Getting started with Sample-based quantum diagonalization (SQD)
This guide demonstrates a simple working example to get started with the qiskit-addons-sqd package. In this example, you can use SQD to obtain an approximation of the ground state of the molecule at equilibrium.
To begin, recall that the generic interacting-electron Hamiltonian has the form:
where and are the Fermionic creation and annihilation operators associated with the -th orbital with spin and and are the one and two-body electronic integrals.
Note
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.
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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-0.05828834141623013 16 16 15 8
-2.004025150191673e-14 16 16 15 9
0.493391201597973 16 16 15 15
-2.887579115224201e-14 16 16 16 1
0.04957040081284462 16 16 16 2
-1.198756349736642e-14 16 16 16 3
-0.09244148681404396 16 16 16 8
3.20929905823288e-14 16 16 16 9
-8.789410358037327e-15 16 16 16 12
0.09929387371351053 16 16 16 15
0.4988558024531632 16 16 16 16
-6.683210272480665 1 1 0 0
-5.05801991735896 2 2 0 0
0.5093740535790473 3 1 0 0
-5.060718004401872 3 3 0 0
-5.241594321289754 4 4 0 0
-5.241594321289755 5 5 0 0
-4.495352077885957 6 6 0 0
1.031276567087045e-15 7 1 0 0
-4.495352077885959 7 7 0 0
-1.852207064622862e-15 8 1 0 0
0.5862891072481656 8 2 0 0
5.217912372939835e-15 8 3 0 0
-4.174967908523783e-15 8 4 0 0
-3.004188106134393 8 8 0 0
0.353765259990788 9 1 0 0
-0.6083289518389058 9 3 0 0
2.666971454264797e-15 9 8 0 0
-3.157845111668903 9 9 0 0
-0.9400165388605516 10 4 0 0
1.175322440382098e-15 10 8 0 0
-3.516783993981039 10 10 0 0
-0.9400165388605513 11 5 0 0
-3.51678399398104 11 11 0 0
0.4229990555408695 12 1 0 0
-1.905974893100913e-15 12 2 0 0
0.626081952156225 12 3 0 0
5.823279594237991e-15 12 8 0 0
-0.2288000350163298 12 9 0 0
-3.333008040651025 12 12 0 0
-1.174115690312534 13 6 0 0
1.462965175656052e-15 13 11 0 0
-3.451001907784892 13 13 0 0
-1.174115690312534 14 7 0 0
2.097688254432345e-15 14 10 0 0
-3.451001907784885 14 14 0 0
1.510890214156748e-15 15 1 0 0
0.4323970482854796 15 2 0 0
-1.262860516547564e-15 15 3 0 0
1.982594591989286e-15 15 4 0 0
0.4973837150599378 15 8 0 0
-4.330965086964359e-15 15 9 0 0
-3.214794676707322e-15 15 12 0 0
-3.961405744595176 15 15 0 0
-0.4289222488017268 16 2 0 0
5.039051098775309e-15 16 3 0 0
-3.293855240294314e-15 16 4 0 0
0.8512656679561205 16 8 0 0
5.29294696293402e-15 16 9 0 0
1.301363795748629e-15 16 10 0 0
2.727924913158556e-15 16 12 0 0
-0.838369929881854 16 15 0 0
-3.317587539899629 16 16 0 0
-77.40622425962903 0 0 0 0
Prepare molecule information
To begin, specify the molecule and its properties using pyscf and the electronic integrals stored in n2_fci.txt.
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()Output:
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.
Note
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)
# rand_seed = 42
# counts = generate_counts_uniform(
# 10_000, num_orbitals * 2, rand_seed=rand_seed
# )
# 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 iterationsNUM_BATCHES: The number of sets of subsamples to diagonalizeSAMPLES_PER_BATCH: The number of samples to include in each batchMAX_DAVIDSON_CYCLES: The number of iterations for the eigenstate solver
# SQD options
ITERATIONS = 5
# Eigenstate solver options
NUM_BATCHES = 10
SAMPLES_PER_BATCH = 300
MAX_DAVIDSON_CYCLES = 200Next, in order to plot the convergence, define arrays to store the approximation of the ground state energy, expectation value of the , 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 = np.zeros((ITERATIONS, 2 * num_orbitals))
occupancies_bitwise = (
None # Orbital i corresponds to column i in bitstring matrix
)Now, run the configuration recovery loop. Each loop consists of three steps:
- Use the
recover_configurations()method to obtain a refined bitstring matrix and probability array based on the average orbital occupancy. - Use the
postselect_and_subsample()to collect batches of subsamples to diagonalize over. - Then use the batches of subsamples as arguments to the
solve_fermion()method to obtain an approximation of the ground state.
One important note here is how to address the very first iteration of the configuration recovery loop. Here the average orbital occupancy is not yet available and so only the postselect_and_subsample() method is called. This removes any non-physical samples (samples with incorrect Hamming weight) before running the eigenstate solver, solve_fermion(). Afterwards, 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. The probability of flipping an individual bit is based on the distance between the value of the bit and the average orbital occupancy obtained. (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 flip_orbital_occupancies, solve_fermion
for i in range(ITERATIONS):
print(f"Starting 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 occupancies_bitwise 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,
occupancies_bitwise,
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=5,
hamming_left=5,
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 = np.zeros((NUM_BATCHES, 2 * num_orbitals))
coeffs = []
for j in range(NUM_BATCHES):
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_davidson=MAX_DAVIDSON_CYCLES,
)
energy_sci += nuclear_repulsion_energy
e_tmp[j] = energy_sci
s_tmp[j] = spin
occs_tmp[j, :num_orbitals] = avg_occs[0]
occs_tmp[j, num_orbitals:] = avg_occs[1]
coeffs.append(coeffs_sci)
# Combine batch results
avg_occupancy = np.mean(occs_tmp, axis=0)
# The occupancies from the solver should be flipped to match the bits in the bitstring matrix.
occupancies_bitwise = flip_orbital_occupancies(avg_occupancy)
# Track optimization history
energy_hist[i, :] = e_tmp
spin_sq_hist[i, :] = s_tmp
occupancy_hist[i, :] = avg_occupancyOutput:
Starting configuration recovery iteration 0
Starting configuration recovery iteration 1
Starting configuration recovery iteration 2
Starting configuration recovery iteration 3
Starting configuration recovery iteration 4
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 200 mH (chemical accuracy is typically accepted to be 1 kcal/mol 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
# Data for energies plot
n2_exact = -109.10288938
x1 = range(ITERATIONS)
# Here we plot the smallest energy obtained across all batches for each iteration
# of the configuration recovery loop.
e_diff = [abs(np.min(energies) - n2_exact) for energies in energy_hist]
yt1 = [1.0, 1e-1, 1e-2, 1e-3, 1e-4]
# Chemical accuracy (+/- 1 milli-Hartree)
chem_accuracy = 0.001
# Data for avg spatial orbital occupancy
y2 = avg_occupancy[:num_orbitals] + avg_occupancy[num_orbitals:]
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 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})
plt.tight_layout()
plt.show()Output:
Next steps
Recommendations
- Read through the SQD tutorial on IBM Quantum Learning