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Implizite Lösungsmittelberechnungen mit Qiskit Serverless

Geschätzte Nutzung: 2 Minuten auf einem Heron r2-Prozessor (HINWEIS: Dies ist nur eine Schätzung. Deine tatsächliche Laufzeit kann variieren.)

Lernziele​

  • Wie du mit Qiskit Serverless einen Remote-Workflow konfigurierst und ausführst

  • Wie du implizite Lösungsmitteleffekte mit einem Quantencomputer berechnest

Voraussetzungen​

Hintergrund​

Implizite Lösungsmittelberechnungen werden häufig in der computergestützten Biophysik eingesetzt. Diese Modelle beschreiben, wie eine gelöste Verbindung mit einem Lösungsmittel interagiert, ohne das Lösungsmittelsystem direkt zu modellieren. Stattdessen wird eine Näherung vorgenommen, bei der das Modell des gelösten Systems in eine mathematische Darstellung eines empirisch charakterisierten dielektrischen Mediums eingebettet wird. Diese dielektrische Näherung interagiert dann mit dem gelösten Stoff, der selbst direkt modelliert wird. Das dielektrische Medium beeinflusst Eigenschaften des gelösten Systems, wie zum Beispiel seine Grundzustandsenergie, indem es mit dessen Elektronenfeld interagiert. Dies ist wichtig für biophysikalische Modelle, die beispielsweise in der Arzneimittelforschung verwendet werden, da sich Verbindungen in verschiedenen dielektrischen Umgebungen unterschiedlich verhalten. Die Modellierung einer Verbindung in der Luft (in vacuo) beschreibt ein anderes Verhalten als die Modellierung in Wasser. Da pharmazeutische Verbindungen in den menschlichen Körper gelangen müssen, der selbst größtenteils aus Wasser besteht, ist es hilfreich, eine Verbindung in einer Lösung wie Wasser statt in vacuo zu modellieren. Mit impliziten Lösungsmittelmodellen können wir dieses Verhalten kostengünstig erreichen, wobei das Endergebnis im Allgemeinen ungenauer ist als das Gegenstück: rechnerisch aufwändigere explizite Lösungsmittelmodelle, die direkte Darstellungen sowohl des gelösten Stoffs als auch der Lösungsmittelmoleküle erzeugen.

In diesem Tutorial zeigen wir, wie ein Quantenalgorithmus, die Sample-basierte Quantendiagonalisierung (SQD), in ein vergleichsweise rechnerisch kostengünstiges implizites Lösungsmittelmodell eingebunden werden kann. Im Beispiel beschreiben wir, wie sich Methylamin verhält, wenn es sich in Wasser löst. Wir vergleichen den Quantenalgorithmus mit einer klassischen, hochmodernen Vergleichsmethode namens CASCI und zeigen eine enge Übereinstimmung zwischen diesen Berechnungen. Wir stellen im Kleinen eine quantenzentrierte Supercomputing-Architektur vor, bei der die rechenintensive klassische Nachbearbeitung des Quanten-Sampling-Teils der Routine in eine cloudbasierte Umgebung innerhalb von Qiskit Serverless ausgelagert wird. Der Code demonstriert außerdem die Parallelisierung über die remote verfügbaren CPU-Kerne, um die Rechenzeit zu verbessern.

Qiskit Serverless ist ein Framework zum Ausführen verteilter Quanten- und klassischer Workloads, ohne dass Infrastruktur verwaltet werden muss. Es gibt keine Serverbereitstellung (kein Hochfahren von EC2-Instanzen, Clustern, Docker-Containern), keine Orchestrierungstools (Kubernetes, Docker Swarm) und keine Überwachung/Wartung. Jeder Serverless-Job läuft in einem sauberen Container, führt deinen Code aus und wird anschließend beendet. Zwischen den Jobs bleibt kein Zustand erhalten. Du schreibst einfach deinen Code und reichst dann deinen Job ein. Innerhalb eines Serverless-Jobs kann ein Programm nahtlos auf IBM Quantum®-Backends zugreifen und die Ergebnisse dort klassisch nachbearbeiten. Mit Qiskit Serverless können Benutzer auf ständig verfügbare Remote-CPU-Kerne und -Speicher zugreifen, was die Verteilung bestimmter klassischer Workloads auf Remote-Ressourcen ermöglicht. Benutzer profitieren außerdem von einem gewissen Vorteil bei der parallelen Verarbeitung von Programmen und vermeiden dabei die üblichen Probleme durch ein Herunterfahren des Geräts während der Ausführung. Weitere Informationen zu Qiskit Serverless findest du in der Dokumentation sowie in zusätzlichem Material auf GitHub.

Dieses Tutorial zeigt eine relevante Anwendung der folgenden Themen:

  • Sample-basierte Quantendiagonalisierung

  • Client-Server-Rechenmodelle für Quantencomputing

Dieses Tutorial wurde von Forschungsarbeiten der Cleveland Clinic inspiriert und basiert auf diesen, wie beschrieben in Kaliakin, Danil, et al. "Implicit solvent sample-based quantum diagonalization." The Journal of Physical Chemistry B 129.23 (2025): 5788-5796. Diese Arbeit legt den vollständigen Workflow für implizite Lösungsmittelberechnungen offen und erweitert ihn um iterative Lösungsmittel-Selbstkonsistenz ("The Heartwood Algorithm", M. Motta, T. Pellegrini, 2025), Geometrieoptimierung und automatische Auswahl des Qubit-Layouts. Weitere Informationen findest du in der SQD IEF-PCM Qiskit Function template, die gemeinsam von der Cleveland Clinic und IBM® auf Basis der Forschung der Cleveland Clinic entwickelt wurde, für eine optimierte Black-Box-Schnittstelle zur Durchführung impliziter Lösungsmittelberechnungen.

Anforderungen​

Stelle sicher, dass du vor Beginn dieses Tutorials Folgendes installiert hast:

  • Qiskit SDK v2.0 oder höher mit Unterstützung für Visualisierung
  • Qiskit Runtime v0.40 oder höher (pip install qiskit-ibm-runtime)
  • Qiskit IBM Catalog pip install qiskit_ibm_catalog
  • Qiskit IBM Serverless pip install qiskit_serverless
  • Qiskit-Addon: Sample-basierte Quantendiagonalisierung (SQD) v0.12.0 pip install qiskit_addon_sqd
  • PySCF pip install pyscf
  • FFSIM pip install ffsim
  • Matplotlib pip install matplotlib
  • Geometric pip install geometric

Setup​

# Added by doQumentation — required packages for this notebook
!pip install -q ffsim matplotlib numpy psutil pyscf qiskit qiskit-addon-sqd qiskit-ibm-catalog qiskit-ibm-runtime qiskit-serverless rustworkx
# Establish Quantum Resource connection
from qiskit_ibm_runtime import QiskitRuntimeService

service = QiskitRuntimeService()

backend = service.least_busy()
print(f"Using backend {backend.name}")
# Establish Classical HPC Resource connection
from qiskit_ibm_catalog import QiskitFunction, QiskitServerless

client = QiskitServerless()

Erstelle direkt neben dem Hauptnotebook-Programm ein Verzeichnis mit dem Namen source_files. In dieses Verzeichnis legst du Python-Dateien, die du mit der Remote-Compute-Umgebung teilen möchtest. Du musst zwei Dateien erstellen:

  • source_files\diagonalization_engine.py

  • source_files\classical_simulation.py

Klicke unten, um den Text der einzelnen Skripte aufzuklappen, und kopiere den Inhalt anschließend in eine lokale Datei mit diesen Pfadnamen.

Klicken, um source_files\diagonalization_engine.py anzuzeigen
Klicken, um source_files\classical_simulation.py anzuzeigen
hinweis

Weitere Informationen findest du im oben referenzierten SQD IEF-PCM Qiskit Function-Vorlagenleitfaden (gemeinsam entwickelt von der Cleveland Clinic und IBM). Siehe auch die Bibliothek qiskit_addon_sqd.

#!/usr/bin/env python3
import numpy as np
from json.encoder import JSONEncoder
from json.decoder import JSONDecoder
from functools import partial
import os

from qiskit_serverless import (
    distribute_task,
    get_arguments,
    get,
    save_result,
    get_runtime_service,
)
from qiskit_addon_sqd.fermion import (
    SCIResult,
    diagonalize_fermionic_hamiltonian,
    solve_sci,
)

### Argument retrieval
args = get_arguments()

data = args["data"]  # Chemistry Data
energy_tol = args["energy_tol"]  # SQD option
occupancies_tol = args["occupancies_tol"]  # SQD option
max_iterations = args["max_iterations"]  # SQD option
symmetrize_spin = args["symmetrize_spin"]  # Eigenstate solver option
carryover_threshold = args["carryover_threshold"]  # Eigenstate solver option
num_batches = args["num_batches"]  # Eigenstate solver option
samples_per_batch = args["samples_per_batch"]  # Eigenstate solver option
max_cycle = args["max_cycle"]  # Eigenstate solver option
mem = args["mem"]  # Memory per Worker

# --- fan‑out target: 1 CPU + mem GB RAM per call -------------
@distribute_task(target={"cpu": 1, "mem": mem * 1024**3})
def _solve_sci_worker(
    ix, ci_strs, one_body_tensor, two_body_tensor, norb, nelec, spin_sq
):
    print(f">>>>> WORKER {ix} INITIATED")
    res = solve_sci(
        ci_strs,
        one_body_tensor,
        two_body_tensor,
        norb=norb,
        nelec=nelec,
        spin_sq=spin_sq,
    )

    print(f">>>>> WORKER {ix} COMPLETE")
    return res

def distribute_solve_sci_batch(
    ci_strings: list[tuple[np.ndarray, np.ndarray]],
    one_body_tensor: np.ndarray,
    two_body_tensor: np.ndarray,
    norb: int,
    nelec: tuple[int, int],
    *,
    spin_sq: float | None = None,
    **kwargs,
) -> list[SCIResult]:
    """Diagonalize Hamiltonian in subspaces, parallelizing across
        vCPUs in the Serverless environment.

    Args:
        ci_strings: List of pairs (strings_a, strings_b) of arrays of
            spin-alpha CI strings and spin-beta CI strings whose Cartesian
            product gives the basis of the subspace in which to perform a
            diagonalization.
        one_body_tensor: The one-body tensor of the Hamiltonian.
        two_body_tensor: The two-body tensor of the Hamiltonian.
        norb: The number of spatial orbitals.
        nelec: The numbers of alpha and beta electrons.
        spin_sq: Target value for the total spin squared for the ground state.
            If ``None``, no spin will be imposed.
        **kwargs: Keyword arguments to pass to
            `pyscf.fci.selected_ci.kernel_fixed_space`
            (https://pyscf.org/pyscf_api_docs/pyscf.fci.html#pyscf.fci.selected_ci.kernel_fixed_space

    Returns:
        The results of the diagonalizations in the subspaces given by ci_strings.
    """
    inputs = [
        (ix, ci_strs, one_body_tensor, two_body_tensor, norb, nelec, spin_sq)
        for ix, ci_strs in enumerate(ci_strings)
    ]

    # fan‑out: spawn one worker per input tuple
    print(">>>>> ENTERING WORKER FAN-OUT")
    refs = [_solve_sci_worker(*input_) for input_ in inputs]
    print(">>>>> WAITING ON WORKERS TO FINISH TASKS")

    # fan‑in: block until every worker finishes
    results = get(refs)
    print(">>>>> DISTRIBUTED JOBS COMPLETED")

    return results

# A caveat of executing a Python program remotely is
# that the inputs to the remote program must be passed
# over an internet network. Similarly, the outputs
# must be passed back to the local program via the same
# structure. Python objects are not always able to be
# passed over a network, and must be encoded in a
# JSON serializable format.
i_data = JSONDecoder().decode(data)

# i_data has all of the information needed from the
# local program to pick up where the computation left off
# after its submission to the remote environment.
[
    job_id,
    hcore,
    eri,
    num_orbitals,
    nuclear_repulsion_energy,
    num_elec_a,
    num_elec_b,
] = i_data

# Re-convert data back into numpy format, after serialization
hcore = np.array(hcore)
eri = np.array(eri)
nuclear_repulsion_energy = np.float64(nuclear_repulsion_energy)

# Instantiate Runtime Service to retrieve the
# bitstrings from the QPU job. We provided these
# credentials upon Serverless setup.
service = get_runtime_service()

# retrieving the QPU job data from the Serverless side
job = service.job(job_id)
primitive_result = job.result()
pub_result = primitive_result[0]
bit_array = pub_result.data.meas  # Getting the bitstrings

# Pass options to the built-in eigensolver
sci_solver = partial(
    distribute_solve_sci_batch, spin_sq=0.0, max_cycle=max_cycle
)

# List to capture intermediate results
result_history = []

def callback(results: list[SCIResult]):
    result_history.append(results)
    iteration = len(result_history)
    print(f">>>>> SQD ITERATION {iteration}")
    for i, result in enumerate(results):
        print(f">>>>> SUBSAMPLE {i}")
        print(f">>>>> \tENERGY: {result.energy + nuclear_repulsion_energy}")
        print(
            f">>>>> \tSUBSPACE DIMENSION: {np.prod(result.sci_state.amplitudes.shape)}"
        )

result = diagonalize_fermionic_hamiltonian(
    hcore,
    eri,
    bit_array,
    samples_per_batch=samples_per_batch,
    norb=num_orbitals,
    nelec=(num_elec_a, num_elec_b),
    num_batches=num_batches,
    energy_tol=energy_tol,
    occupancies_tol=occupancies_tol,
    max_iterations=max_iterations,
    sci_solver=sci_solver,
    symmetrize_spin=symmetrize_spin,
    carryover_threshold=carryover_threshold,
    callback=callback,
    seed=12345,
)

print(">>>>> EXACT DIAGONALIZATION COMPLETE. CLEANING UP, SERIALIZING DATA.")
# Numpy arrays are not JSON serializable.
# Convert them to List objects before using the JSONEncoder
o_data = JSONEncoder().encode(
    [
        result.energy + nuclear_repulsion_energy,
        result.energy,
        result.rdm1.tolist(),
        result.rdm2.tolist(),
        [x.tolist() for x in result.orbital_occupancies],
        [
            result.sci_state.nelec,
            result.sci_state.norb,
            [x.tolist() for x in result.sci_state.orbital_occupancies()],
            [x.tolist() for x in result.sci_state.rdm()],
        ],
    ]
)

# JSON-safe package
save_result({"outputs": o_data})  # single JSON blob returned to client
#!/usr/bin/env python3
from json.encoder import JSONEncoder
from json.decoder import JSONDecoder

from qiskit_serverless import get_arguments, save_result

import pyscf
from pyscf import gto, scf
from pyscf.solvent import pcm
from pyscf.mcscf import avas

import psutil

mem_info = (
    psutil.virtual_memory()
)  # Get information about virtual memory (RAM)
total_ram_gb = mem_info.total / (1024**3)  # Convert bytes to GB
print(f">>>>> SERVERLESS TOTAL RAM: {total_ram_gb:.2f} GB")

### Argument retrieval
args = get_arguments()
data = args["data"]  # Chemistry Data

i_data = JSONDecoder().decode(data)
[mol_geo, eps, ao_labels] = i_data

print(">>>>> DEFINING MOLECULE")
mol = gto.M()
mol.atom = mol_geo
mol.basis = "cc-pVDZ"
mol.unit = "Ang"
mol.charge = 0
mol.spin = 0
mol.verbose = 0

print(">>>>> BUILDING MOLECULE")
mol.build()

print(">>>>> DEFINING PCM")
cm = pcm.PCM(mol)
cm.eps = eps  # for water
cm.method = "IEF-PCM"

print(">>>>> BUILDING RESTRICTED HARTREE FOCK")
mf = scf.RHF(mol).PCM(cm)  # This is the Final SCF object
mf.kernel(verbose=0)

print(">>>>> RUNNING AVAS")
avas_ = avas.AVAS(mf, ao_labels, with_iao=True, canonicalize=True, verbose=0)
avas_.kernel()
norb, ne_act, mo_avas = avas_.ncas, avas_.nelecas, avas_.mo_coeff

print(">>>>> STARTING CASCI")
mc_pcm = pyscf.mcscf.CASCI(mf, norb, ne_act).PCM(
    cm
)  # Make sure to decorate the CASCI object with PCM
mc_pcm.mo_coeff = mo_avas
# mc_pcm.max_memory = 140000

(CASCI_E, _, _, _, _) = mc_pcm.kernel(verbose=0)

print(f">>>>> CASCI_E: {CASCI_E}")
o_data = JSONEncoder().encode([float(CASCI_E)])

# JSON-safe package
save_result({"outputs": o_data})  # single JSON blob returned to client

Wir müssen das Programm, das in der Cloud-Umgebung ausgeführt werden soll, freigeben und es jedes Mal erneut hochladen, wenn wir seinen Quellcode ändern:

client.upload(
    QiskitFunction(
        title="diagonalization_engine",
        entrypoint="diagonalization_engine.py",  # lives in ./source_files
        working_dir="source_files",
    )
)
client.upload(
    QiskitFunction(
        title="classical_simulation",
        entrypoint="classical_simulation.py",  # lives in ./source_files
        working_dir="source_files",
    )
)

Kleinskaliges Simulatorbeispiel​

Dieses Tutorial verwendet keinen kleinskaligen Simulator, da das Ziel darin besteht, eine skalierbare Quantenanwendung zu demonstrieren, die über den Bereich der Simulator-Exploration hinausgeht. Stattdessen zeigen wir später, wie diese Methode mit einer klassischen, hochmodernen Vergleichsmethode namens CASCI umgesetzt werden kann.

Großskaliges Hardwarebeispiel​

# This is a useful helper function that displays
# remote job execution details to the user's local machine
def feedback_serverless(serverless_job):
    import time

    # Wait for the job to execute
    print(f">>>>> Serverless status: {serverless_job.job_id}")
    timer = 0
    while timer < 10000:
        if (
            serverless_job.status() == "QUEUED"
            or serverless_job.status() == "INITIALIZING"
            or serverless_job.status() == "RUNNING"
        ):
            print(f">>>>> [{timer}s] Serverless job {serverless_job.job_id}: \
                {serverless_job.status()}")
            time.sleep(10)
            timer += 10

        elif serverless_job.status() == "ERROR":
            print(
                f">>>>> Serverless job {serverless_job.job_id}: {serverless_job.status()}"
            )
            print(">>>>> Logs:")
            print(serverless_job.logs())
            break

        elif serverless_job.status() == "DONE":
            print(
                f">>>>> Serverless job {serverless_job.job_id}: {serverless_job.status()}"
            )
            break

        else:
            break

    return

Schritt 1: Klassische Eingaben auf ein Quantenproblem abbilden​

1.1: Molekülobjekt mit bekannter a priori\textit{a priori} Molekülgeometrie initialisieren​

# Reference guide for building molecule structures:
# https://pyscf.org/user/gto.html
# Video tutorial on building molecular objects in PySCF:
# https://www.youtube.com/watch?v=cNC2cY9E9j0

molecule_name = "Methylamine"

methylamine_geo = """
    N   -0.7154    0.0000    0.0000;
    C    0.7154    0.0000    0.0000;
    H    1.1069    0.0916    1.0174;
    H    1.0996    0.8349   -0.5930;
    H    1.0996   -0.9274   -0.4345;
    H   -1.0625    0.8564    0.4294;
    H   -1.0625   -0.7661    0.5753;
"""
# Imports
import pyscf
from pyscf import gto  # Deals with molecular initialization
from pyscf import scf  # Solvation methods

# Explicitly defining the Methylamine molecule
mol = gto.M()
mol.atom = methylamine_geo
mol.basis = "cc-pVDZ"
mol.unit = "Ang"
mol.charge = 0
mol.spin = 0
mol.verbose = 0

mol.build()

1.2: Solvatisierungseffekte mit dem Polarizable Continuum Model (PCM) definieren​

# You can explore other solvents (such as methanol) by
# retrieving other dielectric parameters from:
# https://gaussian.com/scrf/
from pyscf.solvent import pcm

eps_water = 78.3553  # If solvating in a different medium,
# set this constant appropriately using a known value
cm = pcm.PCM(mol)
cm.eps = eps_water  # PySCF defaults to water solvation,
# but here we show this solvation parameter explicitly

cm.method = (
    "IEF-PCM"  # Alternative solvation models include C-PCM, SS(V)PE, COSMO
)
# Create a "Restricted Hartree-Fock" object for the solute,
# then wrap the SCF object with a Polarizable Continuum Model
mf_pcm0 = scf.RHF(mol).PCM(
    cm
)  # Restricted Hartree-Fock misses instantaneous correlations,
# post-HF methods like CCSD, CI, MP2 might be worth exploring

1.3: Geometrieoptimierung mit TRIC​

# Geometry optimization with geomeTRIC
from pyscf.geomopt.geometric_solver import (
    optimize,
)  # GeomeTRIC under the hood, for geometry optimization

mol_opt = optimize(
    mf_pcm0, tol_grad=3e-4, verbose=0
)  # Use geomeTRIC/TRIC under the hood

1.4: Kontinuumsmodell und Mean-Field-Objekt mit relevanten Variablen vorbereiten​

from pyscf.mcscf import avas

# Re-define PCM
cm = pcm.PCM(mol_opt)
cm.eps = eps_water  # for water
cm.method = "IEF-PCM"

# Re-build Restricted Hartree Fock object
mf_opt = scf.RHF(mol_opt).PCM(cm)
mf_opt.kernel(verbose=0)

# Run AVAS
ao_labels = ["C 2s", "C 2p", "N 2s", "N 2p", "H 1s"]
avas_ = avas.AVAS(
    mf_opt, ao_labels, with_iao=True, canonicalize=True, verbose=0
)
avas_.kernel()
norb, ne_act, mo_avas = avas_.ncas, avas_.nelecas, avas_.mo_coeff

num_elec_a = (ne_act + mol_opt.spin) // 2
num_elec_b = (ne_act - mol_opt.spin) // 2

Schritt 2: Problem für die Ausführung auf Quantenhardware optimieren​

Weitere Informationen zu den hier gezeigten Hilfsfunktionen findest du im Tutorial Sample-basierte Quantendiagonalisierung eines chemischen Hamiltonoperators.

# Standard SQD helper functions (From SQD Tutorial)

from typing import Sequence
import rustworkx
from qiskit.providers import BackendV2
from qiskit import QuantumCircuit, QuantumRegister

from rustworkx import NoEdgeBetweenNodes, PyGraph

IBM_TWO_Q_GATES = {"cx", "ecr", "cz"}

def create_linear_chains(num_orbitals: int) -> PyGraph:
    """In zig-zag layout, there are two linear chains (with connecting
    qubits between the chains). This function creates those two linear
    chains: a rustworkx PyGraph with two disconnected linear chains.
    Each chain contains `num_orbitals` number of nodes, that is, in the
    final graph there are `2 * num_orbitals` number of nodes.

    Args:
        num_orbitals (int): Number orbitals or nodes in each linear chain.
            They are also known as alpha-alpha interaction qubits.

    Returns:
        A rustworkx.PyGraph with two disconnected linear chains each with
        `num_orbitals` number of nodes.
    """
    G = rustworkx.PyGraph()

    for n in range(num_orbitals):
        G.add_node(n)

    for n in range(num_orbitals - 1):
        G.add_edge(n, n + 1, None)

    for n in range(num_orbitals, 2 * num_orbitals):
        G.add_node(n)

    for n in range(num_orbitals, 2 * num_orbitals - 1):
        G.add_edge(n, n + 1, None)

    return G

def create_lucj_zigzag_layout(
    num_orbitals: int, backend_coupling_graph: PyGraph
) -> tuple[PyGraph, int]:
    """This function creates the complete zigzag graph that 'can be mapped'
    to an IBM QPU with heavy-hex connectivity (the zigzag must be an
    isomorphic sub-graph to the QPU/backend coupling graph for it to be
    mapped). The zigzag pattern includes both linear chains (alpha-alpha
    interactions) and connecting qubits between the linear chains
    (alpha-beta interactions).

    Args:
        num_orbitals (int): Number of orbitals, that is, number of nodes in
            each alpha-alpha linear chain.
        backend_coupling_graph (PyGraph): The coupling graph of the backend
            on which the LUCJ ansatz will be mapped and run. This function takes
            the coupling graph as a undirected `rustworkx.PyGraph` where there
            is only one 'undirected' edge between two nodes, that is, qubits.
            Usually, the coupling graph of an IBM backend is directed (for
            example, Eagle devices such as ibm_brisbane) or may have two edges
            between two nodes (for example, Heron `ibm_torino`). A user
            needs to make such graphs undirected or remove duplicate edges
            (or do both) to make them compatible with this function.

    Returns:
        G_new (PyGraph): The graph with IBM backend compliant zigzag pattern.
        num_alpha_beta_qubits (int): Number of connecting qubits between the
            linear chains in the zigzag pattern. While we want as many
            connecting (alpha-beta) qubits between the linear (alpha-alpha)
            chains, we cannot accommodate all due to qubit and connectivity
            constraints of backends. This is the maximum number of connecting
            qubits the zigzag pattern can have while being backend compliant
            (that is, isomorphic to backend coupling graph).
    """
    isomorphic = False
    G = create_linear_chains(num_orbitals=num_orbitals)

    num_iters = num_orbitals
    while not isomorphic:
        G_new = G.copy()
        num_alpha_beta_qubits = 0
        for n in range(num_iters):
            if n % 4 == 0:
                new_node = 2 * num_orbitals + num_alpha_beta_qubits
                G_new.add_node(new_node)
                G_new.add_edge(n, new_node, None)
                G_new.add_edge(new_node, n + num_orbitals, None)
                num_alpha_beta_qubits = num_alpha_beta_qubits + 1
        isomorphic = rustworkx.is_subgraph_isomorphic(
            backend_coupling_graph, G_new
        )
        num_iters -= 1

    return G_new, num_alpha_beta_qubits

def lightweight_layout_error_scoring(
    backend: BackendV2,
    virtual_edges: Sequence[Sequence[int]],
    physical_layouts: Sequence[int],
    two_q_gate_name: str,
) -> list[list[list[int], float]]:
    """Lightweight and heuristic function to score isomorphic layouts. There
    can be many zigzag patterns, each with different set of physical qubits,
    that can be mapped to a backend. Some of them might include fewer noise
    qubits and couplings than others. This function computes a simple error
    score for each such layout. It sums up 2Q gate error for all couplings
    in the zigzag pattern (layout) and measurement of errors of physical
    qubits in the layout to compute the error score.

    Note:
        This lightweight scoring can be refined using concepts such as
        mapomatic.

    Args:
        backend (BackendV2): A backend.
        virtual_edges (Sequence[Sequence[int]]): Edges in the device-
            compliant zigzag pattern where nodes are numbered from 0 to (2 *
            num_orbitals + num_alpha_beta_qubits).
        physical_layouts (Sequence[int]): All physical layouts of the zigzag
            pattern that are isomorphic to each other and to the larger backend
            coupling map.
        two_q_gate_name (str): The name of the two-qubit gate of the
            backend. The name is used for fetching two-qubit gate error from
            backend properties.

    Returns:
        scores (list): A list of lists where each sublist contains two
            items. First item is the layout, and second item is a float
            representing error score of the layout. The layouts in the `scores`
            are sorted in the ascending order of error score.
    """
    props = backend.properties()
    scores = []
    for layout in physical_layouts:
        total_2q_error = 0
        for edge in virtual_edges:
            physical_edge = (layout[edge[0]], layout[edge[1]])
            try:
                ge = props.gate_error(two_q_gate_name, physical_edge)
            except Exception:
                ge = props.gate_error(two_q_gate_name, physical_edge[::-1])
            total_2q_error += ge
        total_measurement_error = 0
        for qubit in layout:
            meas_error = props.readout_error(qubit)
            total_measurement_error += meas_error
        scores.append([layout, total_2q_error + total_measurement_error])
    return sorted(scores, key=lambda x: x[1])

def _make_backend_cmap_pygraph(backend: BackendV2) -> PyGraph:
    graph = backend.coupling_map.graph
    if not graph.is_symmetric():
        graph.make_symmetric()
    backend_coupling_graph = graph.to_undirected()

    edge_list = backend_coupling_graph.edge_list()
    removed_edge = []
    for edge in edge_list:
        if set(edge) in removed_edge:
            continue
        try:
            backend_coupling_graph.remove_edge(edge[0], edge[1])
            removed_edge.append(set(edge))
        except NoEdgeBetweenNodes:
            pass

    return backend_coupling_graph

def get_zigzag_physical_layout(
    num_orbitals: int, backend: BackendV2, score_layouts: bool = True
) -> tuple[list[int], int]:
    """The main function that generates the zigzag pattern
        with physical qubits that can be used as an `intial_layout` in a
        preset passmanager/transpiler.

    Args:
        num_orbitals (int): Number of orbitals.
        backend (BackendV2): A backend.
        score_layouts (bool): Optional. If `True`, it uses the
            `lightweight_layout_error_scoring` function to score the
            isomorphic layouts and returns the layout with
            fewer erroneous qubits.
            If `False`, returns the first isomorphic subgraph.

    Returns:
        A tuple of device compliant layout (list[int]) with zigzag pattern
        and an int representing number of alpha-beta-interactions.
    """
    backend_coupling_graph = _make_backend_cmap_pygraph(backend=backend)

    G, num_alpha_beta_qubits = create_lucj_zigzag_layout(
        num_orbitals=num_orbitals,
        backend_coupling_graph=backend_coupling_graph,
    )

    isomorphic_mappings = rustworkx.vf2_mapping(
        backend_coupling_graph, G, subgraph=True
    )
    isomorphic_mappings = list(isomorphic_mappings)

    edges = list(G.edge_list())

    layouts = []
    for mapping in isomorphic_mappings:
        initial_layout = [None] * (2 * num_orbitals + num_alpha_beta_qubits)
        for key, value in mapping.items():
            initial_layout[value] = key
        layouts.append(initial_layout)

    two_q_gate_name = IBM_TWO_Q_GATES.intersection(
        backend.configuration().basis_gates
    ).pop()

    if score_layouts:
        scores = lightweight_layout_error_scoring(
            backend=backend,
            virtual_edges=edges,
            physical_layouts=layouts,
            two_q_gate_name=two_q_gate_name,
        )

        return scores[0][0][:-num_alpha_beta_qubits], num_alpha_beta_qubits

    return layouts[0][:-num_alpha_beta_qubits], num_alpha_beta_qubits
from qiskit.transpiler import generate_preset_pass_manager
import ffsim

# Initial LUCJ ansatz layout
initial_layout, _ = get_zigzag_physical_layout(norb, backend=backend)

# Initialize a pass manager
pass_manager = generate_preset_pass_manager(
    optimization_level=3, backend=backend, initial_layout=initial_layout
)

pass_manager.pre_init = ffsim.qiskit.PRE_INIT

Schritte 3 & 4: Mit Qiskit ausführen und mit Qiskit Serverless nachbearbeiten​

Hier kombinieren wir Schritt 3 (Ausführen) und Schritt 4 (Nachbearbeiten), da der Anwendungskontext des impliziten Lösungsmittelmodells eine iterative Rückkopplungsschleife erfordert, die mehrere Zyklen aus Ausführung und Nachbearbeitung durchläuft, um die abschließende Berechnung zu verbessern.

3.1 Die Restricted-Hartree-Fock-Energie (RHF) berechnen​

# Run the kernel to get the RHF energy
mf_opt = scf.RHF(mol_opt).PCM(cm)
hf_e = float(mf_opt.kernel())
print(f"Restricted Hartree-Fock Energy: {hf_e}")

3.2: Klassische Referenzenergie mit CASCI ermitteln​

# Setup a Serverless Client
worker = client.load("classical_simulation")
from json.encoder import JSONEncoder

ao_labels = ["C 2s", "C 2p", "N 2s", "N 2p", "H 1s"]
data_e = JSONEncoder().encode([mol_opt.tostring(), eps_water, ao_labels])
serverless_job = worker.run(data=data_e)
# Optionally, check the Serverless status feedback
# Don't sit here and stare at the feedback unless debugging.
# You can go develop something else while the Serverless job runs.
_ = feedback_serverless(serverless_job)
# If you make a mistake and need to cancel something

# for job in client.jobs():
#     job.cancel()
from json.decoder import JSONDecoder

CASCI_E = JSONDecoder().decode(serverless_job.result()["outputs"])[0]
# We have approximated the red, classical baseline from
# Figure 1 for Methanol (North-West panel)
print(f"CASCI/IEF-PCM(cc-pVDZ): E={CASCI_E}")

Anwendungsparameter konfigurieren​

# Systematically vary these parameters to improve hardware results

# Set to "True" to run on real hardware
use_hardware = True

# Error suppression/mitigation options
# >> Configure within Sampler primitive

# Transpiler Options
optimization_level = 3

# Heartwood algorithm options
n_iter = 15  # How many update loops to run
resample = 1  # (resample=1 -> resample the QPU after every
# update loop; resample=n_iter -> sample QPU only once)
shots = 10000

# SQD options
energy_tol = 1e-4
occupancies_tol = 1e-3
max_iterations = 12

# Eigenstate solver options
num_batches = 5
samples_per_batch = 300
symmetrize_spin = True
carryover_threshold = 1e-5
max_cycle = 200

# Classical post-processing options
local = (
    False  # Remote, Serverless (False) versus Local Post-Processing (True)
)
mem = 16  # Memory allocated to each diagonalization worker (Gb)
# Heartwood algorithm subroutines
import numpy as np
import pyscf
from pyscf import ao2mo, cc
from functools import reduce
import ffsim

from json.encoder import JSONEncoder
from json.decoder import JSONDecoder
import time

from functools import partial
from qiskit_addon_sqd.fermion import (
    SCIResult,
    diagonalize_fermionic_hamiltonian,
    solve_sci_batch,
)

def update_rdm(casci_object, dmas):
    """
    Inputs:
      mc   -> CASCI object
      dmas -> Spin-summed 1-particle reduced density matrix

    This function returns the CASCI/SQD one-body density matrix in
    the full basis of atomic orbitals, written as the sum (last line)
    of two terms:
       - a contribution from the core orbitals,
        np.dot(mocore, mocore.conj().T) * 2, (core = inactive and doubly-occupied)
       - a contribution from the active-space orbitals and electrons (dmas)
        rotated from the active-space to the AO basis (the reduce operation)

    Outputs:
      rho_approximation: The CASCI/SQD one-body density matrix
      in the full basis of atomic orbitals

    """
    mo_coeff = casci_object.mo_coeff
    ncore = casci_object.ncore
    ncas = casci_object.ncas
    mocore = mo_coeff[:, :ncore]
    mocas = mo_coeff[:, ncore : ncore + ncas]
    dm1 = np.dot(mocore, mocore.conj().T) * 2

    rho_approximation = dm1 + reduce(np.dot, (mocas, dmas, mocas.conj().T))
    return rho_approximation

def run_active_space_calculation(
    h1e_cas, h2e_cas, norb, ne_act, orbs, fermilevel, ecore
):
    # ----- perform an HF and a CCSD calculation in the active space
    from pyscf import tools
    from datetime import datetime

    now = datetime.now().strftime("%H:%M:%S")
    print(">>>>> ACTIVE SPACE CALCULATIONS ")
    tools.fcidump.from_integrals(
        f"as_fcidump_{now}.txt",
        h1e_cas,
        h2e_cas,
        norb,
        ne_act,
        ms=0,
        nuc=ecore,
    )  # Forcefully represents the active space in the correct structure
    mf_as = tools.fcidump.to_scf(f"as_fcidump_{now}.txt")
    os.remove(f"as_fcidump_{now}.txt")
    mf_as.kernel()
    print(">>>>> RUNNING CCSD")

    mf_cc = cc.CCSD(mf_as)
    mf_cc.kernel()
    orbts = mf_as.mo_coeff
    t1, t2 = mf_cc.t1, mf_cc.t2
    print(">>>>> UPDATED t1, t2 PARAMETERS")

    # ----- update the HF orbitals
    active = list(
        range(fermilevel - ne_act // 2, fermilevel - ne_act // 2 + norb)
    )
    orbs[:, active] = np.dot(orbs[:, active], orbts)
    return orbs, t1, t2

def get_lucj(norb, num_elec_a, num_elec_b, t1, t2, n_reps=1):
    print(">>>>> CONSTRUCTING LUCJ CIRCUIT")

    alpha_alpha_indices = [(p, p + 1) for p in range(norb - 1)]
    alpha_beta_indices = [(p, p) for p in range(0, norb, 4)]

    ucj_op = ffsim.UCJOpSpinBalanced.from_t_amplitudes(
        t1=t1,  # <---- Update t1 each loop
        t2=t2,  # <---- Update t2 each loop
        n_reps=n_reps,
        interaction_pairs=(alpha_alpha_indices, alpha_beta_indices),
    )
    nelec = (num_elec_a, num_elec_b)

    # create an empty quantum circuit
    qubits = QuantumRegister(2 * norb, name="q")
    circuit = QuantumCircuit(qubits)

    # prepare Hartree-Fock state as the reference state
    # and append it to the quantum circuit
    circuit.append(ffsim.qiskit.PrepareHartreeFockJW(norb, nelec), qubits)

    # apply the UCJ operator to the reference state
    circuit.append(ffsim.qiskit.UCJOpSpinBalancedJW(ucj_op), qubits)
    circuit.measure_all()

    return circuit

# Classical diagonalization engine sent to HPC
def classically_diagonalize(
    bit_array=None,  # Bit string array (only needed if locally processing data)
    nuclear_repulsion_energy=None,  # Electronic energy from the core orbitals
    hcore=None,  # 1-electron hamiltonian integrals
    eri=None,  # 2-electron hamiltonian integrals
    num_orbitals=None,  # Number of spatial orbitals
    nelec=None,  # Number of electrons
    num_elec_a=None,  # Alpha orbitals
    num_elec_b=None,  # Beta orbitals
    job_id=None,  # QPU bitstring Job ID
    client=None,  # Diagonalization engine worker
    energy_tol=1e-4,  # SQD option
    occupancies_tol=1e-3,  # SQD option
    max_iterations=12,  # SQD option
    num_batches=8,  # Eigenstate solver option
    samples_per_batch=300,  # Eigenstate solver option
    symmetrize_spin=False,  # Eigenstate solver option
    carryover_threshold=1e-5,  # Eigenstate solver option
    max_cycle=200,  # Eigenstate solver option
    local=True,  # Remote vs Local Diagonalization
    mem=16.0,  # Memory per Serverless Worker (Gb)
):
    print(">>>>> STARTING DIAGONALIZATION ENGINE ")
    # Pass options to the built-in eigensolver. If you just want to use
    # the defaults, you can omit this step, in which case you would not
    # specify the sci_solver argument in the call to
    # diagonalize_fermionic_hamiltonian below.
    if local:
        sci_solver = partial(
            solve_sci_batch, spin_sq=0.0, max_cycle=max_cycle
        )

        # List to capture intermediate results
        result_history = []

        def callback(results: list[SCIResult]):
            result_history.append(results)
            iteration = len(result_history)
            print(f">>>>> SQD ITERATION {iteration}")
            for i, result in enumerate(results):
                print(f">>>>> SUBSAMPLE {i}")
                print(
                    f">>>>> \tENERGY: {result.energy + nuclear_repulsion_energy}"
                )
                print(
                    f">>>>> \tSUBSPACE DIMENSION: {np.prod(result.sci_state.amplitudes.shape)}"
                )

        result = diagonalize_fermionic_hamiltonian(
            hcore,
            eri,
            bit_array,
            samples_per_batch=samples_per_batch,
            norb=num_orbitals,
            nelec=(nelec // 2, nelec // 2),
            num_batches=num_batches,
            energy_tol=energy_tol,
            occupancies_tol=occupancies_tol,
            max_iterations=max_iterations,
            sci_solver=sci_solver,
            symmetrize_spin=symmetrize_spin,
            carryover_threshold=carryover_threshold,
            callback=callback,
            seed=12345,
        )

        result = (result.energy, result.rdm1, result.rdm2)

    else:
        # Serverless Logic
        print(
            f">>>>> SENDING QISKIT RUNTIME JOB {job_id} TO QISKIT SERVERLESS"
        )

        data = [
            job_id,
            hcore.tolist(),
            eri.tolist(),
            int(num_orbitals),
            float(nuclear_repulsion_energy),
            int(num_elec_a),
            int(num_elec_b),
        ]

        # Encode the execution dependencies with the JSONEncoder
        data_e = JSONEncoder().encode(data)

        # Send to Serverless
        worker = client.load("diagonalization_engine")
        serverless_job = worker.run(
            data=data_e,
            energy_tol=energy_tol,  # SQD option
            occupancies_tol=occupancies_tol,  # SQD option
            max_iterations=max_iterations,  # SQD option
            symmetrize_spin=symmetrize_spin,  # Eigenstate solver option
            carryover_threshold=carryover_threshold,  # Eigenstate solver option
            num_batches=num_batches,  # Eigenstate solver option
            samples_per_batch=samples_per_batch,  # Eigenstate solver option
            max_cycle=max_cycle,  # Eigenstate solver option
            mem=mem,  # Memory per Worker (Gb)
        )

        # Wait for the job to execute
        _ = feedback_serverless(serverless_job)

        o_data = JSONDecoder().decode(serverless_job.result()["outputs"])
        result = (o_data[1], np.array(o_data[2]), np.array(o_data[3]))

        print(f">>>>>>>>>> Active Space Energy: {o_data[1]}")
        print(f">>>>>>>>>> rdm1: {o_data[2]}")
        print(f">>>>>>>>>> rdm2: {o_data[3]}")

    return result
# The Heartwood algorithm
import numpy as np

from qiskit_ibm_runtime import SamplerV2 as Sampler
from qiskit_addon_sqd.counts import generate_bit_array_uniform

mc = pyscf.mcscf.CASCI(mf_opt, ncas=norb, nelecas=ne_act).PCM(cm)
mc.with_solvent.method = mf_opt.with_solvent.method  #  Here we make sure
# that mc is also using the same solvent method defined earlier (IEF-PCM)
mc.with_solvent.eps = mf_opt.with_solvent.eps  # Set the dielectric parameters
mc.mo_coeff = mo_avas.copy()  # Update the molecular orbitals to include
# those computed in the presence of the solvent

h1e_cas, ecore = (
    mc.get_h1eff()
)  # <-- h1eff is the 1-electron hamiltonian integrals. h1e_cas is
# a common alias. ecore is the electronic energy from the core orbitals.
h2e_cas = ao2mo.restore(
    1, mc.get_h2eff(), norb
)  # <-- get the 2-electron hamiltonian integrals

mc.mo_coeff, t1, t2 = run_active_space_calculation(
    h1e_cas,
    h2e_cas,
    norb,
    ne_act,
    mo_avas.copy(),
    mf_opt.mol.nelectron // 2,
    ecore,
)

# Sampler primitive options
sampler = Sampler(mode=backend)

# Explore error suppression techniques and see if they can improve result quality
sampler.options.dynamical_decoupling.enable = True
sampler.options.dynamical_decoupling.sequence_type = "XY4"
sampler.options.twirling.enable_measure = True
sampler.options.environment.job_tags = ["TUT_ISC"]
# sampler.options.twirling.enable_gates = False
# sampler.options.twirling.num_randomizations = 10
# sampler.options.twirling.shots_per_randomization = 1024

# initial approximation for rdm1
with_solvent_e, with_solvent_v = None, None  # Don't touch
data = []
for iiter in range(n_iter):
    print(f">>>>> IMPLICIT SOLVENT ITERATION {iiter+1}/{n_iter}")
    if with_solvent_v is not None:
        # Subsequent update loops enter here
        mc.get_hcore = lambda *args: mc._scf.get_hcore() + with_solvent_v
    else:
        # First update loop starts here
        # hcore is the CAS space (classically computed) 1-electron
        # hamiltonian, which we default to at the start of the routine.
        mc.get_hcore = (
            lambda *args: mc._scf.get_hcore()
        )  # REF: https://pyscf.org/pyscf_api_docs/pyscf.mcscf.html#pyscf.scf.hf.CASBase.get_h1cas

    # Alias mapping
    # hcore : h1e_cas : h1e_eff
    # nuclear_repulsion_energy : ecore
    # eri : h2e_cas : h2e_eff

    h1e_cas, ecore = (
        mc.get_h1eff()
    )  # <-- h1eff is the 1-electron hamiltonian integrals. h1e_cas is a
    # common alias. ecore is the electronic energy from the core orbitals.
    h2e_cas = ao2mo.restore(
        1, mc.get_h2eff(), norb
    )  # <-- get the 2-electron hamiltonian integrals

    mc.mo_coeff, t1, t2 = run_active_space_calculation(
        h1e_cas,
        h2e_cas,
        norb,
        ne_act,
        mo_avas.copy(),
        mf_opt.mol.nelectron // 2,
        ecore,
    )

    if use_hardware:
        if (
            iiter % resample == 0
        ):  # <-- Toggle how often you refresh your bitstrings here. The
            # developer suggests that you do it every time, but benevolently
            # provides the freedom to disagree with him via the resample
            # control variable.
            # The "Quantum-Centric" part
            print(">>>>> GENERATING BITSTRINGS USING QUANTUM HARDWARE")
            # LUCJ Ansatz construction
            circuit = get_lucj(norb, num_elec_a, num_elec_b, t1, t2, n_reps=1)

            print(f">>>>> TRANSPILING LUCJ TO {backend.name}")
            isa_circuit = pass_manager.run(circuit)
            print(f">>>>> SUBMITTING ISA_CIRCUIT TO {backend.name}")
            job = sampler.run(
                [isa_circuit], shots=shots
            )  # <----- Error Suppression/Mitigation configured performed upstream
            job_id = str(job.job_id())

            timer = 0
            while job.status() != "DONE":
                timer += 10
                print(
                    f">>>>> [{timer}s] RUNTIME JOB {job_id}: {job.status()}"
                )
                time.sleep(10)

            primitive_result = job.result()
            print(f">>>>> RETRIEVED {job_id} FROM {backend.name}")

            pub_result = primitive_result[0]
            bit_array = pub_result.data.meas

    else:
        print(">>>>> GENERATING BITSTRINGS CLASSICALLY")
        rng = np.random.default_rng(24)
        bit_array = generate_bit_array_uniform(
            100_000, 2 * norb, rand_seed=rng
        )  # <-- Sample bitstrings from a uniform distribution. This is
        # useful for debug, but runs out of steam on large systems
        job_id = float(
            "nan"
        )  # <-- we will check that valid job_id's were passed during grading
        local = True

    # The "Classical Post-processing" part
    result = classically_diagonalize(
        bit_array=bit_array,
        nuclear_repulsion_energy=ecore,  # Electronic energy from the core orbitals
        hcore=h1e_cas,  # 1-electron hamiltonian integrals
        eri=h2e_cas,  # 2-electron hamiltonian integrals
        num_orbitals=norb,  # Number of spatial orbitals
        nelec=ne_act,  # Number of electrons
        num_elec_a=ne_act // 2,  # Alpha orbitals
        num_elec_b=ne_act // 2,  # Beta orbitals
        job_id=job_id,  # QPU bitstring Job ID
        client=client,  # Diagonalization engine worker
        energy_tol=energy_tol,  # SQD option
        occupancies_tol=occupancies_tol,  # SQD option
        max_iterations=max_iterations,  # SQD option
        num_batches=num_batches,  # Eigenstate solver option
        samples_per_batch=samples_per_batch,  # Eigenstate solver option
        symmetrize_spin=symmetrize_spin,  # Eigenstate solver option
        carryover_threshold=carryover_threshold,  # Eigenstate solver option
        max_cycle=max_cycle,  # Eigenstate solver option
        local=local,  # Remote vs Local Diagonalization
        mem=mem,  # Memory per Worker (Gb)
    )

    # e   : SQD-based estimate of the energy
    # rdm1: Spin-summed 1-particle reduced density matrix
    e, rdm1 = result[0], result[1]
    rho_approximation = update_rdm(
        mc, rdm1
    ).copy()  # <--- Reconstruct the one-body density matrix in the
    # atomic orbital basis to update the external potential due to
    # the solvent

    if with_solvent_e is not None:
        # Subsequent update loops enter here
        edup = np.einsum(
            "ij,ji->", with_solvent_v, rho_approximation
        )  # <-- edup: Incrementing the energy calculation with
        # subsequent iterations
        e += ecore + with_solvent_e - edup

    else:
        # First update loop enters here
        e += (
            ecore  # Pulled from the CAS space object (molecule's core energy)
        )

    # Outputs:
    # with_solvent_e : scalar energy correction due to solvent polarization
    # with_solvent_v : Fock-like matrix to be added to the core Hamiltonian in SCF
    with_solvent_e, with_solvent_v = mc.with_solvent._get_vind(
        rho_approximation
    )
    data.append((iiter, float(e), job_id))
    print(f">>>>> END IITER {iiter}")
    print(f">>>>> TOTAL ENERGY: {e}\n")
import matplotlib.pyplot as plt
from matplotlib.ticker import ScalarFormatter

def plot_data(data, baseline=0, name=None, save=False):
    x_vals, y_vals, job_ids = zip(*data)
    fig, ax = plt.subplots(figsize=(10, 6))

    # Plot line + markers
    ax.plot(
        x_vals,
        y_vals,
        color="navy",
        linewidth=2,
        marker="o",
        markersize=5,
        label="Energy trajectory",
    )
    ax.axhline(
        baseline,
        color="red",
        linestyle="--",
        linewidth=1.5,
        label="Reference energy",
    )

    # Force plain formatting
    ax.yaxis.set_major_formatter(ScalarFormatter(useMathText=True))
    ax.ticklabel_format(style="plain", axis="y")

    # Annotate each point with its exact value
    for x, y, job_id in zip(x_vals, y_vals, job_ids):
        ax.annotate(
            f"{y:.8f}, ID: {job_id}",
            (x, y),
            textcoords="offset points",
            xytext=(0, 8),  # vertical offset
            ha="center",
            fontsize=8,
            rotation=25,
            color="navy",
        )

    # Annotate the Classical Reference line
    for x, y in zip([0.5], [baseline]):
        ax.annotate(
            f"{y:.5f}",
            (x, y),
            textcoords="offset points",
            xytext=(0, 8),  # vertical offset
            ha="center",
            fontsize=8,
            rotation=25,
            color="red",
        )

    # Titles, labels, etc
    ax.set_title(
        f"SQD/IEF-PCM(cc-pVDZ) - {name}\nEnergy Convergence",
        fontsize=14,
        fontweight="bold",
        pad=15,
    )
    ax.set_xlabel("Update Iterations", fontsize=12)
    ax.set_ylabel("Total Energy (Hartrees)", fontsize=12)
    ax.grid(True, linestyle="--", linewidth=0.6, alpha=0.7)
    ax.legend(frameon=True, loc="best")
    plt.tight_layout()

    if save:
        plt.savefig(f"./results/{name}_energy_convergence.png")
    return fig, ax
# Plot your data

fig, ax = plot_data(data, baseline=CASCI_E, name=molecule_name, save=True)
plt.show()

Nächste Schritte​

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Quelle: IBM Quantum docs — aktualisiert 9. Sept. 2026
Englische Version auf doQumentation — aktualisiert 9. Sept. 2026
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