3
$\begingroup$

I want to make a circuit that measures the expected value of a projector. In this case I want to measure the expected value of the singlet projector operator, that is a non-unitary hermitian matrix. How can I do this in Qiskit?

The singlet state is defines as:

$\frac{|01>-|10>}{\sqrt{2}}$

And the matrix of the projector is: \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 1/2 & -1/2 & 0\\ 0 & -1/2 & 1/2 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix}

$\endgroup$

2 Answers 2

2
$\begingroup$

Qiskit contains the the tools to convert your operator from a matrix representation to a sum of Paulis, which you can measure on a quantum circuit. On a high-level you could write

import numpy as np

from qiskit.circuit import QuantumCircuit
from qiskit.opflow import MatrixOp, StateFn

# let's first define your projector via the matrix you specified above
matrix = np.zeros((4,4))
matrix[1:3, 1:3] = np.array([[1, -1], [-1, 1]]) /2
proj = MatrixOp(matrix)

# and now a state (given as a circuit) that we want to project
circuit = QuantumCircuit(2)
circuit.ry(0.2, 0)
circuit.ry(0.3, 1)

# then just evaluate the result
result = (StateFn(proj, is_measurement=True) @ StateFn(circuit)).eval()  # (0.001248958680493557+0j)

If you want to run this on a real backend, you could have a look this question or the Qiskit documentation. Also, here's how to convert your matrix to sums of Paulis:

print(proj.to_pauli_op())  
# SummedOp([
#   0.25 * II,
#   -0.25 * XX,
#   -0.25 * YY,
#   -0.25 * ZZ
# ]) 

Hope that helps!

$\endgroup$
0
$\begingroup$

not sure if that's the preferred way to do it, but you can use a controlled SWAP gate: Prepare an ancilla in the state $|+\rangle$ and use it as the control for a SWAP gate. Since the singlet state is the only eigenstate of the SWAP operator with eigenvalue -1, the ancilla will be equal to -1 at a fraction of cases that is equal to the amplitude squared of the singlet state. So just count only the the outcomes where the ancilla was equal to -1 - that's the expectation value.

Here's an implementation:

from qiskit.circuit import QuantumCircuit
from qiskit import Aer, transpile

c = QuantumCircuit(3, 3)

# prepare a singlet state
c.h(1)
c.x(2)
c.cnot(1, 2)
c.z(1)

# verify it's a singlet
c.h(0)
c.cswap(0, 1, 2)
c.h(0)
c.measure([0, 1, 2], [0, 1, 2])

c.draw()

backend_sim = Aer.get_backend('qasm_simulator')

job_sim = backend_sim.run(transpile(c, backend_sim), shots=10240)
counts = job_sim.result().get_counts()
sum(counts[key] for key in counts if key[-1] == '1')/10240
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.