Coherent control errors and dynamical decoupling

In a rotating frame, suppose an effective two-level Hamiltonian contains a transverse drive of angular frequency $\Omega$, phase $\phi$, and detuning $\delta$. The familiar expression assumes an appropriate rotating-wave approximation and neglects higher levels. With time evolution $U=e^{-iHt/\hbar}$, a positive resonant X drive gives $R_x(\theta)=e^{-i\theta X/2}$ with $\theta=\Omega t$. Acting on zero, it produces a negative imaginary amplitude on one. This fixes the rotation sign before any geometric interpretation.

At an X rotation of pi over two, the resulting Bloch vector points along negative Y. In contrast, positive Z evolution adds a positive relative phase to the one amplitude and rotates an equatorial vector from positive X toward positive Y. Confusing a ket's global phase with its relative phase often reverses this prediction. Angles in exponentials are radians; a frequency quoted in ordinary hertz must be multiplied by two pi before use as an angular frequency.

Consider free evolution under constant detuning alone. An ideal X pulse reverses the sign of Z by conjugation, so two equal free intervals separated by that pulse acquire opposite detuning phases in the toggling frame. Including a final ideal X returns the logical orientation. The two-pulse cycle cancels static detuning exactly in this model. Physical pi pulses have an additional global phase, which does not alter the recovered quantum state.

If detuning changes during the cycle, cancellation depends on the difference between its time integrals in the two intervals. Fast fluctuations need not cancel, and finite-duration pulses experience the unwanted Hamiltonian while being applied. Pulse imperfections can accumulate as well. More elaborate decoupling sequences control additional components only under specified timing, bandwidth, and noise-spectrum assumptions. Refocusing a slowly varying phase does not reverse general irreversible relaxation, measurement backaction, or arbitrary system-environment dynamics.

A systematic pulse-area error changes the intended rotation angle on each application. Repeating the same faulty pulse can amplify its phase error quadratically in state infidelity for small accumulated angles. Randomized error models therefore need justification even when a single-gate average infidelity is small. Calibration estimates amplitude, phase, and drift through repeated prepared experiments; it does not reveal the state of an arbitrary qubit during a computation without disturbing it.

Real devices can leak outside the computational doublet, making a Bloch-sphere picture incomplete. Decoupling pulses can even aggravate leakage if their spectral content excites unwanted transitions. Feedback additionally requires a measured observable, finite latency, and a policy connecting outcomes to control changes. Stabilizer feedback can preserve an encoded state because its checks reveal no logical amplitude in the ideal model; measuring arbitrary single-qubit populations has different backaction. Any claimed improvement must state these measurement and control assumptions alongside the noise model.