Hermitian observables and (avoiding) diagonalization of (large) matrices

Feature details

This is half a question and half a feature / documentation request.

One issue with the qml.Hermitian observable is that (at least when running on default.qubit) its .matrix is diagonalized (via .diagonalizing_gates) even to compute return values such as qml.expval(qml.Hermitiaion(...)) and qml.var(qml.Hermitiaion(...)), which could be computed via a simple matrix vector (inner)products. This is already not ideal performance wise when the qml.Hermitian observable acts on few wires, but gets worse exponentially for large number of wires.

Looking into DefaultQubit.expval() I get the impression that for qml.Hamiltonian and qml.SparseHamiltonian this diagonalization is avoided.

So, am I right that wrapping a qml.Hermitian observable in a qml.Hamiltonian (as also discussed here) avoids the diagonalization?

• Maybe it would be worth explaining this to users?
• Maybe the same trick could be enabled to also speed up var()?
• Maybe this wrapping, which is counter intuitive and difficult to explain to users could be avoided altogether by calculating the diagonalizing gates of qml.Hermitian only when the user executes a QNode with a return type that requires diagonalization such as qml.probs() but not for expval or var?

Implementation

No response

How important would you say this feature is?

1: Not important. Would be nice to have.

Additional information

No response

[cvjjm]

This is also related to #1344

[cvjjm]

Agree, but tape.diagonalizing_gates is not part of the public API, isn’t it? So breaking is not an issue here.