--- jupytext: formats: md:myst,ipynb text_representation: extension: .myst format_name: myst format_version: 0.13 jupytext_version: 1.11.1 kernelspec: display_name: Python 3 (ipykernel) language: python name: python3 --- # Using ZNE to compute the energy landscape of a variational circuit with Qiskit This tutorial shows an example in which the energy landscape for a two-qubit variational circuit is explored with and without error mitigation, using Qiskit as our frontend. {code-cell} ipython3 import matplotlib.pyplot as plt import numpy as np import qiskit from qiskit import QuantumCircuit from qiskit_aer import Aer, AerSimulator from qiskit_aer.noise import NoiseModel from qiskit_aer.noise.errors.standard_errors import ( depolarizing_error, ) from mitiq.zne import mitigate_executor from mitiq.zne.inference import RichardsonFactory  ## Defining the ideal variational circuit in Qiskit We define a function which returns a simple two-qubit variational circuit depending on a single parameter $\gamma$ ("gamma"). {code-cell} ipython3 def variational_circuit(gamma: float) -> QuantumCircuit: """Returns a two-qubit circuit for a given variational parameter. Args: gamma: The variational parameter. Returns: The two-qubit circuit with a fixed gamma. """ circuit = QuantumCircuit(2) circuit.rx(gamma, 0) circuit.cnot(0, 1) circuit.rx(gamma, 1) circuit.cnot(0, 1) circuit.rx(gamma, 0) return circuit  We can visualize the circuit for a particular $\gamma$ as follows. {code-cell} ipython3 circuit = variational_circuit(gamma=np.pi) circuit.draw()  ## Defining the executor functions with and without noise To use error mitigation methods in Mitiq, we define an executor function which computes the expectation value of a simple Hamiltonian $H=Z \otimes Z$, i.e., Pauli-$Z$ on each qubit. To compare to the noiseless result, we define both a noiseless and a noisy executor below. {code-cell} ipython3 # observable to measure z = np.diag([1, -1]) hamiltonian = np.kron(z, z) def noiseless_executor(circuit: QuantumCircuit) -> float: """Simulates the execution of a circuit without noise. Args: circuit: The input circuit. Returns: The expectation value of the ZZ observable. """ # avoid mutating the input circuit circ = circuit.copy() circ.save_density_matrix() # execute experiment without noise job = qiskit.execute( experiments=circ, backend=AerSimulator(method="density_matrix"), noise_model=None, # we want all gates to be actually applied, # so we skip any circuit optimization optimization_level=0, shots=1, ) rho = job.result().data()["density_matrix"] expectation = np.real(np.trace(rho @ hamiltonian)) return expectation # strength of noise channel noise_level = 0.04 def executor_with_noise(circuit: QuantumCircuit) -> float: """Simulates the execution of a circuit with depolarizing noise. Args: circuit: The input circuit. Returns: The expectation value of the ZZ hamiltonian. """ # avoid mutating the input circuit circ = circuit.copy() circ.save_density_matrix() # Initialize qiskit noise model. In this case a depolarizing # noise model with the same noise strength on all gates noise_model = NoiseModel() noise_model.add_all_qubit_quantum_error( depolarizing_error(noise_level, 1), ["rx"] ) noise_model.add_all_qubit_quantum_error( depolarizing_error(noise_level, 2), ["cx"] ) # execute experiment with depolarizing noise job = qiskit.execute( experiments=circ, backend=AerSimulator(method="density_matrix"), noise_model=noise_model, basis_gates=noise_model.basis_gates + ["save_density_matrix"], # we want all gates to be actually applied, # so we skip any circuit optimization optimization_level=0, shots=1, ) rho = job.result().data()["density_matrix"] expectation = np.real(np.trace(rho @ hamiltonian)) return expectation  {note} The above code block uses depolarizing noise, but any Qiskit NoiseModel can be substituted in.  ## Computing the landscape without noise We now compute the energy landscape $\langle H \rangle(\gamma) =\langle Z \otimes Z \rangle(\gamma)$ on the noiseless simulator. {note} The remaining code in this tutorial is generic and does not depend on a particular frontend.  {code-cell} ipython3 gammas = np.linspace(0, 2 * np.pi, 50) noiseless_expectations = [noiseless_executor(variational_circuit(g)) for g in gammas]  The following code plots the values for visualization. {code-cell} ipython3 plt.figure(figsize=(8, 6)) plt.plot(gammas, noiseless_expectations, color="g", linewidth=3, label="Noiseless") plt.title("Energy landscape", fontsize=16) plt.xlabel(r"Ansatz angle $\gamma$", fontsize=16) plt.ylabel(r"$\langle H \rangle(\gamma)$", fontsize=16) plt.legend(fontsize=14) plt.ylim(-1, 1); plt.show()  ## Computing the unmitigated landscape We now compute the unmitigated energy landscape $\langle H \rangle(\gamma) =\langle Z \otimes Z \rangle(\gamma)$ in the following code block. {code-cell} ipython3 gammas = np.linspace(0, 2 * np.pi, 50) expectations = [executor_with_noise(variational_circuit(g)) for g in gammas]  The following code plots these values for visualization along with the noiseless landscape. {code-cell} ipython3 plt.figure(figsize=(8, 6)) plt.plot(gammas, noiseless_expectations, color="g", linewidth=3, label="Noiseless") plt.scatter(gammas, expectations, color="r", label="Unmitigated") plt.title(rf"Energy landscape", fontsize=16) plt.xlabel(r"Ansatz angle $\gamma$", fontsize=16) plt.ylabel(r"$\langle H \rangle(\gamma)$", fontsize=16) plt.legend(fontsize=14) plt.ylim(-1, 1); plt.show()  ## Computing the mitigated landscape We now repeat the same task but use Mitiq to mitigate errors. We initialize a RichardsonFactory with scale factors [1, 3, 5] and we get a mitigated executor as follows. {code-cell} ipython3 fac = RichardsonFactory(scale_factors=[1, 3, 5]) mitigated_executor = mitigate_executor(executor_with_noise, factory=fac)  We then run the same code above to compute the energy landscape, but this time use the mitigated_executor instead of just the executor. {code-cell} ipython3 mitigated_expectations = [mitigated_executor(variational_circuit(g)) for g in gammas]  Let us visualize the mitigated landscape alongside the unmitigated and noiseless landscapes. {code-cell} ipython3 plt.figure(figsize=(8, 6)) plt.plot(gammas, noiseless_expectations, color="g", linewidth=3, label="Noiseless") plt.scatter(gammas, expectations, color="r", label="Unmitigated") plt.scatter(gammas, mitigated_expectations, color="b", label="Mitigated") plt.title(rf"Energy landscape", fontsize=16) plt.xlabel(r"Variational angle $\gamma$", fontsize=16) plt.ylabel(r"$\langle H \rangle(\gamma)$", fontsize=16) plt.legend(fontsize=14) plt.ylim(-1.5, 1.5); plt.show()  Noise usually tends to flatten expectation values towards a constant. Therefore error mitigation can be used to increase the visibility the landscape and this fact can simplify the energy minimization which is required in most variational algorithms such as VQE or QAOA. We also observe that the minimum of mitigated energy approximates well the theoretical ground state which is equal to $-1$. Indeed: {code-cell} ipython3 print(f"Minimum of the noisy landscape: {round(min(expectations), 3)}") print(f"Minimum of the mitigated landscape: {round(min(mitigated_expectations), 3)}") print(f"Theoretical ground state energy: {min(np.linalg.eigvals(hamiltonian))}")