First things first: This question is not about digital computers on the nano scale. It is about the field of molecular logic.
Molecular Logic describes a subfield of chemistry in which information is processed by the interaction of single molecules with their surroundings. In this picture the transistor in a digital computer is replaced by a macromolecule in solution. Sending an electrical pulse could correspond to adding specific ions that can bind to the molecule in the solution. The change in optical properties of the macromolecule would be the output. Other approaches use lasers to induce a conformational transition in the macromolecule instead. Molecular logic building blocks can thus be sensitive to many different forms of input that can be combined. One could for example think about a molecule that can bind more than one ion species. However this versatility has its drawbacks, as the molecules have to be specifically designed to react to a certain input in a well defined way.
The idea of a molecular computer might sound weirder than it is. After all our bodies use molecular logic everyday, e.g. to convert the presence of capsaicin in our food into the sensation of heat in our mouth. For this reason researchers look for inspiration in nature on how to accurately process information using chemistry . For this reason molecular logic is usually mentioned in the context of sensor applications, such as devices for fast and reliable detection of pollutants in the environment.
Another application that has recently attracted attention is the use of molecular logic for keypads . The system can be adjusted to accept different chemical keys so that the end user would not have to design the molecular logic gate itself. Due to the working mechanisms of these molecular keypads they are potentially more secure than digital ones.
Speculations suggest that some molecular logic units can potentially perform more complex operations than regular transistors at a tenth of the size . The mechanisms for manipulating the input and reading the output is still very complicated and an obstacle to miniaturization.
The goal of molecular logic in general is thus not to replace traditional computer systems but rather augment them. It could do this on a similar principle as a quantum computer by tackling problems to which its “hardware” is naturally suited.
There are still many hurdles on the way to molecular computers so it remains to be seen what problems they will tackle in the future.
– Stephan Koehler
 Scholes, “Light-powered molecular logic goes nonlinear ”, Proceedings of the National Academy of Science 110, 17167 (2013)
 Rout et al., “Authorizing Multiple chemical Passwords by a Combinatorial Molecular Keypad Lock”, Journal of the American Chemical Society, 135 (41), 15330 (2013)