Hamiltonian Time Evolution and Expectation Value Computation

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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   331k      0 --:--:-- --:--:-- --:--:--  331k
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 propogation 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' (-0.25731729177667634-0.02206552591519505j)
a'0111'b'0010' (-0.19069981498041644-0.15744080284206466j)
a'0111'b'0100' (-0.13843008024130435+0.6019949651839998j)
a'0111'b'1000' (-0.10069248423101489-0.26258907973635726j)
a'1011'b'0001' (0.01701941604343994+0.11531103507659798j)
a'1011'b'0010' (-0.2470542499114737-0.16189142882301663j)
a'1011'b'0100' (-0.22886695583773375+0.16343476115652358j)
a'1011'b'1000' (-0.17112530964365846+0.14887212980125927j)
a'1101'b'0001' (0.14735656433050406-0.12158475184832696j)
a'1101'b'0010' (0.161121386884907+0.0012034595464655267j)
a'1101'b'0100' (0.05261727070820833-0.08021022138631806j)
a'1101'b'1000' (-0.053664467674974375-0.039818791975797956j)
a'1110'b'0001' (0.007163733471580562-0.116426593460279j)
a'1110'b'0010' (0.12513118505662785+0.0005967634674837372j)
a'1110'b'0100' (-0.021197557000183942+0.10844705229527413j)
a'1110'b'1000' (-0.0437698054486209-0.2461885243951473j)
-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' (-0.2560641884009183-0.03361855885930155j)
a'0111'b'0010' (-0.1853195381107995-0.16373971609697777j)
a'0111'b'0100' (-0.15558817792245747+0.5977902174787312j)
a'0111'b'1000' (-0.09409697627948393-0.26502407481672j)
a'1011'b'0001' (0.012510483985880075+0.1158869411251119j)
a'1011'b'0010' (-0.24250938523577004-0.1686233530631142j)
a'1011'b'0100' (-0.23250182146638482+0.158221072059111j)
a'1011'b'1000' (-0.17392282273811355+0.14559407392819157j)
a'1101'b'0001' (0.15128790458112176-0.11665667087981992j)
a'1101'b'0010' (0.16105827675193568+0.004667024350992057j)
a'1101'b'0100' (0.05394144979209672-0.07932576369597548j)
a'1101'b'1000' (-0.053142303375824186-0.04051304576506322j)
a'1110'b'0001' (0.009490677409496147-0.11626004379977588j)
a'1110'b'0010' (0.1251215922913914+0.0016603441748627057j)
a'1110'b'0100' (-0.021587828185142186+0.10837003851876566j)
a'1110'b'1000' (-0.0437698054486209-0.2461885243951473j)

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: (12.630481923700248-1.3877787807814457e-17j)
Final Energy:   (12.630481923700255+7.632783294297951e-17j)

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: (28.935131308457418+6.938893903907228e-17j)
Final Energy:   (28.935131308457528+9.71445146547012e-17j)

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' (-0.07663823213230848-0.13682257978232756j)
a'0101'b'0000' (-0.06551282411726828-0.039157926588565296j)
a'1001'b'0000' (0.04199937868935139-0.011466770311037199j)
a'0110'b'0000' (-0.04459184393861897-0.07591483244752986j)
a'1010'b'0000' (0.011949278691002417+0.1992676809031353j)
a'1100'b'0000' (0.04921239426447286+0.12713169076138583j)
Sector N = 4 : S_z = 2
a'0111'b'0001' (0.05354748918896002-0.07726642302176052j)
a'0111'b'0010' (0.13653119312298753+0.08148796865438525j)
a'0111'b'0100' (-0.19485603927360895-0.18779356629527197j)
a'0111'b'1000' (-0.06789127678037661+0.06251136767853377j)
a'1011'b'0001' (-0.2045649792967276-0.02160259738462829j)
a'1011'b'0010' (0.00026659787046694433-0.09200558626871955j)
a'1011'b'0100' (-0.3621832207633769-0.026552481320880023j)
a'1011'b'1000' (0.22336431390496428-0.08003045909618679j)
a'1101'b'0001' (-0.1625639215010968-0.09534791041431309j)
a'1101'b'0010' (-0.14881116599045158+0.14459960976767566j)
a'1101'b'0100' (0.12057173909451716-0.031207106803325904j)
a'1101'b'1000' (0.19120060970433495-0.23241718730540672j)
a'1110'b'0001' (0.020160332914659625+0.27828553415071494j)
a'1110'b'0010' (-0.07526549006807054-0.11045706294037846j)
a'1110'b'0100' (-0.007895134303086188+0.022275757428062377j)
a'1110'b'1000' (0.16794678493064905-0.03114947664777408j)
Sector N = 6 : S_z = 2
a'1111'b'0011' (0.0727726028804495+0.083743496580757j)
a'1111'b'0101' (0.01925093225199306-0.002106816018977831j)
a'1111'b'1001' (0.31904566116544464+0.1848542305946012j)
a'1111'b'0110' (-0.050033044766884575+0.1433287963374998j)
a'1111'b'1010' (-0.20743364999919964-0.036578272708865135j)
a'1111'b'1100' (-0.03640524592649554-0.13848962742091117j)
Sector N = 2 : S_z = 2
a'0011'b'0000' (-0.07722113021842385-0.13612487506548568j)
a'0101'b'0000' (-0.06566054308805284-0.03892913554083794j)
a'1001'b'0000' (0.0416071215590376-0.011788985574841489j)
a'0110'b'0000' (-0.04497972178786085-0.07553324677835324j)
a'1010'b'0000' (0.013314509737629755+0.19931514438967535j)
a'1100'b'0000' (0.050127147725368235+0.12678181117275447j)
Sector N = 4 : S_z = 2
a'0111'b'0001' (0.053003974628875725-0.07779354799498137j)
a'0111'b'0010' (0.13706790597616897+0.08079752531710067j)
a'0111'b'0100' (-0.19608361532605442-0.18657109819037743j)
a'0111'b'1000' (-0.06752691511586532+0.06297264868153246j)
a'1011'b'0001' (-0.2046282097312128-0.020177415327989758j)
a'1011'b'0010' (-0.0002933551288283058-0.09213234405893575j)
a'1011'b'0100' (-0.3623000889330873-0.024425836398408392j)
a'1011'b'1000' (0.2229865169415784-0.08143372736589188j)
a'1101'b'0001' (-0.1632121115712832-0.09455077917116718j)
a'1101'b'0010' (-0.1479177650211721+0.14556545694337827j)
a'1101'b'0100' (0.12033230692058951-0.031798047900992776j)
a'1101'b'1000' (0.18970924573871434-0.23361445155287294j)
a'1110'b'0001' (0.021902144066649754+0.2783346944163582j)
a'1110'b'0010' (-0.07593204892539142-0.11007597008452914j)
a'1110'b'0100' (-0.007740666129738713+0.022189430294107672j)
a'1110'b'1000' (0.16789103518091672-0.03210269553300229j)
Sector N = 6 : S_z = 2
a'1111'b'0011' (0.07325180489807873+0.08325490227395989j)
a'1111'b'0101' (0.01933160941670954-0.002100344934771994j)
a'1111'b'1001' (0.3200222141273427+0.18273708155971605j)
a'1111'b'0110' (-0.04931469762581424+0.14358367097520167j)
a'1111'b'1010' (-0.20743532538169057-0.03519670204820731j)
a'1111'b'1100' (-0.03727463848516889-0.13840697734138802j)

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' (-0.17044188783239003+0.015549713711555089j)
a'0111'b'0101' (-0.016353688142264605-0.023995465888992637j)
a'0111'b'1001' (0.026411679481048472+0.2550148809272792j)
a'0111'b'0110' (-0.01772081263333609+0.2742953249494439j)
a'0111'b'1010' (-0.04089100193038861-0.01830297061912792j)
a'0111'b'1100' (-0.12205984401721448-0.16716321188705718j)
a'1011'b'0011' (0.033665463396725874-0.1003209112144722j)
a'1011'b'0101' (-0.1501205613922792+0.22859967143937296j)
a'1011'b'1001' (0.003508701856054892-0.23585969152969283j)
a'1011'b'0110' (0.13625386157816874+0.01089320887489491j)
a'1011'b'1010' (-0.11993673806176988-0.12168361132300079j)
a'1011'b'1100' (-0.00968137721753541-0.0800905544423775j)
a'1101'b'0011' (0.06694907741954417+0.17089304859301946j)
a'1101'b'0101' (0.044962462688345285-0.03544318445969587j)
a'1101'b'1001' (0.39212411149643445-0.1046844731489852j)
a'1101'b'0110' (-0.00620034572910763+0.15757970448407252j)
a'1101'b'1010' (-0.1259386504469936+0.07623397313399599j)
a'1101'b'1100' (-0.11892287869279454-0.1383306009529141j)
a'1110'b'0011' (-0.08500942391394398+0.008232922893746495j)
a'1110'b'0101' (-0.16558435085004325+0.12400633319679112j)
a'1110'b'1001' (-0.1377530594219843+0.42858683226740063j)
a'1110'b'0110' (-0.05753565790805146+0.13175965416632635j)
a'1110'b'1010' (0.1624780194636811-0.018570133549883197j)
a'1110'b'1100' (-0.07711342108527383+0.009851006881928422j)
Sector N = 5 : S_z = 1
a'0111'b'0011' (-0.1710202561101011+0.006656029177801259j)
a'0111'b'0101' (-0.014789544138353216-0.02498989562331057j)
a'0111'b'1001' (0.007075473571678516+0.256281298539516j)
a'0111'b'0110' (-0.03930145989385365+0.2720429152536757j)
a'0111'b'1010' (-0.03902777829960866-0.021997847470884468j)
a'0111'b'1100' (-0.10311484187986188-0.17946998165151457j)
a'1011'b'0011' (0.039832377729151464-0.09803586250480231j)
a'1011'b'0101' (-0.16695744674384097+0.21660841099020098j)
a'1011'b'1001' (0.024468010590334554-0.23461334475035261j)
a'1011'b'0110' (0.13466752305328392+0.023418689742342604j)
a'1011'b'1010' (-0.10618591046754158-0.1338516896407512j)
a'1011'b'1100' (0.00046010111222115346-0.08067226464332201j)
a'1101'b'0011' (0.05385063810677526+0.1754614538872779j)
a'1101'b'0101' (0.04798856333522345-0.03122403184513678j)
a'1101'b'1001' (0.4009889831822513-0.06267370343144771j)
a'1101'b'0110' (-0.023336938841320643+0.15596536422540913j)
a'1101'b'1010' (-0.13452035602530754+0.059803312207848215j)
a'1101'b'1100' (-0.09750838810436026-0.15417561573045502j)
a'1110'b'0011' (-0.08540642706377388+0.00035410415880262995j)
a'1110'b'0101' (-0.17811265936508652+0.1052218061679255j)
a'1110'b'1001' (-0.19052018666266785+0.4078782130086676j)
a'1110'b'0110' (-0.07405185940399396+0.12323668494305318j)
a'1110'b'1010' (0.1634410558771343+0.005565781406800693j)
a'1110'b'1100' (-0.07766949356701514-0.003312373334325396j)

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' (0.25139984159046225+0.03184001791242112j)
a'010'b'111' (0.0187820893784803+0.12584995187149658j)
a'100'b'111' (-0.38347897273537457+0.025235607240446115j)
Sector N = 4 : S_z = 0
a'011'b'011' (-0.09241282144869979+0.3485493592299831j)
a'011'b'101' (-0.01988836730509329+0.2561304053366742j)
a'011'b'110' (-0.21383354526636622+0.15761848192550731j)
a'101'b'011' (0.20632861330849997+0.2044204204517736j)
a'101'b'101' (-0.09720600632594104+0.32489463838557475j)
a'101'b'110' (-0.3039031715966207+0.059304955359077816j)
a'110'b'011' (-0.11907012730671358-0.20763210824617317j)
a'110'b'101' (0.0765476066812338-0.03344466992698934j)
a'110'b'110' (-0.07943228686678636-0.23224693633398655j)
Sector N = 4 : S_z = 2
a'111'b'001' (-0.06417127930613646+0.025920950538833323j)
a'111'b'010' (-0.032528841111842455-0.19104757064186584j)
a'111'b'100' (-0.192768133090633+0.0777576761346049j)
Sector N = 4 : S_z = -2
a'001'b'111' (0.25139984159046225+0.03184001791242112j)
a'010'b'111' (0.0187820893784803+0.12584995187149658j)
a'100'b'111' (-0.38401766831189976+0.02552411467543587j)
Sector N = 4 : S_z = 0
a'011'b'011' (-0.09241282144869979+0.3485493592299831j)
a'011'b'101' (-0.01988836730509329+0.2561304053366742j)
a'011'b'110' (-0.21383354526636622+0.15761848192550731j)
a'101'b'011' (0.20632861330849997+0.2044204204517736j)
a'101'b'101' (-0.09720600632594104+0.32489463838557475j)
a'101'b'110' (-0.3039031715966207+0.059304955359077816j)
a'110'b'011' (-0.11907012730671358-0.20763210824617317j)
a'110'b'101' (0.0765476066812338-0.03344466992698934j)
a'110'b'110' (-0.07943228686678636-0.23224693633398655j)
Sector N = 4 : S_z = 2
a'111'b'001' (-0.06417127930613646+0.025920950538833323j)
a'111'b'010' (-0.03283645010583314-0.1898709448388153j)
a'111'b'100' (-0.192768133090633+0.0777576761346049j)

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' (0.5908062711142874-0.14725857523270403j)
a'01'b'10' (-0.003471410086752633+0.02161237923081287j)
a'10'b'01' (0.6457522689404723+0.21530201001447957j)
a'10'b'10' (-0.013918280398179678-0.40649606335261473j)
Sector N = 2 : S_z = 0
a'01'b'01' (0.5488705264238126-0.2645790387917878j)
a'01'b'10' (0.0013642586550375846+0.01913723432686206j)
a'10'b'01' (0.6784342247800093+0.011010414872908991j)
a'10'b'10' (-0.17416169432186257-0.3710046632568297j)

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]
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     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)