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Entanglement can be performed just between two ** adjacent qubits** (e.g. 1 and 2) or even non-physically adjacent qubits on the processor architecture can be entangled (e.g. 2 and 6).

double T-shape coupling map

How entangled between non-physically adjacent qubits could happen?

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    $\begingroup$ You are using incorrect terminology. Qubits 2 and 6 are connected but not adjacent. Since they are connected you can entangle them by series of swap operations. $\endgroup$
    – MonteNero
    Commented Dec 11, 2022 at 15:25

1 Answer 1

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As @MonteNero mentioned in the comment, it is posible to entangle non-adjacent qubits using $\mathrm{SWAP}$ gates.

In Qiskit, there is a transpilation stage called routing that introduce the $\mathrm{SWAP}$ gates for exactly that propose. Taking the example from your question:

The following circuit entangles the qubits $q_2$ and $q_6$:

from qiskit import QuantumCircuit

circuit = QuantumCircuit(7)
circuit.cx(2,6)
circuit.draw('mpl')

cnot between q2 and q6 The backend you are suggesting has the shape many of the IBM's Falcon r5.11H, such as ibm_oslo

from qiskit.providers.fake_provider import FakeOslo
from qiskit.visualization import plot_gate_map

backend = FakeOslo()
plot_gate_map(backend)

oslo coupling map

The transpiler will insert the $\mathrm{SWAP}# gates to make possible the entanglement between adjacent qubits.

from qiskit import transpile

transpile(circuit, coupling_map=backend.coupling_map, layout_method='trivial').draw('mpl', with_layout=False)

transpiled circuit with swap gates

Let's analyse this case layer by layer:

  • Layer 1, $q_1$ swaps with $q_2$ and $q_5$ with $q_6$. Logically speaking, you can see the connectivity map like this now:

coupling map after layer 1

  • Layer 2, $q_3$ swaps with $q_2$:

coupling map after layer 2

Now, $q_6$ and $q_2$ are adjacent and the $\mathrm{CNOT}$ gate between them works.

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