How to calculate an Expected Value of some operator acting on qubits?

Question

I'm trying to implement the Variational Quantum Eigensolver in Qiskit.

Suppose, I have an operator $A = \sigma_1^z\sigma_2^z$ acting on some two-qubit state $|\psi\rangle$. After a measurement I get a set of probabilities corresponding to states $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$.

My question is: How to calculate $\langle\psi|A|\psi\rangle$ using known set of probabilities?

Answer 1

Qiskit currently supports measurements in the computational basis from Qiskit Terra and Aer, that is, returning 1 if the qubit is in state $|1\rangle$, and 0 if the qubit is measured to be in state $|0\rangle$.

However, it is relatively easy to perform a change of basis unitary to our quantum circuit just prior to measurement, in order to instead measure in the eigenbasis of an arbitrary operator $A$:

1. First, we need to compute the eigendecomposition of our Hermitian observable $A|v_i\rangle=\lambda_i|v_i\rangle$.

2. Since $A$ is Hermitian, we are always able to diagonalize $A$ in this eigenbasis:

   $$A = \sum_{i}\lambda_i |v_i\rangle \langle v_i|= U^\dagger \Lambda U$$

   where $\Lambda = \text{diag}(\lambda_1,\dots,\lambda_n)$ and $U$ is a unitary matrix composed of the eigenvectors down the columns.

3. Next, we apply the unitary operation $U$ to the end of our quantum circuit, using qc.unitary.

   To see why, note that $$\langle\psi|A|\psi\rangle = \langle\psi|U^\dagger \Lambda U|\psi\rangle = \langle\psi'|\Lambda|\psi'\rangle $$ where $|\psi'\rangle = U|\psi\rangle.$ Since $\Lambda$ is diagonal, we have transformed the problem from one where we must perform a measurement in an arbitrary basis, to one where we simply measure in the computational basis.

4. Finally, we measure the state $|\psi'\rangle$ in the computational basis $|i \rangle$ using qc.measure(q, c), and execute the jobs to get the counts and probabilities $\mathbb{P}_i = |\langle i|\psi'\rangle|^2$. Using the previously computed eigenvalues of $A$, we can now reconstruct the expectation value:

   $$\langle \psi | A | \psi\rangle = \langle\psi'|\Lambda|\psi'\rangle = \sum_i \langle\psi'|i\rangle \langle i | \Lambda | i\rangle \langle i|\psi'\rangle = \sum_i \lambda_i \mathbb{P}_i$$

For example,

shots = result.results[0].shots
counts = result.get_counts()
probs = sorted([(i, c/shots) for i, c in counts.items()])
P = np.float64(np.array(probs)[:, 1])
A_expectation = lambda @ p

---

For a more high-level interface to coding and running variational quantum algorithms, you can also check out the PennyLane Python library, which has a Qiskit plugin available for using Qiskit simulators and IBM hardware as a backend.

For example, a expectation values of arbitrary operators in PennyLane using qiskit looks like this:

import pennylane as qml
dev = qml.device('qiskit.basicaer', wires=2)
use 'qiskit.ibm' instead to run on hardware
@qml.qnode(dev)
def circuit(x, y, z):
    qml.RX(x, wires=0)
    qml.RY(y, wires=1)
    qml.RZ(z, wires=0)
    qml.CNOT(wires=[0, 1])
    return qml.expval(qml.Hermitian(A, wires=[0, 1]))
def cost(x, y, z):
    return (1-circuit(x, y, z))**2
optimization follows

You can use NumPy, TensorFlow, or PyTorch to do the optimization - check out some of the tutorials.

Disclaimer: I am one of the developers on PennyLane.

Answer 2

This can actually be easily done using the Qiskit Terra qiskit.quantum_info.analysis.average.average_data function that takes the counts data returned by a backend and the desired observable defined by a dict, list, or ndarray.

The doc-string for that function actually has the ZZ your looking for as an example.

Answer 3

I think it can be done with the method in this question(The first answer).

1. Prepare $|0\rangle$
2. Apply $V$
3. Apply $U$
4. Apply $V^\dagger$
5. Measure in a basis for which $|0\rangle$ is one of the elements
6. Repeat until you have a suitably accurate estimate of the probability of getting the answer $|0\rangle$.

Substitue $V$ with $U(\theta)$ will work. $$U(\theta)|0\rangle = |\psi \rangle$$