260+0.000000j 0.000445+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.001287+0.000000j]
[ 0.000308+0.000000j 0.048269+0.000000j -0.073761+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.010927+0.000000j]
[-0.000437+0.000000j -0.073761+0.000000j 1.987537+0.000000j
-0.000000+0.000000j -0.000000+0.000000j 0.017274+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
1.998412+0.000000j -0.000000+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
-0.000000+0.000000j 1.998412+0.000000j 0.000000+0.000000j]
[ 0.001283+0.000000j -0.010927+0.000000j 0.017274+0.000000j
-0.000000+0.000000j 0.000000+0.000000j 1.967051+0.000000j]]
[[-0.000544+0.000000j -0.096416+0.000000j 0.147514+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.021885+0.000000j]
[ 0.000733+0.000000j 0.000242+0.000000j 0.000079+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000157+0.000000j]
[ 0.000282+0.000000j 0.000213+0.000000j -0.000180+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000110+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000103+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000103+0.000000j -0.000000+0.000000j]
[-0.000012+0.000000j 0.000449+0.000000j 0.000021+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.000447+0.000000j]]
[[ 0.000886+0.000000j 0.147514+0.000000j -3.975076+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.034551+0.000000j]
[ 0.000282+0.000000j -0.000073+0.000000j 0.000684+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.000288+0.000000j]
[-0.007548+0.000000j -0.000414+0.000000j -0.000511+0.000000j
-0.000000+0.000000j -0.000000+0.000000j 0.000102+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000433+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000433+0.000000j 0.000000+0.000000j]
[ 0.000293+0.000000j -0.003503+0.000000j 0.002338+0.000000j
-0.000000+0.000000j -0.000000+0.000000j 0.000595+0.000000j]]
[[-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-3.996823+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000207+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
-0.000866+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.003666+0.000000j -0.006155+0.000000j -0.000351+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.000189+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.002480+0.000000j 0.000000+0.000000j -0.000000+0.000000j]]
[[-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -3.996823+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000207+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.000866+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.003666+0.000000j -0.006155+0.000000j -0.000351+0.000000j
-0.000000+0.000000j 0.000000+0.000000j 0.000189+0.000000j]
[-0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.002480+0.000000j -0.000000+0.000000j]]
[[-0.002572+0.000000j 0.021885+0.000000j -0.034551+0.000000j
0.000000+0.000000j -0.000000+0.000000j -3.934082+0.000000j]
[-0.000012+0.000000j -0.000182+0.000000j -0.000002+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000625+0.000000j]
[ 0.000293+0.000000j 0.000715+0.000000j -0.001159+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.001060+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
-0.001240+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.001240+0.000000j 0.000000+0.000000j]
[-0.002124+0.000000j -0.000521+0.000000j -0.000001+0.000000j
-0.000000+0.000000j -0.000000+0.000000j 0.001252+0.000000j]]]
[[[ 0.000024+0.000000j 0.000733+0.000000j 0.000282+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000012+0.000000j]
[-0.000131+0.000000j -0.000326+0.000000j 0.001095+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.002122+0.000000j]
[ 0.000000+0.000000j 0.000079+0.000000j -0.000180+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000021+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000103+0.000000j 0.000000+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000103+0.000000j -0.000000+0.000000j]
[-0.000011+0.000000j -0.000157+0.000000j -0.000110+0.000000j
0.000000+0.000000j -0.000000+0.000000j -0.000447+0.000000j]]
[[ 0.000127+0.000000j 0.000526+0.000000j -0.000514+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.001103+0.000000j]
[-0.000042+0.000000j -0.003032+0.000000j 0.075980+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.007202+0.000000j]
[-0.000514+0.000000j -0.071534+0.000000j 1.954703+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000282+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
1.951740+0.000000j -0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
-0.000000+0.000000j 1.951740+0.000000j 0.000000+0.000000j]
[ 0.001103+0.000000j -0.014694+0.000000j 0.000282+0.000000j
0.000000+0.000000j 0.000000+0.000000j 1.951786+0.000000j]]
[[-0.000004+0.000000j 0.000079+0.000000j 0.000684+0.000000j
0.000000+0.000000j -0.000000+0.000000j -0.000002+0.000000j]
[ 0.001095+0.000000j 0.149736+0.000000j -3.900509+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000278+0.000000j]
[-0.000414+0.000000j -0.068015+0.000000j -0.076410+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.000251+0.000000j]
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[-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
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[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
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[ 0.000362+0.000000j -0.000052+0.000000j 0.000001+0.000000j
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[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
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[[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[-0.000015+0.000000j 0.000001+0.000000j -0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]]
[[ 0.003666+0.000000j 0.006155+0.000000j 0.000351+0.000000j
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[ 0.006155+0.000000j 0.053634+0.000000j 0.002096+0.000000j
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[ 0.000351+0.000000j 0.002096+0.000000j -0.001890+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.001525+0.000000j 0.000000+0.000000j]
[-0.000189+0.000000j 0.000196+0.000000j 0.003214+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.006178+0.000000j]]
[[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]]
[[-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
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[-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000005+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000091+0.000000j -0.000011+0.000000j 0.000001+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000003+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000002+0.000000j 0.000000+0.000000j 0.000000+0.000000j]]]
[[[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.001497+0.000000j -0.000000+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.000181+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000008+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.002994+0.000000j 0.000362+0.000000j -0.000015+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000091+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.000045+0.000000j 0.000000+0.000000j]]
[[-0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.000181+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000026+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000001+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000362+0.000000j -0.000052+0.000000j 0.000001+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000011+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000005+0.000000j 0.000000+0.000000j]]
[[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000008+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000001+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.000015+0.000000j 0.000001+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.000001+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j]]
[[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]]
[[ 0.003666+0.000000j 0.006155+0.000000j 0.000351+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000189+0.000000j]
[ 0.006155+0.000000j 0.053634+0.000000j 0.002096+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000196+0.000000j]
[ 0.000351+0.000000j 0.002096+0.000000j -0.001890+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.003214+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.001525+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.001525+0.000000j 0.000000+0.000000j]
[-0.000189+0.000000j 0.000196+0.000000j 0.003214+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.006178+0.000000j]]
[[ 0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.000045+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000005+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000091+0.000000j -0.000011+0.000000j 0.000001+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000003+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000002+0.000000j 0.000000+0.000000j]]]
[[[ 0.001287+0.000000j 0.010927+0.000000j -0.017274+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.032857+0.000000j]
[-0.021885+0.000000j 0.000157+0.000000j 0.000110+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000163+0.000000j]
[ 0.034551+0.000000j 0.000288+0.000000j -0.000102+0.000000j
0.000000+0.000000j -0.000000+0.000000j -0.000154+0.000000j]
[-0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000189+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j -0.000189+0.000000j -0.000000+0.000000j]
[-0.065735+0.000000j -0.000057+0.000000j 0.000178+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000033+0.000000j]]
[[ 0.000012+0.000000j 0.002122+0.000000j -0.000021+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000163+0.000000j]
[-0.001103+0.000000j -0.007202+0.000000j -0.000282+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.000020+0.000000j]
[ 0.000002+0.000000j -0.000278+0.000000j 0.000251+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000433+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000196+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000196+0.000000j -0.000000+0.000000j]
[-0.000057+0.000000j -0.000036+0.000000j -0.000433+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.000828+0.000000j]]
[[-0.000293+0.000000j 0.003503+0.000000j 0.000232+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.000154+0.000000j]
[-0.000715+0.000000j -0.117061+0.000000j -0.004564+0.000000j
-0.000000+0.000000j -0.000000+0.000000j -0.000433+0.000000j]
[-0.000126+0.000000j -0.004564+0.000000j 0.004087+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.007032+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.003214+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.003214+0.000000j -0.000000+0.000000j]
[ 0.000178+0.000000j -0.000433+0.000000j -0.007038+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.013471+0.000000j]]
[[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000091+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000011+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000001+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.000045+0.000000j 0.000005+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000002+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.000003+0.000000j 0.000000+0.000000j 0.000000+0.000000j]]
[[ 0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000091+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
-0.000000+0.000000j -0.000011+0.000000j -0.000000+0.000000j]
[-0.000000+0.000000j -0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000001+0.000000j -0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[-0.000045+0.000000j 0.000005+0.000000j -0.000000+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000002+0.000000j]
[ 0.000000+0.000000j -0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.000003+0.000000j 0.000000+0.000000j]]
[[ 0.002124+0.000000j 0.000521+0.000000j 0.000001+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000033+0.000000j]
[ 0.000521+0.000000j 0.224250+0.000000j 0.008743+0.000000j
0.000000+0.000000j 0.000000+0.000000j 0.000828+0.000000j]
[ 0.000001+0.000000j 0.008743+0.000000j -0.007835+0.000000j
0.000000+0.000000j -0.000000+0.000000j 0.013471+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j 0.000000+0.000000j
-0.006178+0.000000j 0.000000+0.000000j 0.000000+0.000000j]
[ 0.000000+0.000000j 0.000000+0.000000j -0.000000+0.000000j
0.000000+0.000000j -0.006178+0.000000j 0.000000+0.000000j]
[ 0.000033+0.000000j 0.000828+0.000000j 0.013471+0.000000j
0.000000+0.000000j 0.000000+0.000000j -0.025808+0.000000j]]]]
2.B.2 Hamiltonian expectations (or any expectation values)
li_h_energy = lihwfn.expectationValue(lih_hamiltonian)
if Print:
print(li_h_energy)
li_h_energy = fqe.expectationValue(lihwfn, lih_hamiltonian, lihwfn)
if Print:
print(li_h_energy)
(-8.87771957038547+0j) (-8.87771957038547+0j)
2.B.3 Symmetry operations
op = fqe.get_s2_operator()
print(lihwfn.expectationValue(op))
op = fqe.get_sz_operator()
print(lihwfn.expectationValue(op))
op = fqe.get_time_reversal_operator()
print(lihwfn.expectationValue(op))
op = fqe.get_number_operator()
print(lihwfn.expectationValue(op))
(-1.8969111426479763e-23+0j) 0j (1.0000000000000018+0j) (4.000000000000007+0j)