Variational Experiment with Batching

Hybrid Quantum-Classical variational quantum algorithms consist of optimizing a trial parametric wavefunction, \(| \psi (\vec{\theta}) \rangle\), to estimate the lowest eigenvalue (or expectation value) of a Hamiltonian, \(\hat{H}\). This could be an Electronic Structure Hamiltonian or a Hamiltonian defining a QUBO (quadratic unconstrained binary optimization) or MAXCUT problem. The optimal parameters of the wavefunction, \((\vec{\theta})\) are an estimation of the lowest eigenvector of the Hamiltonian.

Further details can be found in the following articles:

• A variational eigenvalue solver on a quantum processor

• Towards Practical Quantum Variational Algorithms

For the problem today, we will evaluate the ground-state energy (lowest eigenvalue) of a di-Hydrogen molecule. A Hamiltonian is defined over 2-qubits (PhysRevX.6.031007). A state-preparation (or Ansatz) circuit, a sequence of 1-qubit and 2-qubit gates, is used to generate a trial wavefunction. The wavefunction parameters are rotations on the circuit.

The hardware-efficient state-preparation method is used for today’s problem (nature23879). The variational experiment optimizes the parameters on this circuit, over multiple iterations, in order to minimize the expectation value of the Hamiltonian, \(\langle \psi (\vec{\theta}) | \hat{H} | \psi (\vec{\theta}) \rangle\).

Workflow and Tools

pytket is used to synthesize a state-preparation circuit, prepare measurement circuits with pytket-quantinuum being used to submit (retrieve) jobs in a batch to (from) the H-Series service. The variational experiment requires the following as inputs:

• a symbolic state-preparation circuit.
  

• an Hamiltonian defining the problem to be solved.

The state-preparation, described above, consists of fixed-angle 1-qubit and 2-qubit gates in addition to variable-angle 1-qubit gates. In pytket, variable-angle 1-qubit gates can have two types of parameters:

• numerical parameters (float);
  

• symbolic parameters (sympy.symbol).

Numerical parameters are native python floats. Symbolic parameters require the use of the symbolic library, sympy, which is also a dependency of pytket. Throughout the variational experiment, symbolic parameters on the state-preparation circuit are replaced with additional numerical parameters.

The variational procedure consists of \(n\) iterations until a specific criterion is satisfied. A batch session will run over these \(n\) iterations.
Inactivity for over 1 minutes will lead to the batch session ending, given how the batch feature works for H-Series devices.

During the variational experiment, each iteration updates the numerical values in the parameter set, as described above. Subsequently, these are substituted into a new copy of the original symbolic state-preparation circuit. A set of sub-circuits, each containing measurement information defined by the input Hamiltonian, are appended to the numerical state-preparation circuit, leading to a set of measurement circuits. Finally, these circuits are submitted to H-Series.

Specifically, each iteration consists of:

• classical pre-processing to define measurement circuits;
  

• batch submission to H-Series;
  

• retrieval of measurement results;
  

• classical post-processing to evaluate the cost function;

determining whether to stop or continue the variational procedure.

The SciPy minimizer is used to control the optimization of the cost function. The minimised value of the cost function and the optimal parameters can be retrieved at the end of the variational experiment.

The observable is a sum of Pauli-strings (tensor product over m qubits of Pauli-\(\hat{X}\), Pauli-\(\hat{Y}\), Pauli-\(\hat{Z}\) & Pauli-\(\hat{I}\)) multiplied by numerical coefficients.

• a set of initial numerical parameters to substitute into the symbolic state-preparation circuit. For example, this can be a set of random numerical floating-point numbers.
  

• pytket.backends.Backend object to interface with the H-Series quantum computing service.
  

• Number of shots to simulate each circuit with to generate a distribution of measurements.
  

• Maximum batch cost to limit the credit cost of the variational experiment.

QuantinuumBackend

The QuantinuumBackend is used to submit and retreive all circuits required for the variational experiment. This backend is included in the pytket-quantinuum extension. With this backend, the end-user can access H-series hardware, emulators, syntax checkers. The Quantinuum user portal lists all devices and emulators the end-user can access.

In the code cell below, the instance of QuantinuumBackend uses the H-Series emulator, H1-1E. The H1 syntax checker’s target is H1-1SC and the quantum device’s target is H1-1. The H-Series emulators are a useful utility to test and cost the performance of an algorithm before any hardware session.

The QuantinuumBackend instance requires the user to be authenticated before any jobs can be submitted. The login method will allow authentication.

from pytket.extensions.quantinuum import QuantinuumBackend

quantinuum_backend = QuantinuumBackend(device_name="H1-1E")
quantinuum_backend.login()

1. Synthesize Symbolic State-Preparation Circuit

The code-cell below synthesizes a 2-qubit circuit consisting of arbitrary-angle 2-qubit ZZPhase gates (pytket.circuit.OpType.ZZPhase) and fixed-angle 1-qubit X gate (pytket.circuit.OpType.X). This state-preparation technique is inspired by the Hardware-Efficient Ansatz (HEA) (nature23879), instead of the usual chemistry state-preparation method, Unitary Coupled Cluster (UCC) (arxiv.1701.02691).

The hardware-efficient state-preparation method requires alternating layers of 2-qubit gates and 1-qubit gates. Ultimately, this leads to fewer 2-qubit gates, but requires a greater number of variational parameters, compared to UCC. The optimal parameters for HEA are governed by the noise profile of the device. The HEA circuit used in this example consists of one-layer ZZPhase gates.

from pytket.circuit import Circuit
from sympy import Symbol

symbols = [Symbol(f"p{0}")]
symbolic_circuit = Circuit(2)
symbolic_circuit.X(0)
symbolic_circuit.ZZPhase(symbols[0], 0, 1)

The symbolic state-preparation circuit can be visualised using the pytket.circuit.display submodule.

from pytket.circuit.display import render_circuit_jupyter

render_circuit_jupyter(symbolic_circuit)

2. Hamiltonian Definition and Analysis

A problem hamiltonian is defined using the pytket.utils.operator.QubitPauliOperator class. Each QubitPauliOperator consists of complex coefficients and tensor products of Pauli-operations. The tensor products are referred to as Pauli-strings. This particular Hamiltonian consists of 5 terms operating on qubits q[0] and q[1]. The problem Hamiltonian, \(\hat{H}\), is defined as:

\[\begin{split}\begin{align} \hat{H} &= g_0 \hat{I}_{q[0]} \otimes \hat{I}_{q[1]} + g_1 \hat{Z}_{q[0]} \otimes \hat{I}_{q[1]} + g_2 \hat{I}_{q[0]} \otimes \hat{Z}_{q[1]} \\ &+ g_3 \hat{Z}_{q[0]} \otimes \hat{Z}_{q[1]} + g_4 \hat{X}_{q[0]} \otimes \hat{X}_{q[1]} + g_5 \hat{Y}_{q[0]} \otimes \hat{Y}_{q[1]} \\ \end{align}\end{split}\]

where \(g_0, g_1, g_2\), \(g_3\), \(g_4\) and \(g_5\) are real numercial coefficients.

The QubitPauliOperator is a dictionary mapping pytket.pauli.QubitPauliString to a complex coefficient. These coefficients are sympified (converted from python complex types to sympy complex types).

The QubitPauliString is a map from pytket.circuit.Qubit to pytket.pauli.Pauli.

The coefficients in the Hamiltonian are obtained from PhysRevX.6.031007.

from pytket.utils.operators import QubitPauliOperator
from pytket.pauli import Pauli, QubitPauliString
from pytket.circuit import Qubit

coeffs = [-0.4804, 0.3435, -0.4347, 0.5716, 0.0910, 0.0910]
term0 = {
    QubitPauliString(
        {
            Qubit(0): Pauli.I,
            Qubit(1): Pauli.I,
        }
    ): coeffs[0]
}
term1 = {QubitPauliString({Qubit(0): Pauli.Z, Qubit(1): Pauli.I}): coeffs[1]}
term2 = {QubitPauliString({Qubit(0): Pauli.I, Qubit(1): Pauli.Z}): coeffs[2]}
term3 = {QubitPauliString({Qubit(0): Pauli.Z, Qubit(1): Pauli.Z}): coeffs[3]}
term4 = {QubitPauliString({Qubit(0): Pauli.X, Qubit(1): Pauli.X}): coeffs[4]}
term5 = {QubitPauliString({Qubit(0): Pauli.Y, Qubit(1): Pauli.Y}): coeffs[5]}
term_sum = {}
term_sum.update(term0)
term_sum.update(term1)
term_sum.update(term2)
term_sum.update(term3)
term_sum.update(term4)
term_sum.update(term5)
hamiltonian = QubitPauliOperator(term_sum)
print(hamiltonian)

This Hamiltonian can be converted into a numpy.ndarray instance, and the lowest eigenvalue can be obtained using numpy.linalg.eig. This value is used as a benchmark for the VQE result. The ground-state energy is measured in units of Hartrees (Ha).

from scipy.linalg import eig

sm = hamiltonian.to_sparse_matrix().toarray()
ground_state_energy = eig(sm)[0].real[0]
print(f"{ground_state_energy} Ha")

To measure \(\hat{H}\) on hardware, naively 5 measurement circuits are required. The Identity term does not need to measured, since its expectation value always equals 1.
With pytket, \(\hat{H}\) only requires simulating 2 measurement circuit, thanks to measurement reduction. The four terms \(\hat{X}_{q[0]} \otimes \hat{X}_{q[1]}\), \(\hat{Y}_{q[0]} \otimes \hat{Y}_{q[1]}\), \(\hat{Z}_{q[0]} \otimes \hat{Z}_{q[1]}\), \(\hat{Z}_{q[0]} \otimes \hat{Z}_{q[1]}\) and \(\hat{I}_{q[0]} \otimes \hat{Z}_{q[1]}\), form a commuting set and can be measured with two circuits instead of three. This partitioning can be performed automatically using the measurement_reduction function available in pytket.partition submodule.

The measurement operations for the two commuting set,

• \(\left\{ \hat{X}_{q[0]} \otimes \hat{X}_{q[1]}, \hat{Y}_{q[0]} \otimes \hat{Y}_{q[1]} \right\}\),

• \(\left\{ \hat{Z}_{q[0]} \otimes \hat{Z}_{q[1]}, \hat{Z}_{q[0]} \otimes \hat{I}_{q[1]}, \hat{I}_{q[0]} \otimes \hat{Z}_{q[1]}\right\}\)

include additional 2-qubit gate resources.

from pytket.partition import (
    measurement_reduction,
    PauliPartitionStrat,
)

strat = PauliPartitionStrat.CommutingSets
pauli_strings = [term for term in hamiltonian._dict.keys()]
measurement_setup = measurement_reduction(pauli_strings, strat)

A measurement subcircuit contains the necessary operations to measure the terms in a commuting set. The subcircuit is appended to the numerical state-preparation circuit. Combining the numerical state-preparation circuit and the measurement subcircuits results in a set of measurement circuits required to solve the problem. The MeasurementSetup instance contains all the necessary sub-circuits to measure \(\hat{H}\). The next code cell lists and visualises all measurement subcircuits.

from pytket.circuit.display import render_circuit_jupyter

for measurement_subcircuit in measurement_setup.measurement_circs:
    render_circuit_jupyter(measurement_subcircuit)

Once the quantum computation has been completed, the measurement results can be mapped back to the Pauli-operations and coefficients in the Hamiltonian. This enables calculation of the expectation value for the Hamiltonian. The results attribute in the pytket.partition.MeasurementSetup lists:

• all the Pauli-strings that have been measured;
  

• information to process the quantum computed measurement result in order

to estimate the expectation value of each Pauli-strings.

for i, (term, bitmap_list) in enumerate(measurement_setup.results.items()):
    print(f"{term}\t{bitmap_list}\n")

3. Computing Expectation Values

Once the Hamiltonian has been partitioned into commuting sets, measurement circuits need to be constructed. These measurement circuits are submitted to hardware or emulators for simulation. Once the simulation is complete, a result is available to request, and can be retrieved using pytket. These results are the outcomes
of the measurement circuit simulation. Each result is a distribution of outcomes, specifically, the probability of observing specific bitstring. This distribution is post-processed to compute the expectation value of the Hamiltonian, a necessity to evaluate the cost function in a Hybrid Quantum-Classical variational procedure.

3.1 Computing Expectation Values for Pauli-Strings

The Hamiltonian we are interested in consists of Pauli-strings. The expectation value of the Pauli-string is in the interval \([-1, 1]\).

In the code cell below, a function is provided that calculates the expectation value of Pauli-string from a measured distribution. The MeasurementBitmap is used to extract the necessary data from the measured distribution. The resulting distribution can be summed over to estimate the expectation value of one Pauli-string.

from typing import Dict, Tuple
from pytket.partition import MeasurementBitMap

def compute_expectation_paulistring(
    distribution: Dict[Tuple[int, ...], float], bitmap: MeasurementBitMap
) -> float:
    value = 0
    for bitstring, probability in distribution.items():
        value += probability * (sum(bitstring[i] for i in bitmap.bits) % 2)
    return ((-1) ** bitmap.invert) * (-2 * value + 1)

In the example below, the function compute_expectation_paulistring is called to calculate the expectation for the \(\hat{Z} \otimes \hat{Z}\). First the QubitPauliString is initialised, and that is used to extract the relevant data from the MeasurementSetup object defined in section 2. This data is used for postprocessing.

from pytket.pauli import Pauli, QubitPauliString
from pytket.circuit import Qubit

distribution = {(0, 0): 0.45, (1, 1): 0.3, (0, 1): 0.1, (1, 0): 0.15}
zz = QubitPauliString([Qubit(0), Qubit(1)], [Pauli.Z, Pauli.Z])
bitmap_list = measurement_setup.results.get(zz)
for bitmap in bitmap_list:
    ev = compute_expectation_paulistring(distribution, bitmap)
    print(ev)

3.2 Computing Expectation Values for sums of Pauli-strings multiplied by coefficients

In this step, we will submit circuits to the H-Series emulator (H1-1E). This circuit will produce a result. The result can be retrieved with the ResultHandle object. First, the symbolic circuit is converted into a numerical circuit. The symbols in the circuit are substituted for numerical parameters.

symbol_map = {sym: 0.1 for sym in symbolic_circuit.free_symbols()}
numerical_circuit = symbolic_circuit.copy()
numerical_circuit.symbol_substitution(symbol_map)

3.3 Using QuantinuumBackend

The Quantinuum backend was initialized at the start of the notebook to use the H1-1E emulator. This backend will now be used to calculate the expectation value.

The measurement operations from the MeasurementSetup object are appended to the numerical circuit. Once this step is complete, the circuit is ready for submission if tket optimization in the H-Series stack is selected.

circuit_list = []
for mc in measurement_setup.measurement_circs:
    c = numerical_circuit.copy()
    c.append(mc)
    circuit_list += [c]
compiled_circuit_list = quantinuum_backend.get_compiled_circuits(
    circuit_list, optimisation_level=2
)

Before submitting to the emulator, the total cost of running the set of circuits can be checked beforehand.

cost_list = []

n_shots = 500
for comp_circ in compiled_circuit_list:
    cost = quantinuum_backend.cost(comp_circ, n_shots=n_shots, syntax_checker="H1-1SC")
    cost_list.append(cost)

print("Cost of experiment in HQCs:", sum(cost_list))

Now we run the circuits.

handles = quantinuum_backend.process_circuits(
    compiled_circuit_list, n_shots=10, options={"tket-opt-level": None}
)

The status of the jobs can be checked with ciruit_status method. This method requires the ResultHandle to be passed as input. In this example, the job has completed and the results are reported as being ready to request.

for h in handles:
    circuit_status = quantinuum_backend.circuit_status(h)
    print(circuit_status)

The expectation value of the operator can be evaluated with the function compute_expectation_value in the next code cell. This function requires a list of BackendResult objects, a MeasurementSetup instance, and the QubitPauliOperator instance for the expectation value computation. It is assumed the MeasurementSetup instance contains the measurement info of all the Pauli-strings in the QubitPauliOperator instance. Otherwise the compute_expectation_value function will return zero.

from typing import List
from pytket.utils.operators import QubitPauliOperator
from pytket.partition import MeasurementSetup
from pytket.backends.backendresult import BackendResult

from sympy import Abs

def compute_expectation_value(
    results: List[BackendResult],
    measurement_setup: MeasurementSetup,
    operator: QubitPauliOperator,
) -> float:
    energy = 0
    for pauli_string, bitmaps in measurement_setup.results.items():
        string_coeff = operator.get(pauli_string, 0.0)
        if Abs(string_coeff) > 0:
            value = 0
            for bm in bitmaps:
                index = bm.circ_index
                distribution = results[index].get_distribution()
                value += compute_expectation_paulistring(distribution, bm)
            energy += value * string_coeff / len(bitmaps)
    return energy

The results of the previously submitted circuits can be retrieved with the get_results method on QuantinuumBackend.

results = quantinuum_backend.get_results(handles)

Finally, the expectation value, \(\langle{\psi (\vec{\theta}_r)} | \hat{H} | { \psi (\vec{\theta}_r)} \rangle\), of the QubitPauliOperator instance, \(\hat{H}\), is calculated with respect to \(| { \psi (\vec{\theta}_r)} \rangle\). The state, \(| \psi \rangle\), is prepared with the state-preparation circuit, and \(\vec{\theta}_r\) is a random parameter set.

expectation_value = compute_expectation_value(results, measurement_setup, hamiltonian)
print(f"Expectation Value: {expectation_value}")

4. Variational Procedure

A hybrid quantum-classical variational procedure consists of multiple iterations, controlled by a classical parameter optimizer. The parameters are gate-angles on quantum circuits submitted to H-Series for simulation. In step 3, a procedure is showcased to calculate the expectation value of a Hamiltonian with respect to a quantum state. It is shown how to use the measurement reduction and Pauli-string partitioning facility in pytket to reduce measurement resources for the experiments. For the variational procedure demonstrated below, the cost function calculates the expectation of an input Hamiltonian. The aim is to find the optimal parameters that minimise this expectation value.

4.1. Objective function

The Objective class defined in the code cell performs the following utilities:

• Measurement Reduction;
  

• Creation of a Batch session to use across the variational experiment;
  

• Submission and retrieval of quantum circuits using QuantinuumBackend;
  

• Expectation Value evaluation.

The Objective class requires the following inputs:

• Input symbolic state-preparation circuit;
  

• A QubitPauliOperator instance of the Hamiltonian characterising the use case of interest;
  

• The backend to use. QuantinuumBackend is used to access H-Series service. The backend needs to be instantiated and the user needs to login within previous code cell.
  

• Number of shots to perform per circuit. H-Series devices have an upper limit of 10000 shots per job.
  

• Maximum consumable HQC credit before the batch is terminated.
  

• Number of iterations before the variational experiment is terminated.

The Objective class instance can be passed as a callable to scipy.optimize.minimize.

from typing import Callable
from numpy import ndarray
from numpy.random import random_sample
from pytket.extensions.quantinuum import QuantinuumBackend
from pytket.partition import PauliPartitionStrat
from pytket.backends.resulthandle import ResultHandle

class Objective:
    def __init__(
        self,
        symbolic_circuit: Circuit,
        problem_hamiltonian: QubitPauliOperator,
        quantinuum_backend: QuantinuumBackend,
        n_shots_per_circuit: int,
        max_batch_cost: float = 300,
        n_iterations: int = 10,
    ) -> None:
        r"""Returns the objective function needed for a variational
        procedure on H-Series.
        Args:
            symbolic_circuit
            (pytket.circuit.Circuit): State-preparation
                circuit with symbolic parameters
            problem_hamiltonian (pytket.utils.operators.QubitPauliOperator):
                QubitPauliOperator instance defining the Hamiltonian of the
                problem.
            quantinuum_backend (pytket.extensions.quantinuum.QuantinuumBackend): Backend
                instance to use for the simulation. This will be
                QuantinuumBackend from the pytket.extensions.quantinuum
                package to run experiments on H-Series devices and emulators.
            n_shots_per_circuit (int): Number of shots per circuit.
            max_batch_cost (float): Maximum cost of all jobs in batch. If
                exceeded the batch will terminate.
            n_iterations (int): Total number of iterations before ending
                the batch session.
        Returns:
            Callable[[ndarray], float]
        """
        terms = [term for term in problem_hamiltonian._dict.keys()]
        self._symbolic_circuit: Circuit = symbolic_circuit
        self._symbols: List[Symbol] = symbolic_circuit.free_symbols()
        self._hamiltonian: QubitPauliOperator = problem_hamiltonian
        self._backend: QuantinuumBackend = quantinuum_backend
        self._nshots: int = n_shots_per_circuit
        self._max_batch_cost: float = max_batch_cost
        self._measurement_setup: MeasurementSetup = measurement_reduction(
            terms, strat=PauliPartitionStrat.CommutingSets
        )
        self._iteration_number: int = 0
        self._niters: int = n_iterations

    def __call__(self, parameter: ndarray) -> float:
        value = self._objective_function(parameter, self._iteration_number)
        self._iteration_number += 1
        if self._iteration_number >= self._niters:
            self._iteration_number = 0
        return value

    def circuit_cost(self, syntax_checker: str = "H1-1SC") -> float:
        n = len(self._symbolic_circuit.free_symbols())
        random_parameters = random_sample(n)
        return sum(
            [
                self._backend.cost(c, self._nshots, syntax_checker=syntax_checker)
                for c in self._build_circuits(random_parameters)
            ]
        )

    def _objective_function(
        self,
        parameters: ndarray,
        iteration_number: int,
    ) -> float:
        r"""Substitutes input parameters into the
        symbolic state-preparation circuit, and
        calculates the expectation value.
        Args:
            parameters (ndarray): A list of numpy.ndarray
        Returns:
            float
        """
        assert len(parameters) == len(self._symbolic_circuit.free_symbols())
        circuit_list = self._build_circuits(parameters)
        if not isinstance(self._backend, QuantinuumBackend):
            raise RuntimeError(
                "Batching is not supported for any backend other than QuantinuumBackend."
            )
        if iteration_number == 0:
            self._startjob = self._backend.start_batch(
                self._max_batch_cost,
                circuit_list[0],
                self._nshots,
                noisy_simulation=False,
                options={"tket-opt-level": None},
            )
            handles = [self._startjob] + self._submit_batch(circuit_list[1:])
        else:
            handles = self._submit_batch(circuit_list)
        results = self._backend.get_results(handles)
        expval = compute_expectation_value(
            results, self._measurement_setup, self._hamiltonian
        )
        return expval

    def _build_circuits(self, parameters: ndarray) -> List[Circuit]:
        circuit = self._symbolic_circuit.copy()
        symbol_dict = {s: p for s, p in zip(self._symbols, parameters)}
        circuit.symbol_substitution(symbol_dict)
        circuit_list = []
        for mc in self._measurement_setup.measurement_circs:
            c = circuit.copy()
            c.append(mc)
            circuit_list.append(c)
        cc_list = self._backend.get_compiled_circuits(
            circuit_list, optimisation_level=2
        )
        return cc_list

    def _submit_batch(
        self,
        circuits: List[Circuit],
    ) -> List[ResultHandle]:
        r"""Submit a list of circuits with N shots each
        to the H-Series batch.
        Args:
            circuits (List[Circuit]): A list of circuits
                to submit to the batch on H-Series.
            first job (ResultHandle): The result handle for the
                first job submitted in the batch.
        Returns:
            List[ResultHandle]
        """
        return [
            self._backend.add_to_batch(
                self._startjob,
                c,
                self._nshots,
                options={"tket-opt-level": None},
                noisy_simulation=False,
            )
            for c in circuits
        ]

The Objective class is initialised with the essential data needed to perform the variational experiment. The object contains all the necessary information to compute the value of the objective function.

A convenience method circuit_cost can be used to estimate the total number of HQCs required to estimate the objective function. The variational loop will be multiples of this value (number of function calls across the variational procedure multiplied by the HQC cost of evaluating the objective function).

n_shots_per_circuit = 500
n_iterations = 10
max_batch_cost = 500
objective = Objective(
    symbolic_circuit,
    hamiltonian,
    quantinuum_backend,
    n_shots_per_circuit,
    max_batch_cost=max_batch_cost,
    n_iterations=n_iterations,
)

objective.circuit_cost("H1-1SC")

4.2. Execute the Objective Function

The SciPy minimizer is used to optimize the value of the objective function. Initial parameters are pseudo-random. Passing the Objective instance into scipy.optimize.minimize will start the variational experiment.

The first iteration creates a batch session, and all subsequent circuit submission are added to this batch. If additional circuits are not submitted within 1 minute, the batch session will terminate.

Remember that the status of the batch can be checked at any time on the Quantinuum User Portal.

from scipy.optimize import minimize
from numpy.random import random_sample

method = "COBYLA"
initial_parameters = random_sample(len(symbolic_circuit.free_symbols()))

result = minimize(
    objective,
    initial_parameters,
    method=method,
    options={"disp": True, "maxiter": objective._niters},
    tol=1e-2,
)

The minimal value of the objective function can be retrieved with the fun attribute.

print(f"VQE Energy:\t{result.fun} Ha")

The ground-state energy estimated with VQE can be compared with the value obtained from the numpy.linalg computation. The absolute error and the relative error is calculated here.

import numpy as np

abs_err = lambda experiment, benchmark: np.absolute(experiment - benchmark)
rel_err = (
    lambda experiment, benchmark: abs_err(experiment, benchmark)
    / np.absolute(benchmark)
    * 100
)

ae = abs_err(result.fun, ground_state_energy)
re = rel_err(result.fun, ground_state_energy)
print(f"Absolute error:\t{ae} Ha\nRelative error:\t{re}%")

The optimal parameters can be retrieved with the x attribute.

result.x

The Symbols can be mapped to the optimal parameter by iterating through both lists:

optimal_parameters = {s.name: p for s, p in zip(objective._symbols, result.x)}
print(optimal_parameters)

These symbols can be saved to an output file for further use if necessary using json. See the example below.

import json

json_io = open("parameters.json", "w")
json.dump(optimal_parameters, json_io)