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I am interested in performing a SWAP measurement, namely, measure 2 qubits and project their state onto either the triplet state manifold $\{|00\rangle, |11\rangle, |01\rangle + |10\rangle\}$ or the singlet state $|01\rangle - |10\rangle$ with measurement outcome being +1 in the first case and -1 in the second. The thing is that it needs to be done on a grid of qubits with nearest neighbor coupling only, such as the Sycamore architecture, and the native gateset is CZ + single qubit rotations.

The "canonical" way to do this is a CSWAP with an ancilla as the control prepared in the $|+\rangle$ state and measured in the $X$ basis. However, in this architecture the ancilla is directly connected only one of the qubits and so it is very costly in 2 qubit gates.

My question is: if I have the ability to perform only local gates, namely single qubit gates and 2 qubit gates between nearest neighbor qubits, and I can also measure and perform classically conditioned operations on the qubits, where the condition can be any boolean function of the measured qubits, is there a way to perform this gate using less native gates? It is possible to use more ancillas if needed.

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Measure SWAP from 7 CNOTs, linear connectivity, ancilla outside:

enter image description here

(Note: improved this to not need any feedback anymore, and to just generally be cleaner.)

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  • $\begingroup$ to make sure - the gates that look like 3 qubit gates are a pair of CNOTs from the middle qubit to the 2 outside ones, not a Toffoli with a pair of Hadamards on the top qubit, right? $\endgroup$
    – Lior
    May 2, 2023 at 9:06
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    $\begingroup$ @Lior right. The image is a link to the quirk circuit to check. $\endgroup$ May 2, 2023 at 15:41

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