Add BackendEstimatorV2 interface

[Polarnova]

Closes #145

Summary

Add a BackendEstimatorV2 primitive that mirrors the existing BackendSamplerV2 shape: EstimatorPub carries an ISA circuit and observables, estimator.run() returns a BackendEstimatorJob, and the job result contains expectation values and standard errors.

The estimator uses the same precision-to-shots rule and Pauli expectation/std-error post-processing model as Qiskit Python BackendEstimatorV2 for the supported real-coefficient PauliSparseObservable subset.

The implementation reuses the existing BackendV2 sampler path, QuantumCircuit, SparseObservable/QkObs, BitArray, and standard-library math to extract Pauli terms, build grouped measurement circuits, execute them, and return evs/stds with shots and target_precision.

The patch includes the wrapper fixes required by the estimator and deterministic unit tests for estimator containers, observable extraction, measurement planning, and Pauli statistics.

Details

• Add EstimatorPub, BackendEstimatorV2, BackendEstimatorJob, EstimatorPubResult, and EstimatorResult.
• Implement a generic sampler-backed estimator path for Pauli and identity SparseObservable terms.
• Derive backend shots from precision; the default 0.015625 matches Python BackendEstimatorV2's documented 1 / sqrt(4096) default (source), and Python maps precision to shots with ceil(1 / precision**2) (source).
• Add qubit-wise compatible Pauli grouping, basis-rotation measurement circuits, and BitArray post-processing for evs/stds.

Execution flow

1. BackendEstimatorV2::run() validates each EstimatorPub, resolves precision to a shot count, extracts real-coefficient Pauli terms from SparseObservable, and groups qubit-wise compatible Pauli terms into
   measurement circuits.
2. Measurement circuits with the same shot count are submitted together through the existing BackendV2::run() sampler interface.
3. BackendEstimatorJob::result() reads the returned BitArray samples and
   post-processes them into evs, stds, shots, and target_precision.

Scope notes

• Input circuits with existing measurements report an explicit error in this patch. Matching Python's behavior needs estimator-owned classical registers, compose-with-clbit mapping, measurement-map remapping, and sampler-result routing by register name.
• Measurement circuits with the same shot count are submitted together through the existing BackendV2::run() sampler interface.

Existing wrapper fixes

1. SparseObservable
   Fix bit_terms() and boundaries() to use the QkObs C API array lengths, make indices() and the copy constructor null-safe, and add deep-copy operator=. Estimator term extraction depends on these arrays, and EstimatorPub assignment needs copied QkObs ownership rather than a shared raw handle.

2. QuantumCircuit
   Make copy() const, add const get_measure_map(), and fix compose(circ, ...) to read instruction kinds from the source circuit. Estimator validation is read-only, and measured circuits must replay the input circuit before adding basis rotations.

3. utils.hpp
   Replace the header-level hamming_parity/popcount function pointer variables with C++11 inline functions. This keeps Qiskit::popcount(x) call sites unchanged and avoids duplicate symbols when multiple test translation units include the header.

Testing

• Coverage: deterministic unit tests for estimator containers, observable extraction, measurement planning, and Pauli statistics.

cmake --build test/build -j2
ctest -V -C Debug --test-dir test/build
cmake --build samples/build -j2
git diff --check

AI disclosure: I used ChatGPT/Codex for drafting and review assistance. I reviewed and own the final design, implementation, tests, and PR wording. Local checks I ran are listed above.

[CLAassistant]

All committers have signed the CLA.

[Polarnova]

I kept the default C++ unit tests: estimator containers, QkObs/Pauli extraction, measurement planning, and Pauli statistics.

For end-to-end backend validation, would you prefer a CI test that compares this C++ estimator path against Qiskit Python/Aer on the same circuits and observables, or a separate validation script outside the default CI suite? Thanks

[doichanj]

As I mentioned in the issue #145 (comment)
can you implement backend.run() for EstimatorPub. I think #160 will implement estimator program mode on QRMI

[doichanj]

We can not add h gate here to the circuit, because the input circuit should be ISA circuit here.
I think we do not have to add measure and rotations here if we have backend.run() for estimator

[doichanj]

here also we do not accept sdg gate