Add a CVaR notebook

[a-matsuo]

Summary

This PR adds a CVaR notebook.

Details and comments

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[miamico]

Few initial comments:

1. Most important, this tutorial is meant to reproduce the results in https://arxiv.org/pdf/2312.00733 which show that twirling and over-sampling (by an amount determined by the sampling overhead of PEC) can give mitigated bitstrings and that if we calculate the cost with CVaR we're guaranteed to select the bitstrings with lowest expectation values. Right now the notebook is showcasing only CVaR but we want to include also the estimation of the PEC gamma (can be done using reported device data or code for layer fidelity in qiskit experiments), twirling of the circuit, over-sampling my an amount given by gamma and optimizing the circuit parameters by using CVaR as a cost function with alpha value related to gamma (1/sqrt(gamma)).
2. Can we formulate CVaR as an observable? This would allow us to use the Estimator rather than the Sampler in the optimization of the circuit parameters. There are better error mitigation techniques available for the Estimator
3. Switch to V2 of the Runtime Primitives
4. This is intended to be a utility-scale tutorial. We need to increase the number of qubits to at least 40q (like in the paper)

[miamico]

From our discussion:

• switch to runtime simulator for testing implementation
• implement automatic estimation of gamma_circ (Mirko)
• Use CVaR in the optimization with SamplerV2
• Pass alpha=1/sqrt(gamma_circ) to CVaR
• increase number of shots from shots to shots*sqrt(gamma_circ)
• Set options in SamplerV2 to do DD, Twirling, (look into measurement mitigation). Also look at https://github.com/IBM-Quantum-Technical-Enablement/quantum-enablement/blob/krylov_utility/docs/notebooks/scaling-sampler/scaling-sampler.ipynb (may be outdatad) -See this notebook for utility scale QAOA with latest product tools
  • How does twirling work for Sampler?
  • Twirling options:
  • enable_gates = True
  • enable_measure= False
  • with these assumptions: num_randomizations = 50, shots = 10k
  • shotssqrt(gamma_circ) = num_randomizations shots_per_randomization → shots_per_randomization = shotssqrt(gamma_circ) /num_randomizations = 200sqrt(gamma_circ)
• DD options:
  • enable =True
  • sequence_type = ‘XY4 (fallback to XpXm if it doesn’t work)

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Pull Request Test Coverage Report for Build 9843104612

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Change from base Build 9466320186:	0.0%
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