We introduce a new class of computer called a quantum frequential computer. They harness quantum properties in a different way to conventional quantum computers to generate a quadratic computational run time advantage for all algorithms as a function of the power consumed. They come in two variants: type 1 can process classical algorithms only while type 2 can also process quantum ones. In a type-1 quantum frequential computer, only the control is quantum, while in a type 2 the logical space is also quantum. We also prove that a quantum frequential computer only requires a classical data bus to function. This is useful, because it means that only a relatively small part of the overall architecture of the computer needs to be quantum in a type-1 quantum frequential computer in order to achieve a quadratic run time advantage. As with classical and conventional quantum computers, quantum frequential computers also generate heat and require cooling. We also characterise these requirements.