Former research in quantum computers focussed on quantum dots, which are just 10 atoms across. With quantum dots laser pulses are used to cause an electron to jump between two different excitation states, and while coherent between these two states, perform quantum computations. The problem is the electron loses this coherence in less than a microsecond, and the laser pulse needed was a billionth of a second, which limits the number of computations that can be performed in that short time interval. The power of the quantum dot computer required using more of them, but each one would need its own laser. Packing a thousand lasers into such a small space was impossible, and single lasers that could tune themselves reliably to use different wavelengths do not exist. Quantum dots just didn't work.

Enter the liquid quantum computer. Billions of molecules in a vial are held in a magnetic field, and as such move in circles along magnetic lines of force. This is used to describe their states. Although the molecules bump into one another they do not seem to affect each other's quantum state, and so liquid quantum computers maintained their coherence longer. Nuclear Magnetic Resonance is the tool used to read the quantum states and perform operations on the molecules. The long durations allowed the coupling of multiple qubits for long enough to factor the number 15 using Shor's Algorithm.

Here's the article in this week's Nature journal:

NRM Quantum Computing: Realizing Shor's Algorithm

For a better explanation of quantum computers here's a great website I found explaining Shor's algorithm, and liquid quantum computers:

A Brief History of Quantum Computing