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I have a system of $N$ qubits and want to construct a quantum operator $Z_i Z_j + Z_k$, where $Z_i$ denotes the Pauli-Z operator acting on the $i$th qubit. Is there any direct way in qiskit, how I could implement this?

I know that I can construct an operator by e.g. saying op = Z^Z, if I have a system of 2 qubits and want the operator being the Pauli-Z on each qubit. But I would like to tell qiskit the indices of the qubits that $Z$ should act on (such that on all the other qubits Identity is applied).

My way so far consists of constructing a Quantum Circuit and converting this to an operator by

circZZ = QuantumCircuit(N)  # circuit for Z_i Z_j
circZ = QuantumCircuit(N)  # circuit for Z_k
circZZ.z(i)
circZZ.z(j)
circZ.z(k)
opZZ = CircuitOp(circZZ)  # convert circuit to operator
opZ = CircuitOp(circZ)  # convert circuit to operator
op = opZZ + opZ

But that means I have to create quantum circuits everytime I want to get this operator. Is there any shorter and more elegant way to create such an operator?

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The class Operator has a from_label method: https://qiskit.org/documentation/stubs/qiskit.quantum_info.Operator.html?highlight=operator%20from_label#qiskit.quantum_info.Operator.from_label

That means you could do something like this:

opZZ = Operator.from_label('ZZ')
opZ = Operator.from_label('Z')

It's possible to add opZZ and opZ into N-sized op. However, you have to call the _add by hand:

op = 0 * Operator.from_label('I' * N)  # Set the initial operator to zero
op = op._add(opZZ, qargs=[i,j])
op = op._add(opZ, qargs=[k])
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  • $\begingroup$ Thanks for your answer. I guess that is the same as what I would do with opZZ = Z^Zand opZ = Z. But that's not what I mean. I would like to tell qiskit that $Z_i Z_j$ acts on the $i$th and $j$th qubit, without setting the label to 'II..IZI...IZI...II' with $Z$ being on the $i$th and $j$th position of the string. Is there any way to tell qiskit the index of the qubit the operator should act on? $\endgroup$ – ile2N Dec 7 '20 at 13:07
  • $\begingroup$ If I understand everything correctly, the system has three qubits, $i$, $j$, and $k$. If you want to operatore with those operators, I think they have to have the same size opZ = Operator.from_label('IIZ') and opZZ = Operator.from_label('ZZI'). $\endgroup$ – luciano Dec 7 '20 at 13:19
  • $\begingroup$ No, the system has not only 3 qubits. It has $N$ qubits, and $i,j,k$ are three of these. $N$ can be any number larger than 3. I would like to address the qubit by indices, is this possible? $\endgroup$ – ile2N Dec 7 '20 at 13:25
  • $\begingroup$ I see! The size is an immutable characteristic of an Operator (see Operator.num_qubits). So N needs to be defined at construction time. Then, there is the problem of construction. For that, I think 'II..IZI...IZI...II' is the best way to go so far, because I dont think there is an easy way to modify a specific qubit in an Operator instance. Shall I modify the answer to do it in a programmatically way? $\endgroup$ – luciano Dec 7 '20 at 13:49
  • $\begingroup$ Just to clarify what I said above: $N$ is indeed fixed when the operator is created. I just wanted to point out that it is not necessarily $N=3$, but could also be some arbitrary large number, which is why I wanted to address the qubits to be manipulated by the indices. I think I will stick to my first approach to create a circuit first, since I need to create this operator for different qubits several times. Thank you anyways for your answer! $\endgroup$ – ile2N Dec 7 '20 at 13:58
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You could use the feature that the Opflow in Aqua can take integers as tensorpower, like Z ^ 5 and then fill the blanks with identities. In a short function that could look like

from qiskit.aqua.operators import Z, I

def get_term(i, j, k, n):
    """i, j and k as in your description and n is the number of qubits."""
    zz = (I ^ i) ^ Z ^ (I ^ j - i - 1) ^ Z ^ (I ^ (n - j - i - 1))
    z = (I ^ k) ^ Z ^ (I ^ (n - k - 1))
    return zz + z

print(get_term(0, 2, 4, 5))
# 1.0 * ZIZII
# + 1.0 * IIIIZ

Note that the order of Z's is reversed to what you did with the circuits, so to get the same results you can just call

get_term(n - i - 1, n - j - 1, n - k - 1, n)

But this is just one way to get to the result, I'm sure there are many others! Your method, via circuits, is looks perfectly good to me.

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