General formalism for evaluating the impact of phase noise on Bloch vector rotations

Quantum manipulation protocols for quantum sensors and quantum computation often require many single-qubit rotations. However, the impact of phase noise in the field that performs the qubit rotations is often neglected or treated only for special cases. We present a general framework for calculating the impact of phase noise on the state of a qubit, as described by its equivalent Bloch vector. The analysis applies to any Bloch vector orientation and any rotation axis azimuthal angle for both a single pulse and pulse sequences. Experimental examples are presented for several special cases. We apply the analysis to commonly used composite p pulse sequences for suppression of static amplitude or detuning errors and also to spin-echo sequences. We expect the formalism presented will help guide the development and evaluation of future quantum manipulation protocols.