If your question was about quantum algorithms and the paradigm
of quantum computing, then there is a paper you might be interested in:
"A quantum algorithm for finding the minimum"
by Durr and Hoyer
One pattern in quantum computing algorithms is yes, to set up
the superposition of qubits as states in a register,
operate on those (that is, gates operating on qubits that comprise the states), then take measurements.
The algorithm in the paper is a demonstration of the use of increasingly lowered thresholds and filtered initialized registers that are put through Grover's quantum number finding algorithm. Each iteration is followed by measurement of the first register and comparison to the threshold. The threshold is the minimum at some iteration and the total number of iterations is determined prior to the iterations to ensure success.
Using the algorithm on a list of numbers that have gaps in them, which could be a register holding pre-processed states, seems to need a larger register qubit width so the runtime complexity for input lists is larger than the usual quoted.
It looks like one of the main abilities of quantum computing will be or is combinatorial problems and what steps are needed to put the system into a certain state with high probability.
Note that if your question is hardware based, then the answer presumably depends upon the quantum computer implementation. There is a paper discussing sorting:
"Sorting quantum systems efficiently" by Ionicioiu
If your question was instead about minimum of the amplitudes of all possible states at any given time (rather than the value of a state),
then you'd probably be interested in the papers regarding error analysis in progress because that's an attempt to quantify the lowest probable state...
As a tangent, could consider that one of the goals is storing multiplicity of state. Atoms and molecules have rotational, vibrational, and electronic transitions and any such system needs a decent number of particles in the given state for good statistics and the system needs isolation of its parts.
For example, using sound as a means to facilitate rotational transitions has as its challenge that being a pressure wave, it is quickly everywhere it can be when there is a medium to transport it. Using low frequency radiation has as its challenges the diffraction limit and near field effects... Lots of interesting research happening which may have different answers to your question.