Added an example for VQPE

#include "qcor_qsim.hpp"

// Using noise-mitigation observable evaluation (Verified QPE)

// Compile and run with (e.g. using the 5-qubit Yorktown device)

/// $ qcor -qpu aer:ibmqx2 TrotterTdWorkflow.cpp

/// $ ./a.out

// Note: for simplicity, we use the time-dependent Ising model with only 2

// qubits, hence with QPE, we need a total of 3 qubits. These 3 qubits can be

// directly mapped to the triangular topology of the Yorktown device.

// (Ref: Fig 3 of PhysRevB.101.184305)

int main(int argc, char **argv) {

  // Define the time-dependent Hamiltonian and observable operator using the

  // QCOR API Time-dependent Hamiltonian

  qsim::TdObservable H = [](double t) {

    // Parameters:

    const double Jz = 2 * M_PI * 2.86265 * 1e-3;

    const double epsilon = Jz;

    const double omega = 4.8 * 2 * M_PI * 1e-3;

    return -Jz * Z(0) * Z(1) - epsilon * std::cos(omega * t) * (X(0) + X(1));

  };

  // Observable = average magnetization

  auto observable = (1.0 / 2.0) * (Z(0) + Z(1));

  // Example: build model and TD workflow for Fig. 2 of

  // https://journals.aps.org/prb/pdf/10.1103/PhysRevB.101.184305

  auto problemModel = qsim::ModelBuilder::createModel(observable, H);

  // Trotter step = 3fs, number of steps = 100 -> end time = 300fs

  // Request the Verified QPE observable evaluator:

  auto vqpeEvaluator =

      qsim::getObjEvaluator(observable, "qpe", {{"verified", true}});

  auto workflow =

      qsim::getWorkflow("td-evolution", {{"method", "trotter"},

                                         {"dt", 3.0},

                                         {"steps", 100},

                                         {"evaluator", vqpeEvaluator}});

  // Result should contain the observable expectation value along Trotter steps.

  auto result = workflow->execute(problemModel);

  // Get the observable values (average magnetization)

  const auto obsVals = result.get<std::vector<double>>("exp-vals");

  // Print out for debugging:

  for (const auto &val : obsVals) {

    std::cout << "<Magnetization> = " << val << "\n";

  }

  return 0;

}