Support for time-dependent Hamiltonians and observable evaluation in TrotterQRTE

[Durd3nT]

Summary

This PR adds two features to the TrotterQRTE class.

1. it adds support for time-dependent Hamiltonians (i.e., singly parametrized operators)
2. it extends the evaluation of aux_operators to every timestep.

Details and comments

The above is achieved by constructing the full circuit of a Trotterized time evolution within TrotterQRTE instead of (as was previously the case) in PauliEvolutionGate.
Previously, the number of timesteps was passed to the ProductFormula through the reps argument. PauliEvolutionGate was then called from TrotterQRTE to construct the full circuit corresponding to the full time evolution.
Instead, now TrotterQRTE accepts the argument num_timesteps (analogous to VarQTE). The PauliEvolutionGate is only used to construct a single repetition of a product formula circuit and these single-step evolution gates are chained together in TrotterQRTE.evolve.
This has two benefits:

1. parameter values (the time at the current timestep) can be bound to the Hamiltonian at every time step, allowing to the construction of evolution gates of time-dependent Hamiltonians.
2. aux_operators can be evaluated at every timestep, which is the natural application for any time evolution algorithm.

Since SparsePauliOp does not yet support parameter-assignments, parameter values are assigned to the Hamiltonian at each time step through the helper function _assign_parameters. For this purpose, this helper function was moved to a higher level in the hierarchy of the Algorithms module and the import in VarQTE was adapted.
In the future, a method SparsePauliOp.assign_parameters should be added.

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One or more of the the following people are requested to review this:

• @Cryoris
• @Qiskit/terra-core
• @ajavadia
• @manoelmarques
• @woodsp-ibm

[Cryoris]

Overall looks good, I left some comments below. The main suggestions are

• keeping aux_ops_evaluated as is and instead introduce observables
• don't change the product formula but apply it as the user supplied it

Could also add a releasenote for this new feature? 🙂

[Cryoris]

Where are these numbers coming from? 🙂 Would it be possible to have a manual calculation as reference? 😄

[woodsp-ibm]

One more of a procedural comment is that ideally the copyright dates on these changed files should be updated for 2023,

Eg (C) Copyright IBM 2021, 2022. would become (C) Copyright IBM 2021, 2023. where the last date of copyright is changed to the current year - the format is first date, last date but is they are the same its just put once. So if its a single date now, and not 2023, eg (C) Copyright IBM 2021. it would become (C) Copyright IBM 2021, 2023.

[Cryoris]

LGTM, thanks for adding these features!

[coveralls]

Pull Request Test Coverage Report for Build 4164041180

• 47 of 50 (94.0%) changed or added relevant lines in 4 files are covered.
• 3 unchanged lines in 2 files lost coverage.
• Overall coverage decreased (-0.002%) to 85.272%

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Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/algorithms/utils/assign_params.py	8	9	88.89%
qiskit/algorithms/time_evolvers/trotterization/trotter_qrte.py	34	36	94.44%

Files with Coverage Reduction	New Missed Lines	%
src/vf2_layout.rs	1	86.44%
qiskit/transpiler/passes/synthesis/unitary_synthesis.py	2	93.97%

Totals
Change from base Build 4162466886:	-0.002%
Covered Lines:	67267
Relevant Lines:	78885

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💛 - Coveralls

[noure-beep]

i need i code about trotterization