# qiskit.synthesis.synth_permutation_depth_lnn_kms

`synth_permutation_depth_lnn_kms(pattern)`

Synthesize a permutation circuit for a linear nearest-neighbor architecture using the Kutin, Moulton, Smithline method.

This is the permutation synthesis algorithm from https://arxiv.org/abs/quant-ph/0701194(opens in a new tab), Chapter 6. It synthesizes any permutation of n qubits over linear nearest-neighbor architecture using SWAP gates with depth at most n and size at most n(n-1)/2 (where both depth and size are measured with respect to SWAPs).

**Parameters**

**pattern** (*Union[list[int], np.ndarray]*) – permutation pattern, describing which qubits occupy the positions 0, 1, 2, etc. after applying the permutation. That is, `pattern[k] = m`

when the permutation maps qubit `m`

to position `k`

. As an example, the pattern `[2, 4, 3, 0, 1]`

means that qubit `2`

goes to position `0`

, qubit `4`

goes to position `1`

, etc.

**Returns**

the synthesized quantum circuit.

**Return type**