What utility is provided by the Bloch sphere visualization?

Question

The Bloch sphere is a mainstay of introductions to quantum computing, but what utility does it actually provide? It can't usefully represent multiple qbits because of entanglement and requires a weird transformation to map to & from a state vector. Sure it's a visual representation but I've never seen it actually assist anyone's acquisition of the material. Is it just a holdover from earlier days of the field and the reason why we'll be calling Rx/Ry/Rz with doubled theta values until the stars die out?

Answer 1

Some people really do visualise single qubit operations by thinking about the Bloch sphere, and they work out everything based on that picture. Those who like thinking in that way are often incredibly quick at these sorts of manipulations, and have more intuitive understandings for why things work.

But here's one concrete example where I use it: averaging over single qubit states. Let's say I've implemented some quantum operation on a single qubit state, and I want to compare it to the ideal, so perhaps I calculate a fidelity. (For instance, I might be trying to implement optimal cloning.) Perhaps that fidelity varies depending on what the input state is, and so I want to calculate the average fidelity. How do I average over all possible single-qubit states? It's the same as integrating over the surface of the Bloch sphere, and that conversion helps me to get the correct volume element, with the correct factors of 2.

Of course, I could memorise that function, just as everything that can be done with the Bloch sphere can be done another way. Personally, I don't usually use the Bloch sphere, but this is the most obvious instance where I have stuck with it.

There are other topics where I believe the Bloch sphere is particularly natural. For example, in the topic of magic states - single-qubit states that when combined with Clifford operations provide universal, fault-tolerant quantum computation. There's a lovely convexity argument that eliminates a particular geometric region of the Bloch sphere from consideration, and I think various subsequent results describe regions of the Bloch sphere that either can or can't act as magic states.