Out-of-distribution generalisation for learning quantum channels with low-energy coherent states

TL;DR

This paper demonstrates that learning quantum channels with low-energy coherent states allows for reliable out-of-distribution generalisation to more complex inputs, given sufficient training samples.

Contribution

It establishes theoretical bounds linking low-energy coherent state performance to generalisation on all input states in quantum channel learning.

Findings

Out-of-distribution generalisation is achievable with enough samples.

Error bounds for low-energy inputs can be extended to all inputs.

Channels acting similarly on low-energy states are close on all states.

Abstract

When experimentally learning the action of a continuous variable quantum process by probing it with inputs, there will often be some restriction on the input states used. One experimentally simple way to probe a quantum channel is using low energy coherent states. Learning a quantum channel in this way presents difficulties, due to the fact that two channels may act similarly on low energy inputs but very differently for high energy inputs. They may also act similarly on coherent state inputs but differently on non-classical inputs. Extrapolating the behaviour of a channel for more general input states from its action on the far more limited set of low energy coherent states is a case of out-of-distribution generalisation. To be sure that such generalisation gives meaningful results, one needs to relate error bounds for the training set to bounds that are valid for all inputs. We show that for any pair of channels that act sufficiently similarly on low energy coherent state inputs, one can bound how different the input-output relations are for any (high energy or highly non-classical) input. This proves out-of-distribution generalisation is always possible for learning quantum channels using low energy coherent states, as long as enough samples are used.