Implement SparseObservable.to_matrix

[davidfcohen]

The SparseObservable.to_matrix method expands the observable into its dense matrix representation. The Rust counterpart is SparseObservable::to_matrix.

There are 3 primary motivations for this feature:

1. SparseObservable ought to have a to_matrix method that handles projectors directly, without expanding into Pauli terms.
2. In the future, SparseObservable.to_matrix and SparsePauliOp.to_matrix should share the same implementation.
3. In the future, SparseObservable will be the base for some custom operations. This function is useful for those implementing CustomOperation::to_matrix.

An observable is a linear combination of multi-qubit terms. If the term is composed of Pauli-only operators, its coefficient is added once per row. We take advantage of this property to fill the matrix more efficiently. This part of the solution is derived from the internals of SparsePauliOp.to_matrix.

We decreased the cost of filling the matrix using some clever tricks. Thank you @jakelishman and @Cryoris 😄 However, it's clear that the total time is dominated by the O(4ⁿ) matrix allocation.

closes #13389

AI/LLM disclosure

• I didn't use LLM tooling, or only used it privately.
• I used the following tool to help write this PR description:
• I used the following tool to generate or modify code:

The add_term_kron function is derived from an AI generated algorithm suggested by @Cryoris. The Rust derivative was generated using Claude Sonnet 5. The rest of the PR is hand-written.

[qiskit-bot]

One or more of the following people are relevant to this code:

• @Qiskit/terra-core

[coveralls]

Coverage Report for CI Build 33183909265

Warning

Build has drifted: This PR's base is out of sync with its target branch, so coverage data may include unrelated changes.
Quick fix: rebase this PR. Learn more →

Coverage increased (+0.009%) to 87.76%

Details

• Coverage increased (+0.009%) from the base build.
• Patch coverage: 6 uncovered changes across 1 file (211 of 217 lines covered, 97.24%).
• 40 coverage regressions across 3 files.

Uncovered Changes

File	Changed	Covered	%
crates/quantum_info/src/sparse_observable/matrix.rs	203	197	97.04%
Total (2 files)	217	211	97.24%

Coverage Regressions

40 previously-covered lines in 3 files lost coverage.

File	Lines Losing Coverage	Coverage
crates/providers/src/tensor.rs	30	94.93%
crates/providers/src/program_node.rs	7	67.86%
crates/qasm2/src/lex.rs	3	92.29%

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Coverage Stats

Relevant Lines:	131296
Covered Lines:	115226
Line Coverage:	87.76%
Coverage Strength:	982449.7 hits per line

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💛 - Coveralls