Benjamin Russell, Susan Stepney.
Zermelo Navigation and a Speed Limit to Quantum Information Processing
Physical Review A, 90, 012303, 2014. doi:10.1103/PhysRevA.90.012303
(also available from quant-ph arXiv:1310.6731)
Graf Zeppelin’s circumnavigation of the world, Ernst Zermelo posed his navigation problem back in the late 1920’s: given a Zeppelin travelling at constant speed relative to the air, and given a wind that can vary in time and space, what is the optimal way to steer the Zeppelin to reach its destination D in minimum time?
Mathematicians have, in the way of mathematicians, been generalising this problem. Now there are results for general differentiable manifolds, not just mere Euclidean space.
Taking what seems to be an unrelated tack, one thing we are interested in is ultimate speed limits in quantum computers. These speed limits come from physical constraints on the speed with which a quantum system can move from one state to another.
Here’s the connection to Zeppelins: Quantum mechanics can be expressed geometrically. In the paper, we cast the quantum speed limit problem as a (generalised) Zermelo navigation problem. The manifold is given by the special unitary group SU(N), and the wind is given by the time-independent “drift Hamiltonian” (what the quantum system would do if it wasn’t being controlled). The system is affected by both this drift Hamiltonian and some specific control Hamiltonian (the analogue of the Zeppelin's engine).
We consider the case where the control Hamiltonian is also time-independent (the Zeppelin is steered in a fixed manner). This allows us to calculate the optimal time taken to reach our desired D, a specific target quantum state. From this general closed-form result (if you want to see what it looks like, see the paper!), we can then calculate specific times for our specific quantum setups and specific target quantum states.
This particular approach gives us the optimal time, but does not give the actual control Hamiltonian needed to achieve that time. It’s like knowing that there is a direction you could steer your Zeppelin to reach your destination in a day, that you can't get there any faster than that, but you have no idea which direction to go.
Finding the optimal control (the direction) that achieves the optimum time: that’s the next paper!