Dependence of the Quantum Speed Limit on System Size and Control Complexity

TL;DR

This paper derives a lower bound on the minimum time required to implement quantum unitary transformations, analyzing how system size and control complexity influence this bound, with implications for quantum control and computation.

Contribution

It extends previous work by explicitly deriving a bound that depends on system dimension and control number, and systematically compares it with actual minimum gate times.

Findings

The bound depends on system size and control number.

Numerical analysis shows the bound accurately captures the scaling of minimum time.

The bound is correct in order of magnitude for studied systems.

Abstract

We extend the work in New J. Phys. 19, 103015 (2017) by deriving a lower bound for the minimum time necessary to implement a unitary transformation on a generic, closed quantum system with an arbitrary number of classical control fields. This bound is explicitly analyzed for a specific N-level system similar to those used to represent simple models of an atom, or the first excitation sector of a Heisenberg spin chain, both of which are of interest in quantum control for quantum computation. Specifically, it is shown that the resultant bound depends on the dimension of the system, and on the number of controls used to implement a specific target unitary operation. The value of the bound determined numerically, and an estimate of the true minimum gate time are systematically compared for a range of system dimension and number of controls; special attention is drawn to the relationship between these two variables. It is seen that the bound captures the scaling of the minimum time well for the systems studied, and quantitatively is correct in the order of magnitude.