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 -q fqe --quiet
Print = True
from openfermion import FermionOperator, MolecularData
from openfermion.utils import hermitian_conjugated
import numpy
import fqe
from fqe.unittest_data import build_lih_data, build_hamiltonian
numpy.set_printoptions(floatmode='fixed', precision=6, linewidth=80, suppress=True)
numpy.random.seed(seed=409)

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_ncr(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' (-0.9832017407384288-0.08751592993809962j)
a'000011'b'000101' (-0.038333927909412015-0.0034121477949351545j)
a'000011'b'100001' (-0.003633381455202132-0.00032341158146809j)
a'000101'b'000011' (-0.038333927909406935-0.0034121477949347087j)
a'000101'b'000101' (0.03431518282571416+0.003054434433325243j)
a'000101'b'100001' (-0.05906185803441181-0.005257164690403509j)
a'000101'b'100010' (-0.001073525344578889-9.555564554363788e-05j)
a'001001'b'001001' (0.02705871481472957+0.0024085276354275376j)
a'001001'b'001010' (-0.003108703183821363-0.00027670930255657017j)
a'010001'b'010001' (0.02705871481472957+0.0024085276354275376j)
a'010001'b'010010' (-0.003108703183821363-0.00027670930255657017j)
a'100001'b'000011' (-0.0036333814552024306-0.0003234115814681175j)
a'100001'b'000101' (-0.0590618580344128-0.005257164690403599j)
a'100001'b'100001' (0.11314187441703756+0.010070889930469399j)
a'000110'b'000110' (0.003747532670094563+0.00033357218982965186j)
a'001010'b'001001' (-0.003108703183821363-0.00027670930255657076j)
a'001010'b'001010' (0.0017871363875789756+0.00015907509845870595j)
a'010010'b'010001' (-0.003108703183821363-0.0002767093025565707j)
a'010010'b'010010' (0.0017871363875789756+0.00015907509845870595j)
a'100010'b'000101' (-0.0010735253445789887-9.555564554364532e-05j)
a'100010'b'100010' (0.0010474346607506374+9.323348063182956e-05j)
Sector N = 4 : S_z = 0
a'000011'b'000011' (-0.9832017407384288-0.08751592993809962j)
a'000011'b'000101' (-0.038333927909412015-0.0034121477949351545j)
a'000011'b'100001' (-0.003633381455202132-0.00032341158146809j)
a'000101'b'000011' (-0.038333927909406935-0.0034121477949347087j)
a'000101'b'000101' (0.03431518282571416+0.003054434433325243j)
a'000101'b'100001' (-0.05906185803441181-0.005257164690403509j)
a'000101'b'100010' (-0.001073525344578889-9.555564554363788e-05j)
a'001001'b'001001' (0.02705871481472957+0.0024085276354275376j)
a'001001'b'001010' (-0.003108703183821363-0.00027670930255657017j)
a'010001'b'010001' (0.02705871481472957+0.0024085276354275376j)
a'010001'b'010010' (-0.003108703183821363-0.00027670930255657017j)
a'100001'b'000011' (-0.0036333814552024306-0.0003234115814681175j)
a'100001'b'000101' (-0.0590618580344128-0.005257164690403599j)
a'100001'b'100001' (0.11314187441703756+0.010070889930469399j)
a'000110'b'000110' (0.003747532670094563+0.00033357218982965186j)
a'001010'b'001001' (-0.003108703183821363-0.00027670930255657076j)
a'001010'b'001010' (0.0017871363875789756+0.00015907509845870595j)
a'010010'b'010001' (-0.003108703183821363-0.0002767093025565707j)
a'010010'b'010010' (0.0017871363875789756+0.00015907509845870595j)
a'100010'b'000101' (-0.0010735253445789887-9.555564554364532e-05j)
a'100010'b'100010' (0.0010474346607506374+9.323348063182956e-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' (-0.2560641884009184-0.03361855885930155j)
a'0111'b'0010' (-0.18531953811079951-0.16373971609697777j)
a'0111'b'0100' (-0.15558817792245747+0.5977902174787313j)
a'0111'b'1000' (-0.09409697627948393-0.2650240748167201j)
a'1011'b'0001' (0.012510483985880075+0.11588694112511189j)
a'1011'b'0010' (-0.24250938523576998-0.16862335306311424j)
a'1011'b'0100' (-0.23250182146638484+0.158221072059111j)
a'1011'b'1000' (-0.17392282273811355+0.1455940739281916j)
a'1101'b'0001' (0.15128790458112176-0.11665667087981993j)
a'1101'b'0010' (0.16105827675193568+0.004667024350992057j)
a'1101'b'0100' (0.05394144979209672-0.07932576369597548j)
a'1101'b'1000' (-0.053142303375824186-0.04051304576506321j)
a'1110'b'0001' (0.009490677409496147-0.11626004379977589j)
a'1110'b'0010' (0.12512159229139136+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: (12.630481923700248-9.71445146547012e-17j)
Final Energy:   (12.630481923700263-1.8041124150158794e-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: (141.2704138222322+3.1086244689504383e-15j)
Final Energy:   (141.27041382223277+2.886579864025407e-15j)

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' (-0.24418877035671677-0.4359513076502115j)
a'0101'b'0000' (-0.2087404095670353-0.1247670474297033j)
a'1001'b'0000' (0.1338206317206431-0.036536027310508076j)
a'0110'b'0000' (-0.14208088099568209-0.24188383619281842j)
a'1010'b'0000' (0.038073420915662064+0.6349171766849513j)
a'1100'b'0000' (0.1568031217239613+0.4050736868094568j)
Sector N = 4 : S_z = 2
a'0111'b'0001' (0.065667289481117-0.09475470548834775j)
a'0111'b'0010' (0.16743331046524537+0.09993174484750536j)
a'0111'b'0100' (-0.23895925153409597-0.23029827667709676j)
a'0111'b'1000' (-0.08325761287979849+0.07666002905796641j)
a'1011'b'0001' (-0.2508656879461394-0.026492073437737934j)
a'1011'b'0010' (0.00032693894335967416-0.11282989284647038j)
a'1011'b'0100' (-0.44415883477082774-0.03256230130953867j)
a'1011'b'1000' (0.2739200153566201-0.09814434634331096j)
a'1101'b'0001' (-0.1993582192944168-0.11692870999985559j)
a'1101'b'0010' (-0.18249270064995415+0.17732791167784964j)
a'1101'b'0100' (0.14786163486436185-0.03827044269229047j)
a'1101'b'1000' (0.2344764614847571-0.28502189272242795j)
a'1110'b'0001' (0.024723370555637374+0.3412719626310678j)
a'1110'b'0010' (-0.09230088654198047-0.1354576290541601j)
a'1110'b'0100' (-0.009682098593708626+0.027317594785402195j)
a'1110'b'1000' (0.2059594261188556-0.0381997686762949j)
Sector N = 6 : S_z = 2
a'1111'b'0011' (0.14962168947061893+0.1721780305312678j)
a'1111'b'0101' (0.039580238900613-0.004331648995447881j)
a'1111'b'1001' (0.6559632190189102+0.3800633918892853j)
a'1111'b'0110' (-0.10286877741171899+0.29468640407210506j)
a'1111'b'1010' (-0.4264870560824111-0.0752055408765342j)
a'1111'b'1100' (-0.07484979491614394-0.2847369917895301j)
Sector N = 2 : S_z = 2
a'0011'b'0000' (-0.24650978043349706-0.4341895955941334j)
a'0101'b'0000' (-0.20937972581394831-0.12371481583485076j)
a'1001'b'0000' (0.1332031971877493-0.037424580613475505j)
a'0110'b'0000' (-0.143447663435852-0.240888692482901j)
a'1010'b'0000' (0.042090548068496964+0.634827740537068j)
a'1100'b'0000' (0.15941820546542598+0.4040614429143586j)
Sector N = 4 : S_z = 2
a'0111'b'0001' (0.06492882852351695-0.09559275388194698j)
a'0111'b'0010' (0.16816567068713825+0.09922590103630588j)
a'0111'b'0100' (-0.24060968411556843-0.22880363918723792j)
a'0111'b'1000' (-0.08273618725613303+0.07733048416939783j)
a'1011'b'0001' (-0.2509032478157556-0.024572046712502266j)
a'1011'b'0010' (-0.00035068539071255237-0.1131194747511108j)
a'1011'b'0100' (-0.4442916699997619-0.029943930720641436j)
a'1011'b'1000' (0.27356792849104916-0.09998846521835247j)
a'1101'b'0001' (-0.20022523284081128-0.11608926178024193j)
a'1101'b'0010' (-0.18137606192493846+0.17854603046321618j)
a'1101'b'0100' (0.1475294681453879-0.03883875156453887j)
a'1101'b'1000' (0.2325623467096962-0.2865999756605273j)
a'1110'b'0001' (0.026918612988161505+0.34146090349801655j)
a'1110'b'0010' (-0.0931040513139298-0.13501796473772448j)
a'1110'b'0100' (-0.00951178481074542+0.027052833849691163j)
a'1110'b'1000' (0.20605628528172587-0.03925986293880907j)
Sector N = 6 : S_z = 2
a'1111'b'0011' (0.1506250927572696+0.17121529895649787j)
a'1111'b'0101' (0.03966849180347327-0.004419428533078575j)
a'1111'b'1001' (0.6580760930486189+0.3758759789900442j)
a'1111'b'0110' (-0.10127363882553753+0.2952458106297045j)
a'1111'b'1010' (-0.4266680080737957-0.07247806734509565j)
a'1111'b'1100' (-0.07660485965656404-0.2844522232715525j)

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' (-0.1710202561101012+0.006656029177801262j)
a'0111'b'0101' (-0.01478954413835322-0.024989895623310575j)
a'0111'b'1001' (0.007075473571678512+0.256281298539516j)
a'0111'b'0110' (-0.03930145989385367+0.2720429152536758j)
a'0111'b'1010' (-0.03902777829960868-0.021997847470884475j)
a'0111'b'1100' (-0.10311484187986189-0.1794699816515146j)
a'1011'b'0011' (0.039832377729151464-0.09803586250480231j)
a'1011'b'0101' (-0.166957446743841+0.216608410990201j)
a'1011'b'1001' (0.024468010590334554-0.23461334475035261j)
a'1011'b'0110' (0.13466752305328392+0.023418689742342604j)
a'1011'b'1010' (-0.10618591046754156-0.13385168964075117j)
a'1011'b'1100' (0.0004601011122211552-0.080672264643322j)
a'1101'b'0011' (0.05385063810677526+0.1754614538872779j)
a'1101'b'0101' (0.04798856333522345-0.031224031845136776j)
a'1101'b'1001' (0.4009889831822512-0.06267370343144768j)
a'1101'b'0110' (-0.02333693884132066+0.15596536422540921j)
a'1101'b'1010' (-0.13452035602530754+0.059803312207848215j)
a'1101'b'1100' (-0.09750838810436027-0.15417561573045507j)
a'1110'b'0011' (-0.08540642706377385+0.0003541041588026282j)
a'1110'b'0101' (-0.17811265936508647+0.10522180616792545j)
a'1110'b'1001' (-0.1905201866626678+0.40787821300866756j)
a'1110'b'0110' (-0.07405185940399397+0.1232366849430532j)
a'1110'b'1010' (0.1634410558771343+0.005565781406800696j)
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' (0.30390798276421177+0.03849022160763121j)
a'010'b'111' (0.022704974112154053+0.15213535840863748j)
a'100'b'111' (-0.4635735659147738+0.030506393487579976j)
Sector N = 4 : S_z = 0
a'011'b'011' (-0.06440203545045874+0.24290231417536567j)
a'011'b'101' (-0.01386010421665774+0.17849600505477198j)
a'011'b'110' (-0.1490195337276497+0.10984353579388341j)
a'101'b'011' (0.14378938398839422+0.14245957388110925j)
a'101'b'101' (-0.06774238214202745+0.22641745691735018j)
a'101'b'110' (-0.21178860815906603+0.04132932830693202j)
a'110'b'011' (-0.08297937926453704-0.14469778312467776j)
a'110'b'101' (0.053345646219331536-0.023307424061958057j)
a'110'b'110' (-0.055355965487381587-0.1618517343434305j)
Sector N = 4 : S_z = 2
a'111'b'001' (-0.12666672895403738+0.05116497678766766j)
a'111'b'010' (-0.06420819321126867-0.3771059438035301j)
a'111'b'100' (-0.38050213630121726+0.15348471455665633j)
Sector N = 4 : S_z = -2
a'001'b'111' (0.30390798276421177+0.03849022160763121j)
a'010'b'111' (0.022704974112154053+0.15213535840863748j)
a'100'b'111' (-0.4646383559085599+0.031075970529626587j)
Sector N = 4 : S_z = 0
a'011'b'011' (-0.06440203545045874+0.24290231417536567j)
a'011'b'101' (-0.01386010421665774+0.17849600505477198j)
a'011'b'110' (-0.1490195337276497+0.10984353579388341j)
a'101'b'011' (0.14378938398839422+0.14245957388110925j)
a'101'b'101' (-0.06774238214202745+0.22641745691735018j)
a'101'b'110' (-0.21178860815906603+0.04132932830693202j)
a'110'b'011' (-0.08297937926453704-0.14469778312467776j)
a'110'b'101' (0.053345646219331536-0.023307424061958057j)
a'110'b'110' (-0.055355965487381587-0.1618517343434305j)
Sector N = 4 : S_z = 2
a'111'b'001' (-0.12666672895403738+0.05116497678766766j)
a'111'b'010' (-0.06457992593593184-0.37568283355286103j)
a'111'b'100' (-0.38050213630121726+0.15348471455665633j)

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' (-0.9870501056639676-0.00876298425084819j)
a'000011'b'000101' (-0.0384839713479148-0.0003416589118862754j)
a'000011'b'100001' (-0.003647602935297334-3.238325409440235e-05j)
a'000101'b'000011' (-0.0384839713479097-0.0003416589118862308j)
a'000101'b'000101' (0.03444949641617426+0.0003058410149547989j)
a'000101'b'100001' (-0.059293032957665366-0.0005264007526380628j)
a'000101'b'100010' (-0.0010777272393492372-9.568002262034465e-06j)
a'001001'b'001001' (0.02716462574072948+0.00024116626254639198j)
a'001001'b'001010' (-0.003120871007714629-2.770694732178501e-05j)
a'010001'b'010001' (0.02716462574072948+0.000241166262546392j)
a'010001'b'010010' (-0.003120871007714629-2.770694732178501e-05j)
a'100001'b'000011' (-0.0036476029352976343-3.2383254094405115e-05j)
a'100001'b'000101' (-0.059293032957666365-0.0005264007526380719j)
a'100001'b'100001' (0.11358472476587893+0.0010083998412408193j)
a'000110'b'000110' (0.0037622009416752988+3.340063743457013e-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' (-0.0010777272393493371-9.568002262035203e-06j)
a'100010'b'100010' (0.0010515344463539172+9.335483532366945e-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' (0.5908062711142873-0.147258575232704j)
a'01'b'10' (-0.0034714100867526327+0.021612379230812866j)
a'10'b'01' (0.6457522689404722+0.21530201001447952j)
a'10'b'10' (-0.013918280398179675-0.4064960633526146j)
Sector N = 2 : S_z = 0
a'01'b'01' (0.5488705264238125-0.26457903879178774j)
a'01'b'10' (0.0013642586550375831+0.019137234326862058j)
a'10'b'01' (0.6784342247800091+0.011010414872908991j)
a'10'b'10' (-0.17416169432186251-0.37100466325682957j)

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]]
(-0.07375405619488316-0.0006547850894203786j)

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
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   [-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)