 # Potential error in problem statement

Hello,

It may be a nitpick, but i think there is an error in the definition of the error metric. In general the difference of two unitary operators may not be diagonalizable, so the error may differ from the biggest eigenvalue. You probably mean something like $|A| = max_{v \neq 0} \frac{|Av|}{|v|}$ which is equal to the definition stated if $A$ can be diagonalized.

Or maybe actually all operators of the form $U - U’$ can be diagonalized and I’m gravely wrong here.

@JanczarKurek Thanks for pointing this out. Yes, the metric you described captures the intent

@JanczarKurek For the error calculation, I think the problem is asking for the euclidean norm / 2-matrix norm, which simplifies the $|A| = max_{v \neq 0} \frac{|Av|}{|v|}$ generalized $L^p$ space norm to the operator norm stated.

Hi,
In quantum computing we usually disregard the global phase of a unitary U.
For example, in Qiskit transpilation to QASM often changes the global phase.
Will you take care of this global phase when evaluating the submissions?

More precisely:
Suppose the unitary U = e^-iH is the unitary we should simulate.
And my submission is a unitary V, such that ||U - cV|| < 0.1 for a certain complex coefficient c, |c|=1.
Will the submission be deemed correct?

When we check the solutions, We compare the matrices, while ignoring the global phase

Just to be sure we are on the same page (as ‘global phase’ is, to the best of my knowledge, not a well defined term for an arbitrary unitary matrix):
If you say ‘ignoring the global phase’, do you mean, instead of calculating the norm of (U - e^-iH) you calculate the norm of (e^-ic U - e^-iH) where c is the coefficient of the identity operator in the Hamiltonian?

Hi @grossardt,

You are correct. In quantum computing the global phase is not important, so to be fair to all contestants we are ignoring it.

To be exact, we are eliminating the global phase of both the submitted solution and the exact solution. By that we are comparing special unitary matrices.