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Getting started with Sample-based quantum diagonalization (SQD)

Package versions

The code on this page was developed using the following requirements. We recommend using these versions or newer.

qiskit-addon-sqd~=0.10.0
pyscf~=2.8.0; sys.platform != 'win32'

Note for Windows users

This example uses the pyscf package, a Python-based chemistry simulation library, which is not natively supported on Windows; therefore, the examples below will not work.

In order to execute the code snippets below, Windows users need to create a Python environment using the Windows Subsystem for Linux, as suggested in the pyscf install guide.

This guide demonstrates a simple working example to get started with the qiskit-addon-sqd package. In this example, you can use SQD to obtain an approximation of the ground state of the $N_2$ molecule at equilibrium.

To begin, recall that the generic interacting-electron Hamiltonian has the form:

$\hat{H} = \sum_{\substack{pr\\ \sigma}}h_{pr} \hat{a}_{p\sigma}^\dagger \hat{a}_{r\sigma} + \sum_{\substack{prqs\\ \sigma\tau}} \frac{1}{2}\left(pr|qs\right)\hat{a}_{p\sigma}^\dagger \hat{a}_{q\tau}^\dagger \hat{a}_{s\tau}\hat{a}_{r\sigma}$

where $\hat{a}_{p\sigma}^\dagger$ and $\hat{a}_{p\sigma}$ are the Fermionic creation and annihilation operators associated with the $p$-th orbital with spin $\sigma$ and $h_{pr}$ and $\left(pr|qs\right)$ are the one and two-body electronic integrals.

Prepare molecule information

To begin, specify the molecule and its properties using pyscf and the electronic integrals stored in n2_fci.txt.

tala

This example ingests pre-generated data in a file called n2_fci.txt, which contains one- and two-body electronic integrals. To run the code cells below on a local machine, copy and paste the data into a file with the same name. This example also requires the pyscf package to be installed.

Click here to examine the orbital information used to generate the ansatz.

&FCI NORB=  16,NELEC=10,MS2=0,
  ORBSYM=1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  ISYM=1,
 &END
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# Added by doQumentation — installs packages not in the Binder environment
%pip install -q qiskit-addon-sqd

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 12   13    1\n -0.0006877470781163472   13   12   13    3\n -0.00706138745189237   13   12   13    9\n 0.03095947911215346   13   12   13   12\n 0.5050262226298633   13   13    1    1\n 0.4389181655673818   13   13    2    2\n -0.03281907698126586   13   13    3    1\n 0.4438172532847061   13   13    3    3\n 0.439717348069014   13   13    4    4\n 0.4678487895461089   13   13    5    5\n 0.444061375818176   13   13    6    6\n 0.4154111112115643   13   13    7    7\n -0.05046002042170361   13   13    8    2\n 0.3352105865180082   13   13    8    8\n -0.03060992899015901   13   13    9    1\n 0.04979106181529502   13   13    9    3\n 0.3564016583483769   13   13    9    9\n 0.0823476578938087   13   13   10    4\n 0.3867621774153833   13   13   10   10\n 0.08746627524833039   13   13   11    5\n 0.4142004572909493   13   13   11   11\n -0.04074553881022964   13   13   12    1\n -0.05711544314260815   13   13   12    3\n 2.073081039098704e-15   13   13   12    8\n 0.01211268605646502   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 16    9   15   12\n 6.286637876939949e-15   16    9   15   15\n -0.0346223583056286   16    9   16    1\n -2.808777214152596e-15   16    9   16    2\n 0.0170445544789866   16    9   16    3\n 4.15319597915637e-14   16    9   16    8\n 0.02380874268885576   16    9   16    9\n 0.02769440758746226   16   10    4    2\n -0.02673016631071906   16   10    7    1\n -0.0161557132592215   16   10    7    3\n -0.01638813739815387   16   10    8    4\n -0.0007699984003947267   16   10    9    7\n 0.007442968458760703   16   10   10    2\n -0.009889489083785945   16   10   10    8\n 0.001554393503338174   16   10   12    7\n -0.01735043648355827   16   10   14    1\n 0.002753286902048062   16   10   14    3\n 0.001494807418268969   16   10   14    9\n 0.02335754497241757   16   10   14   12\n 0.01404471420438996   16   10   15    4\n -0.006396325132918946   16   10   15   10\n 0.01091245741709193   16   10   16    4\n 0.0277313078959266   16   10   16   10\n 0.02769440758746226   16   11    5    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-0.001519629208699993   16   12    9    2\n 0.01384026476000111   16   12    9    8\n 1.776653005017684e-15   16   12    9    9\n 0.01460262989473362   16   12   10    7\n 0.01460262989473362   16   12   11    6\n 0.01891237059503479   16   12   12    2\n -0.01498043615347002   16   12   12    8\n -1.457920716678454e-15   16   12   12    9\n -3.051532156117631e-15   16   12   12   12\n 0.02386454932121355   16   12   13    5\n 0.04683595810183153   16   12   13   11\n 1.383016952962567e-15   16   12   13   13\n 0.02386454932121322   16   12   14    4\n 0.04683595810183153   16   12   14   10\n 1.276115389668579e-15   16   12   14   14\n -0.02179737382377572   16   12   15    1\n 1.856803597162178e-15   16   12   15    2\n 0.0264119391333841   16   12   15    3\n -1.25333291974904e-14   16   12   15    8\n 0.01288433222568699   16   12   15    9\n -0.03421263480964471   16   12   15   12\n 3.604755383079805e-15   16   12   15   15\n -0.02855049523302051   16   12   16    1\n -5.219375524274725e-15   16   12   16    2\n 0.01651322398521536   16   12   16    3\n 4.362257616865854e-14   16   12   16    8\n 0.01542447990351162   16   12   16    9\n 0.02682900793297205   16   12   16   12\n -0.05668006138333965   16   13    5    1\n -0.0161562579681439   16   13    5    3\n 0.0458493427584348   16   13    6    2\n -0.02133831631236967   16   13    8    6\n -0.01242301321668599   16   13    9    5\n -0.01829474708738727   16   13   11    1\n 0.007411196438025029   16   13   11    3\n 0.0195624809948784   16   13   11    9\n -0.00273197443016854   16   13   12    5\n 0.04251675853009274   16   13   12   11\n 0.01216894721678041   16   13   13    2\n -0.02205337167814066   16   13   13    8\n 0.02403042546124195   16   13   15    6\n -0.008850108812868383   16   13   15   13\n 0.007384989569455208   16   13   16    6\n 0.04513224232659596   16   13   16   13\n -0.05668006138333961   16   14    4    1\n -0.01615625796814387   16   14    4    3\n 0.04584934275843478   16   14    7    2\n -0.02133831631236992   16   14    8    7\n -0.01242301321668604   16   14    9    4\n -0.01829474708738724   16   14   10    1\n 0.007411196438025063   16   14   10    3\n 0.01956248099487841   16   14   10    9\n -0.002731974430168641   16   14   12    4\n 0.04251675853009265   16   14   12   10\n 0.01216894721678029   16   14   14    2\n -0.02205337167814028   16   14   14    8\n 0.02403042546124205   16   14   15    7\n -0.00885010881286854   16   14   15   14\n 0.00738498956945486   16   14   16    7\n 0.04513224232659604   16   14   16   14\n 0.1569633113847971   16   15    1    1\n 0.05787714956534407   16   15    2    2\n -0.04994490156642798   16   15    3    1\n 0.07686411480382442   16   15    3    3\n 0.08303790680849328   16   15    4    4\n 0.08303790680849335   16   15    5    5\n 0.06081298889881077   16   15    6    6\n 0.06081298889880927   16   15    7    7\n 5.310399066097946e-15   16   15    8    1\n -0.0231604828365148   16   15    8    2\n 2.712708624576935e-15   16   15    8    3\n 0.03639814722352972   16   15    8    8\n -0.03902419268036143   16   15    9    1\n 1.59899267460415e-15   16   15    9    2\n 0.0351701600430642   16   15    9    3\n -4.436474560991833e-15   16   15    9    8\n 0.04604524611495352   16   15    9    9\n 0.0525136519141776   16   15   10    4\n 0.05544937108918365   16   15   10   10\n 0.05251365191417747   16   15   11    5\n 0.05544937108918409   16   15   11   11\n -0.02729364114267291   16   15   12    1\n -0.02046989051649213   16   15   12    3\n 3.954528327059162e-15   16   15   12    8\n 0.004497148195492229   16   15   12    9\n 0.04149040805239823   16   15   12   12\n 0.05596924463339107   16   15   13    6\n 0.0635965839010509   16   15   13   13\n 0.05596924463339334   16   15   14    7\n 0.06359658390104776   16   15   14   14\n -1.950499113974221e-15   16   15   15    1\n -0.01336194832617421   16   15   15    2\n -1.272433537729245e-15   16   15   15    3\n -0.0252597323102154   16   15   15    8\n 4.745472317309655e-15   16   15   15    9\n -1.032702527524231e-15   16   15   15   12\n 0.0930165819216586   16   15   15   15\n 8.703197119210286e-15   16   15   16    1\n 0.03023672607424808   16   15   16    2\n 4.801543401410668e-15   16   15   16    3\n -0.04266373857922931   16   15   16    8\n -8.944651560152458e-15   16   15   16    9\n 4.54937645014102e-15   16   15   16   12\n 0.06578256216923312   16   15   16   15\n 0.6095177502791381   16   16    1    1\n 3.413477667354715e-15   16   16    2    1\n 0.4514856598796156   16   16    2    2\n -0.07283207717602366   16   16    3    1\n 0.4694776049347383   16   16    3    3\n 0.499413913175186   16   16    4    4\n 0.4994139131751858   16   16    5    5\n 0.4561590215966879   16   16    6    6\n 1.668732234451617e-15   16   16    7    4\n 0.4561590215966919   16   16    7    7\n -2.137924509739039e-14   16   16    8    1\n -0.0645202770416191   16   16    8    2\n -5.628194789149302e-15   16   16    8    3\n 0.3505526213192748   16   16    8    8\n -0.05255663699969248   16   16    9    1\n -4.88449799296871e-15   16   16    9    2\n 0.07246707264790894   16   16    9    3\n 2.338442757089573e-14   16   16    9    8\n 0.388479126897645   16   16    9    9\n 0.09884064916343566   16   16   10    4\n 0.4293099836227132   16   16   10   10\n 0.09884064916343638   16   16   11    5\n 1.741943194642991e-15   16   16   11    6\n 0.4293099836227113   16   16   11   11\n -0.0489141873120035   16   16   12    1\n -0.05002398657977705   16   16   12    3\n -7.816833871831079e-15   16   16   12    8\n 0.0299487240229355   16   16   12    9\n 0.4061588202601188   16   16   12   12\n 0.1158501412475767   16   16   13    6\n -3.710521830287963e-15   16   16   13   11\n 0.4362464030193691   16   16   13   13\n -1.1187828317558e-15   16   16   14    4\n 0.1158501412475714   16   16   14    7\n -1.418465610520137e-15   16   16   14   10\n 0.4362464030193751   16   16   14   14\n 7.77876570191495e-15   16   16   15    1\n -0.02772008861884064   16   16   15    2\n 4.027542590396144e-15   16   16   15    3\n -0.05828834141623013   16   16   15    8\n -2.004025150191673e-14   16   16   15    9\n 0.493391201597973   16   16   15   15\n -2.887579115224201e-14   16   16   16    1\n 0.04957040081284462   16   16   16    2\n -1.198756349736642e-14   16   16   16    3\n -0.09244148681404396   16   16   16    8\n 3.20929905823288e-14   16   16   16    9\n -8.789410358037327e-15   16   16   16   12\n 0.09929387371351053   16   16   16   15\n 0.4988558024531632   16   16   16   16\n -6.683210272480665    1    1  0  0\n -5.05801991735896    2    2  0  0\n 0.5093740535790473    3    1  0  0\n -5.060718004401872    3    3  0  0\n -5.241594321289754    4    4  0  0\n -5.241594321289755    5    5  0  0\n -4.495352077885957    6    6  0  0\n 1.031276567087045e-15    7    1  0  0\n -4.495352077885959    7    7  0  0\n -1.852207064622862e-15    8    1  0  0\n 0.5862891072481656    8    2  0  0\n 5.217912372939835e-15    8    3  0  0\n -4.174967908523783e-15    8    4  0  0\n -3.004188106134393    8    8  0  0\n 0.353765259990788    9    1  0  0\n -0.6083289518389058    9    3  0  0\n 2.666971454264797e-15    9    8  0  0\n -3.157845111668903    9    9  0  0\n -0.9400165388605516   10    4  0  0\n 1.175322440382098e-15   10    8  0  0\n -3.516783993981039   10   10  0  0\n -0.9400165388605513   11    5  0  0\n -3.51678399398104   11   11  0  0\n 0.4229990555408695   12    1  0  0\n -1.905974893100913e-15   12    2  0  0\n 0.626081952156225   12    3  0  0\n 5.823279594237991e-15   12    8  0  0\n -0.2288000350163298   12    9  0  0\n -3.333008040651025   12   12  0  0\n -1.174115690312534   13    6  0  0\n 1.462965175656052e-15   13   11  0  0\n -3.451001907784892   13   13  0  0\n -1.174115690312534   14    7  0  0\n 2.097688254432345e-15   14   10  0  0\n -3.451001907784885   14   14  0  0\n 1.510890214156748e-15   15    1  0  0\n 0.4323970482854796   15    2  0  0\n -1.262860516547564e-15   15    3  0  0\n 1.982594591989286e-15   15    4  0  0\n 0.4973837150599378   15    8  0  0\n -4.330965086964359e-15   15    9  0  0\n -3.214794676707322e-15   15   12  0  0\n -3.961405744595176   15   15  0  0\n -0.4289222488017268   16    2  0  0\n 5.039051098775309e-15   16    3  0  0\n -3.293855240294314e-15   16    4  0  0\n 0.8512656679561205   16    8  0  0\n 5.29294696293402e-15   16    9  0  0\n 1.301363795748629e-15   16   10  0  0\n 2.727924913158556e-15   16   12  0  0\n -0.838369929881854   16   15  0  0\n -3.317587539899629   16   16  0  0\n -77.40622425962903  0  0  0  0"

with open("n2_fci.txt", "w") as output_file:
    output_file.write(orbital_string)

from pyscf import ao2mo, tools

# Specify molecule properties
num_orbitals = 16
num_alpha = num_beta = 5
open_shell = False
spin_sq = 0

# Read in molecule from disk
molecule_scf = tools.fcidump.to_scf("n2_fci.txt")

# Core Hamiltonian representing the single-electron integrals
core_hamiltonian = molecule_scf.get_hcore()

# Electron repulsion integrals representing the two-electron integrals
electron_repulsion_integrals = ao2mo.restore(
    1, molecule_scf._eri, num_orbitals
)

# Nuclear repulsion energy of the molecule
nuclear_repulsion_energy = molecule_scf.mol.energy_nuc()

Parsing n2_fci.txt

Obtain samples from an ansatz

Once the molecule information has been loaded in, you should specify an ansatz circuit and optimize it for execution. This involves considering any symmetries of your problem, the choice of ansatz, and optimize it to run on quantum hardware.

However, these choices are problem specific and out of scope for this guide. Instead, to demonstrate the SQD workflow, we will generate a random set of counts for the configuration recovery loop to post-process. This simulates the measurement data of a 32-qubit circuit sampled with 10,000 shots. After the count data has been generated, the recover_configurations() method requires that you convert the count data into a matrix of the bitstrings measured at each shot, as well as an array of probabilities for each state that was measured.

tala

The artificial generation of samples is only used as a means to demonstrate the tooling of qiskit-addon-sqd. The package is meant to be used as part of a larger workflow wherein an ansatz or other circuit is defined, optimized, and executed using the Sampler primitive. The code cell below contains commented-out code demonstrating what a typical workflow might look like given a circuit ansatz transpiled to an ISA circuit.

# from qiskit_ibm_runtime import SamplerV2 as Sampler

# sampler = Sampler(mode=backend)
# job = sampler.run([isa_circuit], shots=10_000)
# primitive_result = job.result()
# pub_result = primitive_result[0]
# counts = pub_result.data.meas.get_counts()

from qiskit_addon_sqd.counts import generate_counts_uniform
from qiskit_addon_sqd.counts import counts_to_arrays
import numpy as np

rng = np.random.default_rng(24)
counts = generate_counts_uniform(10_000, num_orbitals * 2, rand_seed=rng)

# Convert counts into bitstring and probability arrays
bitstring_matrix_full, probs_array_full = counts_to_arrays(counts)

Run configuration recovery loop

The measurement samples can then be refined by repeating the configuration recovery and diagonalizing sets of subsamples to approximate the ground state until convergence.

First, specify the following options:

• ITERATIONS: The number of self-consistent configuration recovery iterations
• NUM_BATCHES: The number of sets of subsamples to diagonalize
• SAMPLES_PER_BATCH: The number of samples to include in each batch
• MAX_DAVIDSON_CYCLES: The number of iterations for the eigenstate solver

# SQD options
ITERATIONS = 5

# Eigenstate solver options
NUM_BATCHES = 1
SAMPLES_PER_BATCH = 500
MAX_DAVIDSON_CYCLES = 200

Next, in order to plot the convergence, define arrays to store the approximation of the ground state energy, expectation value of the $\langle S \rangle ^2$, and the orbital occupancy of the molecule.

# Self-consistent configuration recovery loop
energy_hist = np.zeros((ITERATIONS, NUM_BATCHES))  # energy history
spin_sq_hist = np.zeros((ITERATIONS, NUM_BATCHES))  # spin history
occupancy_hist = []
avg_occupancy = None

Now, run the configuration recovery loop. Each loop consists of three steps:

1. Use the recover_configurations() function to obtain a refined bitstring matrix and probability array based on the average orbital occupancy.
2. Use the postselect_and_subsample() to collect batches of subsamples to diagonalize over.
3. Then use the batches of subsamples as arguments to the solve_fermion() function to obtain an approximation of the ground state.

It is important to note how to address the first iteration of the configuration recovery loop. Since the average orbital occupancy is not yet available, only the postselect_and_subsample() function is called. This removes any non-physical samples (samples with incorrect Hamming weight) before running the eigenstate solver, solve_fermion(). Afterward, the average orbital occupation is calculated across all batches and used as input to the recover_configurations() method, which flips individual bits probabilistically based on this average. See Section II-A of the supplementary information in the SQD paper for more information.

from qiskit_addon_sqd.configuration_recovery import recover_configurations
from qiskit_addon_sqd.subsampling import postselect_and_subsample
from qiskit_addon_sqd.fermion import (
    bitstring_matrix_to_ci_strs,
    solve_fermion,
)

for i in range(ITERATIONS):
    print(f"\nStarting configuration recovery iteration {i}")
    # On the first iteration, we have no orbital occupancy information from the
    # solver, so we just post-select from the full bitstring set based on Hamming weight.
    if avg_occupancy is None:
        bitstring_matrix_tmp = bitstring_matrix_full
        probs_array_tmp = probs_array_full

    # If there is average orbital occupancy information, use it to refine the full set of noisy configurations
    else:
        bitstring_matrix_tmp, probs_array_tmp = recover_configurations(
            bitstring_matrix_full,
            probs_array_full,
            avg_occupancy,
            num_alpha,
            num_beta,
            rand_seed=rng,
        )

    # Throw out configurations with an incorrect particle number in either the spin-up or spin-down systems
    batches = postselect_and_subsample(
        bitstring_matrix_tmp,
        probs_array_tmp,
        hamming_right=num_alpha,
        hamming_left=num_beta,
        samples_per_batch=SAMPLES_PER_BATCH,
        num_batches=NUM_BATCHES,
        rand_seed=rng,
    )

    # Run eigenstate solvers in a loop. This loop should be parallelized for larger problems.
    e_tmp = np.zeros(NUM_BATCHES)
    s_tmp = np.zeros(NUM_BATCHES)
    occs_tmp = []
    coeffs = []
    for j in range(NUM_BATCHES):
        strs_a, strs_b = bitstring_matrix_to_ci_strs(batches[j])
        print(f"Batch {j} subspace dimension: {len(strs_a) * len(strs_b)}")
        energy_sci, coeffs_sci, avg_occs, spin = solve_fermion(
            batches[j],
            core_hamiltonian,
            electron_repulsion_integrals,
            open_shell=open_shell,
            spin_sq=spin_sq,
            max_cycle=MAX_DAVIDSON_CYCLES,
        )
        energy_sci += nuclear_repulsion_energy
        e_tmp[j] = energy_sci
        s_tmp[j] = spin
        occs_tmp.append(avg_occs)
        coeffs.append(coeffs_sci)

    # Combine batch results
    avg_occupancy = tuple(np.mean(occs_tmp, axis=0))

    # Track optimization history
    energy_hist[i, :] = e_tmp
    spin_sq_hist[i, :] = s_tmp
    occupancy_hist.append(avg_occupancy)
    print()

Starting configuration recovery iteration 0
Batch 0 subspace dimension: 6400

Starting configuration recovery iteration 1
Batch 0 subspace dimension: 555025

Starting configuration recovery iteration 2
Batch 0 subspace dimension: 573049

Starting configuration recovery iteration 3
Batch 0 subspace dimension: 518400

Starting configuration recovery iteration 4
Batch 0 subspace dimension: 562500

Visualize the results

Lastly, the results can be visualized by examining the approximated energy and average orbital occupancy at each iteration of the configuration recovery loop. The first plot shows that after a few iterations, the ground state energy is estimated to within approximately 27 mH (chemical accuracy is typically accepted to be 1 kcal/mol $\approx$ 1.6 mH). Recall that this demonstration used pure noise, and that the ability to approximate the energy to this degree comes from prior knowledge about the molecule and its electronic structure.

The second plot shows the average occupancy of each spatial orbital after the final iteration. Notice that both the spin-up and spin-down electrons occupy the first five orbitals with high probability in the solutions.

import matplotlib.pyplot as plt

# Chemical accuracy (+/- 1 milli-Hartree)
chem_accuracy = 0.001
exact_energy = -109.04667

# Data for energies plot
x1 = range(ITERATIONS)
min_e = [np.min(e) for e in energy_hist]
e_diff = [abs(e - exact_energy) for e in min_e]
yt1 = [1.0, 1e-1, 1e-2, 1e-3, 1e-4]

# Data for avg spatial orbital occupancy
y2 = occupancy_hist[-1][0] + occupancy_hist[-1][1]
x2 = range(len(y2))

fig, axs = plt.subplots(1, 2, figsize=(12, 6))

# Plot energies
axs[0].plot(x1, e_diff, label="energy error", marker="o")
axs[0].set_xticks(x1)
axs[0].set_xticklabels(x1)
axs[0].set_yticks(yt1)
axs[0].set_yticklabels(yt1)
axs[0].set_yscale("log")
axs[0].set_ylim(1e-4)
axs[0].axhline(
    y=chem_accuracy,
    color="#BF5700",
    linestyle="--",
    label="chemical accuracy",
)
axs[0].set_title("Approximated Ground State Energy Error vs SQD Iterations")
axs[0].set_xlabel("Iteration Index", fontdict={"fontsize": 12})
axs[0].set_ylabel("Energy Error (Ha)", fontdict={"fontsize": 12})
axs[0].legend()

# Plot orbital occupancy
axs[1].bar(x2, y2, width=0.8)
axs[1].set_xticks(x2)
axs[1].set_xticklabels(x2)
axs[1].set_title("Avg Occupancy per Spatial Orbital")
axs[1].set_xlabel("Orbital Index", fontdict={"fontsize": 12})
axs[1].set_ylabel("Avg Occupancy", fontdict={"fontsize": 12})

print(f"Exact energy: {exact_energy:.5f} Ha")
print(f"SQD energy: {min_e[-1]:.5f} Ha")
print(f"Absolute error: {e_diff[-1]:.5f} Ha")
plt.tight_layout()
plt.show()

Exact energy: -109.04667 Ha
SQD energy: -109.02383 Ha
Absolute error: 0.02284 Ha

Next steps

Recommendations

• Read through the SQD tutorial on IBM Quantum Learning