Chapter 12 Approximation

About quantifying precision in implementations of quantum circuits using the notion of metrics — more specifically, the trace distance. Also about the practical feasibility of universal sets of gates and correctly distinguishing non-orthogonal states described by density operators.

We have talked a lot about preparing specific quantum states and constructing specific unitary operations, but the space of states of any quantum system is a continuous space, and the set of unitary transformations is also continuous. It is entirely unrealistic to imagine that in the actual world we will be able to prepare, for example, a qubit precisely in the state |0\rangle, or to perform a unitary transformation that is exactly equal to the controlled-not gate. We never have infinite precision in our manipulations of the physical world. The good news is that, for all practical purposes, infinite precision is not actually necessary, and we can achieve most of our goals by preparing quantum states and performing quantum operations that are “close enough” to the desired ones. But what is “close enough”, and how do we quantify it?