Another noteworthy example of an early quantum computer may be Kwiat, Weinfurter, Herzog, Zeilinger, and Kasevic's early-90's instantiation of their improved bomb-tester, based on the quantum Zeno effect. See a pdf of their Phys Rev Letters article. See also a pdf of their Scientific American article, from which the below image was taken:
This device uses interaction-free measurement to freeze a polarized photon in a particular orientation, or otherwise to let the photon slowly rotate to a different basis, depending on whether a bomb is present or not.
I think it can be argued that this is a "quantum computer" that uses "qubits" in the sense that it performs quantum operations (rotations) to make a decision on whether or not a bomb is present, in a manner that is completely non-classical.
Also, although an idealized version has been described as a "one-qubit" computer, at least the specific implementation of Kwiat et al. utilized spontaneous parametric down-conversion that generated two entangled photons/qubits, with one photon being measured to herald the presence of the other.
This is from the mid-90's and doesn't pre-date Aspect, Grangier, and Roger's Bell violation experiment of the early 80's, but I would argue it is a bit more computational in a certain sense, and hints at more to come of the potential power of quantum computation. The experimental results were submitted to PRL in '94, which makes it pretty contemporaneous with Shor's algorithm.
Both Zeilinger and Aspect are Nobel laureates, for good reason! But Zeilinger's work here on the bomb tester wasn't explicitly mentioned by the Nobel committee, although Aspect's work mentioned by @JamesWootton on the Bell violations was.