QURI Chemistry: a Qiskit Function by QunaSys

Note

Qiskit Functions are an experimental feature available only to IBM Quantum™ Premium Plan users. They are in preview release status and subject to change.

Overview

This function helps you solve the quantum chemistry ground state estimation problem by using one of two algorithms based on the Quantum Selected Configuration Interaction (QSCI) algorithm:

QSCI
Optimization-based QSCI

The overall procedure for using this function is summarized in the following flow chart:

Function description

The QSCI algorithm samples from a specific ansatz on quantum computers. Each sampled bitstring corresponds to an electron configuration, and the number of times a bitstring is sampled represents the importance of that electron configuration. Choose a number, $R$ , to select the $R$ most important electron configurations from the sample. The function constructs and diagonalizes the subspace Hamiltonian, depending on your choice of $R$ . The smallest eigenvalue is called "QSCI energy" and is the estimation of the QSCI algorithm's true ground state energy.

One important QSCI ingredient is its initial state preparations. Several possible initial state preparations are provided in this function. These include the hardware efficient, UCCSD, kUpCCGSD, and kuCJ ansatzes. A CCSD amplitude-based double excitation ansatz is provided specifically to be executed on near-term devices. Ansatz-specific settings are also provided for you to customize according to your needs.

In addition to returning the QSCI energy, the function also computes the estimated ground state wave function to help you identify the important electron configurations.

Get started

First, authenticate using your IBM Quantum™ API token and select the Qiskit Function as follows:

from qiskit_ibm_catalog import QiskitFunctionsCatalog
 
catalog = QiskitFunctionsCatalog()
 
function = catalog.load("qunasys/quri-chemistry")

Inputs

The function is called with the following arguments:

function.run(
    method = ...,               # Allow only "QSCI" or "OPT_QSCI"
    molecule = ...,             # The molecule
    circuit_options = {...},    # Options for customizing the circuits
    qsci_setting = {...},       # Parameters of the QSCI algorithm
    mitigation_setting = {...}  # Specifies the error mitigation method
    max_iter = ...              # Maximum number of iterations when method = "OPT_QSCI".
    instance = ...              # The instance to use.
    backend_name = ...          # The backend name to use.
)

Each argument is described in following table.

Name	Type	Description	Required	Default	Example
method	str	The algorithm name	Yes	-	“QSCI” or "OPT_QSCI"
mole	json	The molecule setting.	Yes		The molecule setting. See the examples that follow the table.
circuit_options	json	The settings for the circuit.	Yes		See the examples following the table.
qsci_setting	dict[str, int]	Specify the number of shots and the size of the subspace. See the following section for details.	Yes		See the QSCI setting table
mitigation_setting	json	Settings for configuring error mitigation.	No	None	See the mitigation section.
max_iter	int	The maximum number of iterations when the method is "OPT_QSCI"	No	2000	5
backend_name	str	The backend name to use	No	The least busy one available	“ibm_torino”
instance	str	The instance to use	No	"hub1/group1/project1"

Note

Refer to the Specify an instance in your code guide to learn more.

`molecule`

Specify the details about the molecule here. The input is the same as the pyscf.gto.M with an additional active_space option. Detailed settings for configuring a molecule are summarized in the following table.

Name	Type	Description	Required	Default	Example
atom	str	The list of atom coordinates	Yes	-	“H 0 0 0; H 0 0 1”
basis	str	The basis set to represent the electronic wave function. Allowed values are the same as those supported by PySCF	No	“sto-3g”	“sto-3g”
spin	float	The $s_z$ quantum number of the molecule. Only spin=0.0 is supported	No	0.0	0.0
charge	int	The total charge of the molecule	No	0	0
active_space	json	The active space you want to choose. Review the “Active space” table for more information	No	None	Review the “Active space” table for more information

The active space settings are summarized below.

Name	Type	Description	Required	Default	Example
n_active_ele	int	The number of active electrons	Yes	-	4
n_active_orb	int	The number of active spatial orbitals	Yes	-	3
active_orbs_indices	list[int]	The list of active spatial orbital indices	No	None	[0, 1, 2]

The `circuit_options` field

The circuit_options field should be a dictionary containing detailed settings of the circuit for running the QSCI algorithm - for example, the state preparation settings. In general it should take this form:

"circuit_options": {
    "ansatz": ...,
    "state_prep_method": ...,
    "ansatz_setting": ...,
}

Name	Type	Description	Required	Default	Example
ansatz	str	The name of the ansatz to use. Only “DoubleExcitation” and “UCCSD” are supported. `DoubleExcitation` is highly recommended to obtain a stable result, especially for systems larger than eight qubits.	Yes		“DoubleExcitation”	Yes	“DoubleExcitation”
state_prep_method	str	How to prepare the circuit parameter of the ansatz. Only CCSD is allowed.	No	“CCSD"	“CCSD”
ansatz_setting	str	Specific setting to configure the ansatz with	No	None	See the examples following the table.

Each field is explained in the subsequent sections.

`ansatz`, `state_prep_method` and `ansatz_setting`

The ansatz string specifies the ansatz to use. This specifies a type of parametric circuit without specifying what the concrete circuit parameters are. state_prep_method specifies how the circuit parameter is prepared. Finally, the ansatz_setting lets you customize the ansatz.

Ansatz	Preparation method	Ansatz Setting Default	Note
Double Excitation	CCSD	`{n_amplitudes: 10}`	Lets you optimize from the result in the last example.
UCCSD	CCSD (default), Random	`{use_singles: True, n_trotter: 1, reduce_parameter: True}`	Not recommended for systems with eight or more qubits.
KuCJ	Random	`{k: 1}`	Recommended for system up to 16 qubits
KUpCCGSD	Random	`{k: 1, n_trotter: 1, reduce_parameter: False}`
Hardware Efficient	Random	`{n_layers: depends on qubit size}`

Caution

When method is "QSCI", only the "CCSD" preparation method is supported. The "Random" preparation method is only supported when method = "OPT_QSCI".

Note

Detailed settings for UCCSD ansatz can be found in the QURI Parts documentation. The reduce_parameter option is the same as the singlet_excitation option in the QURI Parts documentation.

Note

Detailed settings for KUpCCGSD ansatz can be found in the QURI Parts documentation.

`qsci_setting`

Name	Type	Description	Required	Example
n_shots	int	The total number of shots to sample from the ansatz	Yes	100000
number_of_states_pick_out	int	The size of the subspace Hamiltonian is $R$ . It affects the classical resource, where classical diagonalization of the $R \times R$ subspace Hamiltonian is performed if the number of bitstrings ( $n$ ) sampled from the ansatz is larger than $R$ . Otherwise, the $n \times n$ subspace matrix is diagonalized.	Yes	10000

`mitigation_setting`

Usually, the problem at hand needs to respect some symmetry, especially particle number and spin conservation. To restore symmetry, use symmetry post-selection or configuration recovery error mitigation. You can choose either of them with the mitigation_setting field in either algorithm. Symmetry post-selection is the default value.

post-selection removes the erroneous states completely, leaving only the basis states to construct the subspace Hamiltonian. This usually saves classical resources, but might miss many states. Configuration recovery attempts to flip individual bits in erroneous basis states with respect to some probability distribution. Using configuration recovery increases the number of basis states to build a larger subspace Hamiltonian, but it uses a lot more classical resources.

The options are explained in the following table.

Name	Type	Description	Required	Default
particle_number_symmetry	bool	If True, remove bitstrings that do not respect particle number conservation from sampling result.	No	True
spin_symmetry	bool	If True, remove bitstrings that do not respect $s_z$ conservation from the sampling result.	No	True
configuration_recovery	Union[bool, dict[str, int]]	Activate configuration recovery if set to a value other than False. Refer to the configuration recovery section for dict[str, int] input	No	False

Caution

You cannot set spin symmetry to True when particle symmetry is False.

Symmetry post-selection

These keys control the symmetry post-selection: particle_number_symmetry and spin_symmetry. Setting particle_number_symmetry to True selects those states that conserve the total electron numbers. Setting spin_symmetry to True selects those states that conserve the $s_z$ quantum number. Currently only $s_z = 0$ is supported. The default is:

mitigation_setting = {
    "particle_number_symmetry": True,
    "spin_symmetry": True,
}

Configuration recovery

Instead of the symmetry post-selection, you can choose to do configuration recovery. To turn it on, set it to True, and optionally customize it with a dictionary containing the following parameters:

Name	Type	Description	Required	Default
n_recover_iteration	int	The number of recovery iterations.	No	5
n_batch	int	The number of batches generated per iteration.	No	1
number_of_states_pick_out	int or None	The dimension of the QSCI vector in the recovery process. If None, it is set to the same value that was used for QSCI.	No	None
recovery_r_multiple	int	The multiplier for number_of_states_pick_out. This is the number of states selected from the raw sample.	No	10

# Use the default configuration recovery setting
default_mitigation_setting = {"configuration_recovery": True}
 
# Customized configuration recovery setting
customized_mitigation_setting = {
    "configuration_recovery": {
        "n_recover_iteration": 20,
        "n_batch": 10,
        "number_of_states_pick_out": 5000,
    }
}

Caution

Other error mitigations are not allowed when configuration recovery is turned on.

Outputs

The output of the function contains these fields:

Name	Type	Description
qsci_energy	float	The QSCI energy that acts as the estimation of the ground state energy.
state_vector	dict	A dictionary representing the QSCI estimated ground state vector.

The state_vector field is a dictionary containing these fields:

Name	Type	Description
bits	list[int]	A list of integers representing the electron configurations
reals	list[float]	Real parts of the amplitudes of the electron configurations at the corresponding positions
imags	list[float]	Imaginary parts of the amplitudes of the electron configurations at the corresponding positions

Example

This example shows you how to compute the ground state energy of the water molecule using the QSCI algorithm.

Set up the molecule

First, configure the setting for a $H_2O$ molecule in JSON format. The coordinates of the water molecule can be loaded from OpenFermion.

molecule = {
    "atom": "O 0 0 0; H 0.2774 0.8929 0.2544; H 0.6068 -0.2383 -0.7169",
    "basis": "6-31g",  # default to "sto-3g"
    "spin": 0.0,  # default to 0.0. Current code forces spin = 0.0
    "charge": 0,  # default to 0
    "active_space": {  # default to None. Equivalent to no active space chosen.
        "n_active_ele": 10,
        "n_active_orb": 10,
        "active_orbs_indices": [i for i in range(10)],
    },
}

Execute algorithms

All the function input is in JSON format. You will set up a JSON string that specifies the options for running an algorithm. As stated previously, the two types of algorithms provided are "QSCI" and "OPT_QSCI".

QSCI

First, set up the number of shots and the maximum subspace Hamiltonian size with the qsci_setting field.

qsci_setting = {"n_shots": 1e5, "number_of_states_pick_out": 12000}

Next, set up the ansatz. This example uses the DoubleExcitation as the ansatz because it carries the most non-trivial information with the lowest circuit depth. You can specify how many excitation amplitudes to include with the n_amplitudes key in the ansatz_setting field. The default value is 10. The function sorts the CCSD amplitudes according to their magnitude and chooses the largest n_amplitudes to construct the ansatz. An example JSON dictionary using this ansatz follows:

qsci_double_exc_json = {
    "ansatz": "DoubleExcitation",
    "state_prep_method": "CCSD",  # This is optional with a default of "CCSD".
    "ansatz_setting": {
        "n_amplitudes": 20
    },  # This is optional with a default of {"n_amplitudes": 10}.
}

Turn on the configuration recovery algorithm for error mitigation.

mitigation_setting = {  # Refer to the "Error mitigation" section for details.
    "configuration_recovery": {"number_of_states_pick_out": 10000}
}

Finally, execute the function.

job = function.run(
    method="QSCI",
    molecule=molecule,
    circuit_options=qsci_double_exc_json,
    qsci_setting=qsci_setting,
    mitigation_setting=mitigation_setting,
    backend_name="ibm_torino",
)

Print out the result from the run:

result = job.result()
print(result)

Output:

{'qsci_energy': -76.05069405129802, 'state_vector': {'bits': [1023, 131583, 66303, 2943, 9087, 9183, 5103, 3039, 9207, 525183, 5055, 525279, 263103, 2031, 3063, 33663, 263151, 133503, 1983, 17391, 139647, 33759, 17343, 5115, 525303, 12543, 6399, 133599, 655743, 139743, 2043, 68079, 196863, 655839, 164223, 3327, 33783, 263163, 264447, 24831, 133623, 164319, 528639, 11103, 70383, 141663, 527199, 37119, 68031, 9471, 18687, 74223, 270591, 328383, 17403, 70335, 67263, 533343, 135663, 139767, 786687, 74175, 655863, 68223, 74367, 14463, 67311, 540927, 525567, 11127, 135615, 328431, 663903, 590271, 6975, 49407, 11223, 590319, 82671, 164343, 35679, 13263, 13215, 70395, 132591, 132543, 98799, 135903, 393663, 295167, 135807, 41823, 68091, 6063, 172383, 657759, 267183, 536703, 7023, 13167, 557919, 132831, 74487, 147903, 7119, 99039, 74463, 82623, 13119, 166239, 3903, 14559, 393855, 3999, 527319, 197583, 529359, 268479, 529263, 264111, 37743, 147951, 35703, 68319, 530655, 265023, 527223, 271263, 590559, 10095, 274671, 197487, 533463, 68343, 6075, 7071, 198879, 67323, 21423, 272511, 141783, 529311, 788607, 28911, 10143, 198783, 590583, 143727, 34047, 529215, 197535, 21483, 265119, 536799, 526287, 25503, 393711, 395583, 132603, 268527, 137631, 557943, 527487, 45183, 74235, 7143, 50031, 141687, 533367, 271167, 37839, 41943, 7035, 41847, 137583, 76239, 148191, 279471, 787407, 13179, 10191, 6123, 530559, 657783, 541503, 132735, 204927, 98751, 7131, 526143, 137535, 26751, 39039, 328443, 18351, 271311, 541599, 25455, 13275, 295791, 14583, 549087, 271215, 135927, 7095, 148095, 663927, 28863, 541551, 98943, 274623, 548991, 82683, 721119, 267195, 21435, 135675, 4047, 200943, 265167, 25551, 19407, 13551, 265071, 664023, 3951, 264123, 794751, 10047, 590463, 37791, 458943, 11487, 295743, 393951, 19311, 25407, 197499, 688479, 99063, 86703, 558039, 303231, 787263, 393975, 137823, 26847, 22767, 143823, 19359, 264171, 76191, 148215, 272607, 39135, 393723, 147963, 57471, 7407, 50079, 526239, 76383, 13503, 13239, 794847, 788703, 145695, 787311, 11391, 590331, 527583, 592335, 787359, 49983, 205023, 271287, 35799, 542943, 98811, 45279, 19323, 197607, 51327, 19383, 4059, 172503, 529275, 529371, 295887, 332463, 37863, 57567, 35967, 267243, 10203, 10215, 72303, 140607, 344751, 140703, 526299, 71343, 790719, 134559, 78255, 271335, 76095, 37695, 25575, 401727, 721023, 166263, 11511, 541659, 529335, 657879, 69183, 659823, 143679, 535383, 526191, 395679, 536823, 72111, 598479, 265083, 78399, 42111, 106911, 18363, 917919, 529383, 15183, 542847, 140895, 591279, 13287, 592287, 197595, 295839, 401871, 533727, 198903, 3963, 200955, 72255, 330399, 78543, 279531, 50127, 197439, 39159, 688503, 34671, 10227, 819423, 659835, 53487, 529599, 852543, 395727, 592191, 271551, 26871, 561375, 134511, 299199, 12063, 598335, 76143, 137679, 299247, 45303, 34767, 279483, 37851, 100767, 401775, 10107, 34719, 47199, 790767, 794871, 106959, 137847, 69039, 7359, 271323, 206943, 397887, 137595, 19263, 168399, 530679, 140751, 265191, 156063, 7155, 541671, 787431, 34623, 541647, 13563, 156015, 533631, 397935, 140655, 115119, 149967, 72351, 4023, 273183, 229599, 86763, 53439, 197559, 10167, 529647, 143739, 172407, 561279, 592479, 573663, 280767, 330303, 917823, 107103, 598623, 803055, 688599, 45855, 78315, 27471, 665943, 69279, 200895, 271599, 303327, 541887, 55503, 803007, 100791, 100839, 787323, 83631, 672063, 136623, 84687, 273231, 397983, 37815, 659919, 106863, 197871, 529395, 659775, 25467, 75183, 526263, 132855, 541623, 594639, 100671, 174423, 18411, 43863, 395631, 796767, 100815, 297087, 296127, 581727, 667983, 102831, 594543, 143967, 143835, 401823, 541935, 168351, 402015, 286911, 15135, 75423, 143847, 531615, 134463, 148911, 265407, 45903, 90735, 789279, 4083, 917967, 134607, 149823, 656751, 71355, 72171, 205047, 787419, 166359, 398031, 199455, 134751, 8079, 229503, 69327, 622959, 201615, 78567, 535647, 397743, 180831, 276543, 199527, 12111, 140763, 149871, 38079, 38127, 149919, 594495, 156255, 592239, 265179, 544959, 156111, 534351, 4071, 659931, 37755, 180687, 594351, 14127, 787695, 526203, 230223, 332475, 84639, 137691, 336303, 156087, 672159, 563295, 531231, 78519, 660063, 545007, 13299, 460863, 25851, 656859, 42207, 145743, 561399, 296175, 819327, 276687, 50415, 287631, 787383, 622911, 329403, 106983, 38703, 180543, 90687, 275367, 672207, 99759, 84591, 329391, 598491, 25563, 25527, 541563, 266127, 295803, 787647, 852399, 549711, 149943, 202863, 394815, 286959, 723039, 403791, 592347, 76155, 14223, 600351, 51447, 180639, 269115, 592359, 395739, 168303, 656991, 136815, 332523, 14235, 149991, 143775, 28923, 29583, 275343, 136767, 15303, 303903, 152271, 22719, 297183, 426447, 623055, 84543, 271611, 330159, 274683, 76503, 110991, 29487, 75243, 533751, 537375, 51423, 42783, 295863, 265203, 19431, 690519, 268539, 526311, 107127, 7419, 47223, 136827, 265143, 23439, 134583, 151983, 336447, 176463, 537663, 25839, 592599, 55407, 368943, 271227, 271347, 428367, 143991, 656703, 199503, 137703, 788727, 573687, 136683, 180591, 537447, 58191, 78447, 721143, 72423, 59487, 917871, 297807, 721695, 527607, 197823, 336543, 672111, 534303, 925983, 103023, 395991, 280815, 672231, 115407, 34779, 19419, 34791, 25791, 623007, 71403, 295911, 209055, 100731, 399663, 528159, 598743, 57591, 545583, 674079, 72123, 207063, 134619, 919839, 36063, 598455, 598383, 75471, 39711, 692559, 80271, 307359, 426303, 553071, 680271, 27423, 266031, 395703, 201531, 221343, 108831, 165471, 606951, 276639, 336495, 394911, 330447, 15195, 106815, 796791, 458991, 268203, 15471, 530319, 594615, 8091, 281391, 69231, 795807, 136863, 303351, 531291, 330351, 551007, 795423, 297759, 198543, 265455, 594591, 656847, 75195, 50103, 410031, 426351, 34743, 19647, 14247, 9213, 526323, 138543, 574287, 209103, 265467, 538719, 209007, 205599, 657111, 143859, 15423, 288927, 789327, 659871, 90783, 426591, 80463, 805023, 796887, 467007, 272631, 206031, 50043, 38139, 361023, 72411, 661851, 807087, 100071, 76407, 562119, 434511, 459003, 106971, 273207, 142623, 560223, 146007, 233679, 558303, 15219, 542967, 23343, 543519, 266043, 22779, 361167, 45975, 205647, 152187, 15207, 90555, 197883, 90543, 39759, 152223, 280827, 213231, 721755, 103035, 541683, 100959, 275403, 337323, 37875, 52059, 156027, 725103, 269199, 528207, 573567, 723159, 426399, 202935, 725055, 397947, 551127, 789399, 819999, 917943, 145719, 165591, 203535, 152175, 144087, 84399, 594663, 180711, 39735, 275631, 12231, 667935, 78495, 661791, 403743, 156123, 467055, 537543, 115311, 104751, 106875, 55359, 170319, 170271, 150111, 344763, 659895, 623079, 624927, 90747, 538743, 799503, 215103, 103095, 460959, 537399, 108879, 819447, 42831, 591471, 92583, 46143, 100719, 530379, 208959, 215199, 336615, 168315, 27855, 852687, 152127, 795543, 330171, 606831, 72375, 549663, 119439, 727119, 558327, 591291, 76251, 165279, 141015, 84711, 19443, 115131, 299919, 72399, 791439, 205659, 151995, 230175, 50991, 76263, 410223, 75495, 102975, 205767, 7983, 3069, 280491, 295923, 658767, 205719, 737391, 753759, 168543, 919887, 199575, 537435, 631119, 44127, 313551, 77391, 534483, 550047, 405903, 168663, 27483, 598503, 558207, 789663, 672219, 137655, 656799, 46239, 103119, 237687, 402039, 600399, 137715, 566103, 395643, 311535, 69099, 594651, 672375, 395871, 273471, 205983, 528327, 137943, 155967, 919911, 23403, 165327, 199887, 792807, 26511, 182631, 36087, 788427, 168411, 599631, 394959, 661911, 275247, 406047, 26523, 531519, 202983, 798951, 614943, 689487, 591423, 167199, 336507, 399903, 80535, 8139, 606927, 231543, 134523, 303951, 410175, 276591, 304083, 59511, 394683, 26475, 140775, 459663, 330423, 918111, 198603, 198555, 269103, 197619, 598515, 58143, 75483, 467127, 213243, 188775, 721815, 136887, 661863, 156135, 463035, 410295, 401787, 398007, 606699, 45879, 14187, 795591, 50163, 791451, 201675, 12123, 795471, 83643, 34683, 165231, 51999, 983199, 206967, 659943, 73131, 201519, 403767, 270219, 658719, 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-9.541030732486924e-05, 3.054619574173801e-05, 1.5463671325285635e-11, -3.055938739998159e-05, -1.5511514017007016e-11, 2.800773299641902e-06, -1.225924088156057e-09, 1.2259156790368839e-09, -0.0005968716260040485, -0.0003456669493577969, -3.139409719589229e-11, 9.35719448667909e-05, 3.7118896006081897e-11, -5.28999292918467e-11, 9.179053453559835e-05, 8.271246575612983e-11, -0.00018520634609095805, 2.93214576683212e-11, -8.552043342278602e-12, 6.510877733915124e-11, -0.0002543117467145152, -0.0001102736957661947, 2.460199771739412e-11, -1.4401170899244096e-11, 0.0001455337804541825, 3.111041943949224e-05, -7.231848537307431e-05, 7.898294807369167e-05, -3.406193510163393e-05, -5.619476670428404e-11, -1.2615077012048177e-10, 5.3669429646808315e-05, -9.226870467319667e-11, -0.0001324648281067364, 3.3323854367918714e-11, 0.0001035639593142253, 1.6300552906960716e-11, -5.090580064638691e-05, 1.4418135710290948e-10, -3.415568415993609e-05, 1.1200352371399826e-10, -1.0394601001926161e-10, -6.965905933514025e-05, 8.683087987951588e-05, 2.3590340314827922e-11, -8.83009318030425e-12, -8.227757761036788e-11, 0.00019779461689454795, 4.447820677870685e-05, 3.481725572629829e-12, 0.00036986967539154753, 7.649649849257777e-11, -2.543941723581303e-11, -0.00029624206392609803, -7.365339771504106e-05, -1.900358756888232e-12, 8.313348589747688e-11, -0.00018530954529882013, -5.415721841943634e-11, 9.16872557835155e-05, 0.00017327734434793268, -7.995185645316653e-12, 3.9013065524794406e-05, 2.956480684691724e-11, -1.1754155071978494e-06, 6.511078410262327e-11, 0.0002541508394766667, -2.5835296535610352e-11, -3.0973420149183026e-11, 9.350925802693954e-05, 1.6427138631033757e-10, -5.622474169791539e-11, 3.130275523077011e-05, -7.224433345424201e-05, 4.476405755644026e-11, -6.47947877409895e-11, 4.985428486805699e-11, 1.5295196602768335e-05, -1.2971929152863143e-05, -6.880885111009792e-05, 1.1265160281421642e-10, -6.959203118452615e-05, 1.44489601425176e-10, -3.416023080320932e-05, 3.183761839071842e-05, -0.00011017450685604003, -3.452561549936053e-05, 9.287950550354598e-11, -0.0001979984788783047, -4.3300178181601385e-11, 1.1725336812655948e-11, -7.369880523585031e-05]}}

Use the "qsci_energy" key to see the QSCI energy:

CASCI_ENERGY = -76.0508056368444  # Exact result
qsci_energy = result["qsci_energy"]
print(f"CASCI energy: {CASCI_ENERGY} Ha")
print(f"QSCI energy: {qsci_energy} Ha")
print(f"Energy error: {(qsci_energy - CASCI_ENERGY) * 1000: .2e} mHa")

Output:

CASCI energy: -76.0508056368444 Ha
QSCI energy: -76.05069405129802 Ha
Energy error:  1.12e-01 mHa

The returned ground state energy estimation of an $\text{H}_2 \text{O}$ molecule is very close to the exact result!

The state_vector field contains three values: bits, reals, and imags. The bits value is a list of integers representing electron configurations. The $i$ -th position of the reals (imags) field represents the real (imaginary) part of the amplitude of the electron configuration on position $i$ . You can sort it by magnitude and print out the top 10 most important electron configurations.

import numpy as np
 
amps = np.array(result["state_vector"]["reals"]) + 1j * np.array(
    result["state_vector"]["imags"]
)
sort_idx = np.argsort(np.abs(amps))
 
for i, (bit, real, imag) in enumerate(
    zip(
        np.array(result["state_vector"]["bits"])[sort_idx][::-1],
        np.array(result["state_vector"]["reals"])[sort_idx][::-1],
        np.array(result["state_vector"]["imags"])[sort_idx][::-1],
    )
):
    if i > 20:
        break
    print(
        f"Electron configuration: {bin(bit)[2:].zfill(20)}, "
        f"amplitude: {real + 1j * imag: .3e}, "
        f"magnitude: {real**2 + imag**2: .3e}"
    )

Output:

Electron configuration: 00000000001111111111, amplitude: -6.114e-01+7.731e-01j, magnitude:  9.715e-01
Electron configuration: 00110000000011111111, amplitude:  3.561e-02-4.503e-02j, magnitude:  3.296e-03
Electron configuration: 00000011001111001111, amplitude:  2.643e-02-3.341e-02j, magnitude:  1.815e-03
Electron configuration: 00001100001111001111, amplitude:  1.789e-02-2.262e-02j, magnitude:  8.315e-04
Electron configuration: 00100001000111101111, amplitude: -1.717e-02+2.171e-02j, magnitude:  7.662e-04
Electron configuration: 00010010001011011111, amplitude: -1.713e-02+2.166e-02j, magnitude:  7.623e-04
Electron configuration: 00000001101101101111, amplitude: -1.577e-02+1.994e-02j, magnitude:  6.463e-04
Electron configuration: 00000010011110011111, amplitude: -1.575e-02+1.992e-02j, magnitude:  6.450e-04
Electron configuration: 11000000001111001111, amplitude:  1.548e-02-1.957e-02j, magnitude:  6.228e-04
Electron configuration: 00000000111100111111, amplitude:  1.536e-02-1.942e-02j, magnitude:  6.129e-04
Electron configuration: 00010001001011101111, amplitude:  1.417e-02-1.791e-02j, magnitude:  5.216e-04
Electron configuration: 00100010000111011111, amplitude:  1.404e-02-1.775e-02j, magnitude:  5.120e-04
Electron configuration: 00000000111111001111, amplitude:  1.291e-02-1.632e-02j, magnitude:  4.329e-04
Electron configuration: 00010000101011110111, amplitude: -1.267e-02+1.602e-02j, magnitude:  4.172e-04
Electron configuration: 11000000001100111111, amplitude:  1.266e-02-1.601e-02j, magnitude:  4.166e-04
Electron configuration: 00100000010111111011, amplitude: -1.260e-02+1.593e-02j, magnitude:  4.124e-04
Electron configuration: 10000001001101101111, amplitude: -1.225e-02+1.549e-02j, magnitude:  3.902e-04
Electron configuration: 10010000001001111111, amplitude: -1.222e-02+1.545e-02j, magnitude:  3.878e-04
Electron configuration: 01000010001110011111, amplitude: -1.217e-02+1.538e-02j, magnitude:  3.846e-04
Electron configuration: 01100000000110111111, amplitude: -1.215e-02+1.536e-02j, magnitude:  3.835e-04
Electron configuration: 00001001001111001111, amplitude: -1.141e-02+1.442e-02j, magnitude:  3.381e-04

Optimization-based QSCI

If you don't want to use CCSD as the state preparation method, or if you want to optimize further from the CCSD initial state, you can use the optimization-based QSCI. This algorithm gives you access to more chemistry-inspired ansatz with random initial parameters. For this algorithm, the QSCI energy is used as the cost function for the COBYLA optimizer. In each iteration, the circuit parameters are updated and the QSCI energy is evaluated with the circuit carrying the new set of circuit parameters.

The JSON dictionary used to run this algorithm is similar to that of QSCI. It only contains one additional field, "max_iter", which lets you limit the resource usage. The default is 2000 iterations.

Example of running this algorithm with the 1-uCJ ansatz:

opt_qsci_1ucj_json = {
    "ansatz": "KuCJ",
    "state_prep_method": "RANDOM",  # This is optional with a default of "RANDOM" for "KuCJ".
    "ansatz_setting": {
        "k": 1
    },  # This is optional with the default described in the previous table.
}

opt_job = function.run(
    method="OPT_QSCI",
    molecule=molecule,
    parameters=opt_qsci_1ucj_json,
    qsci_setting={"n_shots": 1e5, "number_of_states_pick_out": 5e4},
    mitigation_setting={
        "configuration_recovery": {"number_of_states_pick_out": 10000}
    },
    max_iter=5,
    backend_name="ibm_strasbourg",
)

Print out the result.

opt_result = opt_job.result()
print(opt_result)

Output:

{'qsci_energy': -76.05051672600099, 'state_vector': {'bits': [1023, 1983, 2943, 33783, 3039, 263151, 33663, 2031, 5103, 9087, 525183, 33759, 2043, 17343, 9207, 9213, 525279, 17406, 131583, 5115, 525303, 263163, 525309, 17391, 9183, 3063, 263103, 66303, 3069, 2046, 17403, 263166, 5118, 33789, 5055, 67323, 133503, 70335, 264123, 655863, 67263, 279471, 139743, 6138, 67311, 655743, 133599, 139647, 11127, 164223, 267195, 527223, 18411, 18426, 533343, 558039, 82671, 21486, 264111, 267183, 11223, 328443, 328446, 70383, 328431, 35799, 533373, 6063, 527199, 11103, 6126, 21483, 533469, 18363, 279534, 67326, 41847, 533463, 279486, 655839, 533367, 133629, 70398, 11229, 21498, 558045, 11133, 82683, 21423, 41823, 139767, 527319, 35679, 557919, 264126, 164343, 10095, 35703, 267243, 655869, 267198, 21435, 328383, 279531, 527325, 271356, 6075, 18366, 83631, 164349, 264171, 557943, 271311, 83691, 139773, 527229, 6078, 133623, 13215, 82686, 6123, 68319, 41949, 279546, 35805, 21438, 18414, 270591, 279483, 18351, 271323, 13287, 164319, 70395, 264186, 533493, 41853, 557949, 41973, 529371, 82623, 6975, 541563, 541683, 529359, 7038, 196863, 6399, 10215, 529374, 86703, 541659, 527349, 267258, 264174, 295923, 265086, 535389, 141789, 98751, 34779, 295911, 265182, 529263, 25503, 98811, 329391, 41943, 688479, 35829, 283563, 271326, 132591, 393663, 43863, 34671, 37869, 787311, 295167, 13167, 786687, 558069, 50139, 295806, 526206, 74487, 25470, 43893, 172383, 4077, 295899, 135663, 99039, 141663, 37791, 25575, 197595, 657783, 332463, 265143, 74367, 166239, 525567, 4071, 535413, 25455, 264447, 7035, 141693, 11253, 132831, 529215, 10206, 68091, 166365, 657789, 68349, 271230, 526311, 7158, 688599, 10173, 13299, 590271, 3966, 19422, 7023, 19311, 74223, 34743, 10236, 34047, 83646, 19437, 268266, 267246, 25587, 271227, 271167, 13182, 529389, 526263, 657909, 529311, 19323, 4062, 83706, 265203, 344766, 271599, 393723, 3327, 132855, 43869, 25596, 50043, 132735, 529395, 74463, 13278, 4086, 663927, 172413, 7161, 10167, 68223, 787446, 529401, 141687, 83643, 529404, 10227, 35709, 295791, 265197, 540927, 4047, 526287, 37839, 657759, 332478, 526326, 265071, 10203, 135678, 295926, 148191, 590463, 529275, 135933, 526143, 664029, 526239, 172509, 787449, 787383, 50127, 590583, 13302, 13119, 664053, 174423, 329403, 19263, 283566, 541503, 787326, 528639, 332538, 559965, 265023, 197439, 198459, 271287, 50079, 135675, 12543, 559989, 541629, 4089, 172503, 37758, 271347, 37755, 329406, 7149, 788427, 590319, 690549, 74493, 229503, 295887, 665973, 345834, 10047, 37119, 197499, 68094, 265212, 393726, 98799, 135615, 34803, 4083, 10107, 657885, 541677, 34623, 68343, 535383, 4023, 86778, 166263, 197487, 37881, 265167, 135807, 590559, 787443, 4029, 3903, 332523, 688605, 7143, 7101, 7155, 25593, 74235, 13179, 50031, 13308, 143838, 268218, 271350, 295932, 37851, 34683, 90870, 348843, 401790, 541566, 688503, 541689, 526203, 268206, 663933, 25407, 99069, 19407, 7134, 50172, 529278, 3963, 197598, 344811, 19359, 86763, 265206, 25467, 688629, 7095, 135903, 526332, 22446, 197613, 197622, 787407, 295929, 526323, 19431, 34782, 197625, 545595, 19443, 10191, 394941, 666069, 265179, 295863, 25590, 37854, 34809, 526269, 37863, 137583, 9471, 7131, 426363, 83694, 25551, 25527, 147966, 134589, 24831, 795483, 13263, 38778, 426459, 295743, 197607, 50142, 529341, 332526, 344763, 271335, 393975, 688509, 348906, 280494, 37815, 141783, 690519, 197535, 526299, 393855, 344814, 787437, 787263, 295803, 787359, 590334, 132603, 19326, 529398, 663903, 98814, 393711, 136767, 530379, 198606, 52089, 3999, 696693, 329454, 34686, 3951, 132861, 787422, 143967, 590589, 287547, 34791, 140727, 50046, 528183, 36723, 51150, 656823, 533727, 86715, 71418, 19389, 397803, 71355, 541551, 295839, 49407, 147963, 172533, 199605, 213918, 787323, 348858, 10143, 25533, 271215, 271293, 592239, 283578, 49983, 37875, 34749, 50109, 7164, 656871, 19419, 10233, 197502, 271263, 134523, 690645, 591603, 541647, 329466, 397806, 344751, 166269, 143805, 265083, 103023, 46035, 560085, 34806, 134751, 280491, 197628, 200943, 90861, 168429, 10110, 531255, 37878, 527607, 594555, 75258, 528219, 143862, 25566, 12135, 295902, 660087, 789351, 76263, 71403, 333546, 665943, 68079, 296946, 13305, 303927, 541599, 558999, 344826, 19707, 788409, 791358, 272235, 22458, 295917, 265209, 71406, 271341, 141813, 166389, 566133, 268203, 7071, 535509, 345771, 134511, 296826, 13245, 529335, 548991, 172407, 275307, 19446, 99063, 98943, 657879, 794847, 265119, 304083, 143775, 14190, 549711, 336363, 38079, 566103, 38142, 531381, 52179, 459747, 529662, 106971, 38874, 12153, 917943, 69357, 918135, 336606, 287694, 90876, 76281, 545007, 544959, 543699, 795495, 205767, 134643, 10221, 199635, 147951, 721782, 165591, 75423, 42873, 19383, 132543, 197583, 574332, 7995, 330486, 266091, 13275, 918231, 15303, 168318, 205689, 25563, 42111, 526329, 168411, 141021, 558879, 265191, 75183, 541662, 672189, 50157, 13239, 393951, 27510, 336591, 197619, 566109, 166359, 592365, 787431, 393981, 330471, 534303, 789357, 606699, 140790, 52062, 199542, 559065, 287643, 280506, 12111, 541671, 794877, 76215, 591579, 526302, 132606, 558942, 459579, 42855, 401883, 401787, 230358, 590331, 20286, 147903, 84591, 664023, 265149, 135927, 37695, 296763, 197559, 721821, 68031, 90855, 394671, 136926, 591582, 205659, 788334, 545679, 591612, 273567, 205773, 594615, 275247, 534396, 789303, 534429, 299259, 168435, 11391, 659964, 75486, 567153, 100047, 791454, 213822, 538719, 75375, 201579, 330222, 299958, 791802, 819327, 20472, 820023, 199623, 87738, 312243, 143835, 330489, 395679, 360894, 558969, 268539, 272511, 100851, 591279, 537525, 528246, 53439, 69246, 672159, 869043, 69372, 45942, 10230, 536823, 69327, 789462, 38139, 27606, 104046, 312270, 75510, 623031, 562077, 721743, 273273, 311487, 205686, 134619, 788283, 789477, 296934, 104115, 598653, 156135, 559053, 803691, 78459, 459003, 12063, 336495, 20346, 426399, 148221, 299247, 658932, 198633, 330399, 795609, 156015, 297087, 534453, 598647, 53487, 528348, 168567, 12141, 201720, 149883, 594414, 345774, 198507, 272334, 852723, 15483, 562125, 4059, 34767, 72435, 51099, 39159, 852543, 75507, 58206, 592605, 43989, 273372, 537447, 852399, 543705, 19449, 598749, 7119, 269229, 90813, 197565, 559959, 13293, 45183, 78567, 721119, 45915, 266127, 275403, 803055, 201675, 269235, 197823, 14235, 18687, 273267, 13503, 27423, 361167, 394734, 143727, 592311, 803826, 140667, 34797, 137823, 50163, 76275, 54126, 591471, 534366, 115431, 140655, 530286, 852669, 527583, 463086, 72429, 795486, 134775, 345786, 36702, 917919, 852603, 336543, 534381, 136911, 22506, 790767, 156063, 29607, 158031, 526191, 591342, 304092, 303903, 297807, 592350, 795513, 395643, 531291, 69039, 534390, 39711, 149238, 29547, 689757, 722886, 341610, 137973, 19647, 296175, 149967, 639687, 273213, 102975, 656751, 198447, 459630, 199455, 174453, 410355, 12261, 144867, 414123, 45855, 591543, 434646, 115179, 110193, 106998, 541686, 713052, 205662, 39039, 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6.660995935444294e-06, -1.9402544961938593e-08, 2.0366275043358935e-12, -7.312879322864346e-12, 1.5162794773908183e-07, -3.415413646975435e-12, 8.98757455233241e-13, -8.476760639840727e-06, -5.1564519993196285e-08, -1.6176294812427447e-12, 6.523071960002894e-06, -2.7927363874160497e-08, 7.833539382316912e-12, -2.7538469257331537e-06, -5.940250489804229e-06, -1.718282553329805e-10, 2.7203042740939206e-11, -3.6518226663888154e-08, 3.19080288216423e-06, 1.5408440953805957e-07, 5.210256235313012e-10, -3.5542765283341295e-10, 2.8863362874545314e-11, 7.0177601253177585e-06, 2.9201353131882256e-09, 2.4107805161934457e-07, 0.0003744647124119423, 1.2731983744640741e-12, 1.1599145030646262e-12, 3.106673751538566e-06, -3.098293680188467e-07, 1.9165260761514887e-07, 0.0004249150586240644, 1.5810848287719173e-12, 6.289749236912827e-07, 1.1279309410590716e-11, -7.134242173320597e-08, -4.3102602456970685e-05, 5.734423418338453e-08, -1.1837611813797578e-07, -1.3329699999646749e-08, 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3.590421991769529e-12, 5.342580808419347e-12, 2.009879912147823e-12, -4.134176023637082e-10, 6.976602234041744e-08, -3.952966936331898e-09, -6.356303066884324e-12, 2.180964288080103e-08]}}

The returned result has the same fields as the one in the QSCI example.

CASCI_ENERGY = -76.0508056368444  # Exact result
opt_qsci_energy = opt_result["qsci_energy"]
print(f"CASCI energy: {CASCI_ENERGY} Ha")
print(f"QSCI energy: {opt_qsci_energy} Ha")
print(f"Energy error: {(opt_qsci_energy - CASCI_ENERGY) * 1000: .2e} mHa")
print()
print("Top 20 important configurations:")
amps = np.array(opt_result["state_vector"]["reals"]) + 1j * np.array(
    opt_result["state_vector"]["imags"]
)
sort_idx = np.argsort(np.abs(amps))
 
for i, (bit, real, imag) in enumerate(
    zip(
        np.array(opt_result["state_vector"]["bits"])[sort_idx][::-1],
        np.array(opt_result["state_vector"]["reals"])[sort_idx][::-1],
        np.array(opt_result["state_vector"]["imags"])[sort_idx][::-1],
    )
):
    if i > 20:
        break
    print(
        f"Electron configuration: {bin(bit)[2:].zfill(20)}, "
        f"amplitude: {real + 1j * imag: .3e}, "
        f"magnitude: {real**2 + imag**2: .3e}"
    )

Output:

CASCI energy: -76.0508056368444 Ha
QSCI energy: -76.05051672600099 Ha
Energy error:  2.89e-01 mHa

Top 20 important configurations:
Electron configuration: 00000000001111111111, amplitude: -1.934e-01-9.664e-01j, magnitude:  9.713e-01
Electron configuration: 00110000000011111111, amplitude:  1.127e-02+5.629e-02j, magnitude:  3.295e-03
Electron configuration: 00000011001111001111, amplitude:  8.434e-03+4.214e-02j, magnitude:  1.847e-03
Electron configuration: 00001100001111001111, amplitude:  5.640e-03+2.818e-02j, magnitude:  8.258e-04
Electron configuration: 00010010001011011111, amplitude: -5.445e-03-2.721e-02j, magnitude:  7.698e-04
Electron configuration: 00100001000111101111, amplitude: -5.418e-03-2.707e-02j, magnitude:  7.622e-04
Electron configuration: 00000010011110011111, amplitude: -5.053e-03-2.524e-02j, magnitude:  6.628e-04
Electron configuration: 00000001101101101111, amplitude: -5.023e-03-2.510e-02j, magnitude:  6.551e-04
Electron configuration: 00000000111100111111, amplitude:  4.978e-03+2.487e-02j, magnitude:  6.434e-04
Electron configuration: 11000000001111001111, amplitude:  4.918e-03+2.457e-02j, magnitude:  6.280e-04
Electron configuration: 00100010000111011111, amplitude:  4.486e-03+2.241e-02j, magnitude:  5.224e-04
Electron configuration: 00010001001011101111, amplitude:  4.458e-03+2.227e-02j, magnitude:  5.160e-04
Electron configuration: 00000000111111001111, amplitude:  4.109e-03+2.053e-02j, magnitude:  4.383e-04
Electron configuration: 00010000101011110111, amplitude: -4.008e-03-2.002e-02j, magnitude:  4.170e-04
Electron configuration: 11000000001100111111, amplitude:  4.006e-03+2.001e-02j, magnitude:  4.166e-04
Electron configuration: 00100000010111111011, amplitude: -4.002e-03-1.999e-02j, magnitude:  4.157e-04
Electron configuration: 01000010001110011111, amplitude: -3.889e-03-1.943e-02j, magnitude:  3.927e-04
Electron configuration: 01100000000110111111, amplitude: -3.856e-03-1.927e-02j, magnitude:  3.860e-04
Electron configuration: 10010000001001111111, amplitude: -3.856e-03-1.927e-02j, magnitude:  3.860e-04
Electron configuration: 10000001001101101111, amplitude: -3.848e-03-1.923e-02j, magnitude:  3.845e-04
Electron configuration: 00000110001111001111, amplitude:  3.611e-03+1.804e-02j, magnitude:  3.386e-04

Performance

N2 dissociation curve

As a performance benchmark, the 20-qubit $\text{N}_2$ dissociation curve is shown.

The red curve in the plot is generated by the QSCI method with the DoubleExcitation ansatz. You can reproduce it by using the following code with different choices of d, the distance between the nitrogen atoms.

d = 1.0
 
job = function.run(
    method="QSCI",
    molecule={"atom": f"N 0 0 0; N 0 0 {d}"},
    parameters={"ansatz": "DoubleExcitation", "state_prep_method": "CCSD"},
    qsci_setting={"n_shots": 100000, "number_of_states_pick_out": 50000},
    mitigation_setting={"configuration_recovery": True},
    backend="ibm_strasbourg",
)

Note that each point typically takes about one minute of QPU usage. On the classical side, where $R$ is chosen to be 50000, it doesn't necessarily mean that diagonalization of a $50000 \times 50000$ matrix is done. The number of samples after configuration recovery in this case is approximately 5000 - 7000, which sets the dimension of the subspace Hamiltonian. Thus, the classical diagonalization cost is much smaller than that of typical FCI computation.

The key takeaway of this result is that QSCI with suitable configuration recovery outperforms scalable classical methods such as Hartree-Fock (HF), CCSD, and CISD at all choices of d. This implies that at a scale where FCI energy is no longer available, QSCI serves as a reliable method to estimate the ground state energy at all distances. For example, using the 6-31g basis, a 36-qubit $\text{N}_2$ dissociation curve produced by QURI Function QSCI is given by

Here, the FCI curve is not available on a typical laptop (M2 Pro Chip, 16G RAM). However, QSCI can produce a qualitatively correct dissociation curve compared to the classical methods on the same machine with the assistance of the ibm_strasbourg quantum computer.

Azobenzene

Azobenzene has two isomers, trans-azobenzene and cis-azobenzene. The energy difference between the ground states of the two isomers plays an important role in the photoisomerization of azobenzene. This example benchmarks the ground state energy difference between QSCI and FCI with different active space settings up to 28 qubits. With configuration recovery, QURI Function QSCI yields error less than 5 mHa with only $10^5$ shots, 30 to 40 seconds of QPU usage, and under 10 minutes of classical post-processing time per point.

Support

For running systems larger than 20 qubits, please contact sales@qunasys.com.

Next steps

Recommendations

QURI Chemistry: a Qiskit Function by QunaSys

Overview

Function description

Get started

Inputs

molecule

The circuit_options field

ansatz, state_prep_method and ansatz_setting

qsci_setting

mitigation_setting

Symmetry post-selection

Configuration recovery

Outputs

Example

Set up the molecule

Execute algorithms

QSCI

Optimization-based QSCI

Performance

N2 dissociation curve

Azobenzene

Support

Next steps

`molecule`

The `circuit_options` field

`ansatz`, `state_prep_method` and `ansatz_setting`

`qsci_setting`

`mitigation_setting`