Gate fidelity of self-adjoint gates constrained by conservation laws

Recent investigations show that conservation laws limit the accuracy of gate operations in quantum computing. The inevitable error under the angular momentum conservation law has been evaluated so far for the CNOT, Hadamard, and NOT gates for spin 1/2 qubits, while the SWAP gate has no constraint. Here, we consider arbitrary self-adjoint gates under arbitrary conservation laws from a geometrical point of view. A lower bound of the gate infidelity, one minus the squared gate fidelity, is obtained to be inversely proportional to the squared size of the controller and proportional to the squared sine of 2 times the relative angle between the axes of rotations on the Bloch sphere generated by the conserved quantity and the gate operation. Taking all the directions into account, we conclude that no self-adjoint gate on a spin 1/2 qubit can be implemented with a gate infidelity less than (4 + 4N) by any rotationally invariant interaction with a spin N/2 ancilla. In the case where the above axes are orthogonal like the NOT gate, the above bound vanishes, but by further elaborations the gate infidelity is shown to have another non vanishing lower bound. PACS numbers: 03.67.Lx, 03.67.-a, 03.65.Yz, 03.65.Ta Gate fidelity of self-adjoint gates constrained by conservation laws 2