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Is it possible to apply a CNOT gate in MBQC to qubits that aren´t next to each other?

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  • $\begingroup$ Please state what is behind abbrev. MQBC. $\endgroup$ Nov 27, 2023 at 7:08

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Maybe you should clarify exactly what you mean by MBQC, because to me this seems like a trivial "of course you can".

For example, a CNOT can be implemented up to Pauli tracking by this circuit:

enter image description here

import stim
c = stim.Circuit("""
    MZ 99
    MXX 1 99
    MZZ 0 99
    MX 99
""")

# Verify the stabilizer flow generators of a CNOT are present.
# (requires stim~=1.13.dev)
assert c.has_flow("X_ -> XX", measurements=[1, 3])
assert c.has_flow("Z_ -> Z_")
assert c.has_flow("_X -> _X")
assert c.has_flow("_Z -> ZZ", measurements=[0, 2])

The right hand side only contains measurements (the Paulis happen in the classical control system, not on the quantum computer). Pretty much as measurement based as you can possibly get. So requiring the use of measurements places no constraints on which qubits you can do a CNOT between. That will have to flow from things like constraints on which qubits you can do Mxx and Mzz between.

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  • $\begingroup$ MBQC stands for measurement based quantum computation. $\endgroup$
    – Manuel
    Nov 27, 2023 at 15:05
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    $\begingroup$ @Manuel yes I know what the acronym expands into. But which specific flavor of MBQC did you have in mind? Cluster states? Lattice surgery? Honeycomb codes? Delayed choice gates? It's a whole world out there. $\endgroup$ Nov 27, 2023 at 16:19
  • $\begingroup$ I first thought of Cluster states. I just need the best paradigm to apply Cnot gates between far distanced qubits in MBQC without needing to swap them in place. It would be also practical to have a Toffoli gate for MBQC that doesn´t need to be decomposed like in qiskit quantum circuits. $\endgroup$
    – Manuel
    Nov 27, 2023 at 16:46

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