Ensure the result of simulation is normalized

[NoureldinYosri]

• Normalize the final state vector of a simulation, fixes #cirq.Simulator final_state_vector is not normalized #6402

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.78%. Comparing base (f955650) to head (7856e81).
Report is 1 commits behind head on main.

❗ Current head 7856e81 differs from pull request most recent head 9aac89e. Consider uploading reports for the commit 9aac89e to get more accurate results

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #6522      +/-   ##
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- Coverage   97.78%   97.78%   -0.01%     
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  Files        1107     1105       -2     
  Lines       95182    95113      -69     
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- Hits        93073    93004      -69     
  Misses       2109     2109              

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[daxfohl]

I considered doing this in the past when the earlier issue came up, but was concerned it was like sweeping dirt under the rug. Does doing this then make other invariants start failing? I think it's worth looking at what other projects do here, and try to follow prior art on the subject.

[NoureldinYosri]

@daxfohl for state vector there is not an issue

for density matrix it doesn't work (I reverted my changes and restricted this PR to state vector simulation). the invariant of density matrix is that $Tr[\rho^2] \leq 1$ with equality if and only if $\rho$ is pure. I don't think we should implicitly add the computation $Tr[\rho^2]$

[pavoljuhas]

Can we raise UserWarning if abs(norm(state_vector) - 1) > np.sqrt(np.finfo(state_vector.dtype).eps)?

That should be sufficient for an accumulated round off from ~1/eps operations, but give some warning in case the state vector would be widely off.

[pavoljuhas]

I think we should have some check here so we don't re-normalize incorrect state vector.