Autonomous Quantum Machines and Finite-Sized Clocks

Processes such as quantum computation, or the evolution of quantum cellular automata, are typically described by a unitary operation implemented by an external observer. In particular, an interaction is generally turned on for a precise amount of time, using a classical clock. A fully quantum mechanical description of such a device would include a quantum description of the clock whose state is generally disturbed because of the back-reaction on it. Such a description is needed if we wish to consider finite-sized autonomous quantum machines requiring no external control. The extent of the back-reaction has implications on how small the device can be, on the length of time the device can run, and is required if we want to understand what a fully quantum mechanical treatment of an observer would look like. Here, we consider the implementation of a unitary by a finite-sized device and show that the back-reaction on it can be made exponentially small in the device’s dimension while its energy only increases linearly with dimension. As a result, an autonomous quantum machine need only be of modest size and energy. We are also able to solve a long-standing open problem by using a finite-sized quantum clock to approximate the continuous evolution of an idealised clock. The result has implications for how well quantum devices can be controlled and on the equivalence of different paradigms of control.