Suppose you have a standard spin qubit: spin "up" is $|1\rangle$ and spin "down" is $|0\rangle$ . Your qubit is in some storage location which does not couple the two spin directions to each other or have any different energy, you store $|\psi\rangle = a |0\rangle + b |1\rangle$ in this qubit.
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Now some proton flies by, or a minor fluctuation in the Earth's magnetic field occurs, or someone in the next door facility engages the magnets on the MRI machine. These will typically couple with the spin by some sort of tiny local Hamiltonian $\hat h /\hbar = \alpha |0\rangle\langle 0| + \beta |1\rangle\langle 1|$ for some time $\tau$. The state of the qubit is therefore changed to:$$|\psi'\rangle = a ~ e^{-i\alpha\tau} ~ |0\rangle + b ~e^{-i\beta\tau}~|1\rangle.$$A global phase factor is unobservable, but this system will behave as if $b\rightarrow b~e^{-i(\beta - \alpha)\tau}$, which could impact further experimentation.
These errors can also occur if the electronics for "switching on" or "switching off" an operational Hamiltonian are not 100% instantaneous and predictable; you will in general get either a little bit too little $e^{-i ~\hat h ~t/\hbar}$ or too much of it. So if the interaction Hamiltonian that was supposed to create $\psi$ worked by inducing $e^{-i ~ \omega ~ t ~ \hat x}$ on some previous state, where $\hat x$ is the usual $|0\rangle\langle 1| + |1\rangle\langle 0|$ quadrature, the shift from $t \rightarrow t + dt$ causes an approximate error:$$|\psi'\rangle = e^{-i~\omega~dt~\hat x} \left( a ~ |0\rangle + b ~|1\rangle \right) \approx |\psi\rangle - i ~\omega~dt~\left(b|0\rangle + a |1\rangle\right)$$That in turn could mess up not just the phases but the relative amplitudes of each term.
