Added an example to verify the five-qubit code

#include <qcor_qec>

// Note: although QEC encode and syndrome decode kernels don't need FTQC

// runtime, in order to feed-forward syndrome, we need to use FTQC runtime.

// Compile with:

// $ qcor -qrt ftqc -qpu qpp five_qubit_qec_code.cpp

// Helper kernel to verify error-correcting procedure

__qpu__ void applyError(qreg q, int qIdx, int opId) {

  if (opId == 1) {

    std::cout << "Apply X error @ q[" << qIdx << "].\n";

    X(q[qIdx]);

  }

  if (opId == 2) {

    std::cout << "Apply Y error @ q[" << qIdx << "].\n";

    Y(q[qIdx]);

  }

  if (opId == 3) {

    std::cout << "Apply Z error @ q[" << qIdx << "].\n";

    Z(q[qIdx]);

  }

}

using namespace ftqc;

int main() {

  // Five-qubit code + 1 scratch qubit for syndrome measurement

  auto q = qalloc(6);

  const std::vector<int> LOGICAL_REG{0, 1, 2, 3, 4};

  const int ANC_QUBIT = 5;

  // Retrieve "five-qubit" code:

  auto [stabilizers, encodeFunc, recoverFunc] = getQecCode("five-qubit");

  // Encode the qubit: from qubit 0 to qubit register [0-4]

  encodeFunc(q, 0, {1, 2, 3, 4});

  // Test all possible *single-qubit* error

  for (int qId = 0; qId < 5; ++qId) {

    for (int opId = 1; opId <= 3; ++opId) {

      // If using a perfec simulator, apply a random error to observe syndrome

      // changes.

      applyError(q, qId, opId);

      std::vector<int> syndromes;

      // Measure the stabilizer syndromes:

      measure_stabilizer_generators(q, stabilizers, LOGICAL_REG, ANC_QUBIT,

                                    syndromes);

      assert(syndromes.size() == 4);

      std::cout << "Syndrome: <X0Z1Z2Z3> = " << syndromes[0]

                << "; <X1Z2Z3X4> = " << syndromes[1]

                << "; <X0X2Z3Z4> = " << syndromes[2]

                << "; <Z0X1X3Z4> = " << syndromes[3] << "\n";

      // Recover:

      recoverFunc(q, LOGICAL_REG, syndromes);

      // Measure again to check:

      syndromes.clear();

      measure_stabilizer_generators(q, stabilizers, LOGICAL_REG, ANC_QUBIT,

                                    syndromes);

      std::cout << "AFTER CORRECTION: \nSyndrome: <X0Z1Z2Z3> = " << syndromes[0]

                << "; <X1Z2Z3X4> = " << syndromes[1]

                << "; <X0X2Z3Z4> = " << syndromes[2]

                << "; <Z0X1X3Z4> = " << syndromes[3] << "\n";

    }

  }

}

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