Fixing and documenting how the Estimator calculates stds

[SamFerracin]

Summary

This PR slightly modifies the formula used by the Estimator to calculate stds, and adds details on how the stds are computed.

Details and comments

Addresses issue 1751.

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Thank you for opening a new pull request.

Before your PR can be merged it will first need to pass continuous integration tests and be reviewed. Sometimes the review process can be slow, so please be patient.

While you're waiting, please feel free to review other open PRs. While only a subset of people are authorized to approve pull requests for merging, everyone is encouraged to review open pull requests. Doing reviews helps reduce the burden on the core team and helps make the project's code better for everyone.

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

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[coveralls]

Pull Request Test Coverage Report for Build 9681788876

Details

• 2 of 2 (100.0%) changed or added relevant lines in 1 file are covered.
• 15 unchanged lines in 2 files lost coverage.
• Overall coverage increased (+0.01%) to 89.761%

---

Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/lex.rs	3	93.13%
crates/qasm2/src/parse.rs	12	97.15%

Totals
Change from base Build 9680727079:	0.01%
Covered Lines:	63786
Relevant Lines:	71062

---

💛 - Coveralls

[coveralls]

Pull Request Test Coverage Report for Build 9715313071

Warning: This coverage report may be inaccurate.

This pull request's base commit is no longer the HEAD commit of its target branch. This means it includes changes from outside the original pull request, including, potentially, unrelated coverage changes.

• For more information on this, see Tracking coverage changes with pull request builds.
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• For a quick fix, rebase this PR at GitHub. Your next report should be accurate.

Details

• 3 of 3 (100.0%) changed or added relevant lines in 1 file are covered.
• 384 unchanged lines in 12 files lost coverage.
• Overall coverage decreased (-0.006%) to 89.744%

---

Files with Coverage Reduction	New Missed Lines	%
qiskit/circuit/library/standard_gates/r.py	1	97.73%
crates/qasm2/src/expr.rs	1	94.02%
crates/accelerate/src/isometry.rs	1	99.65%
crates/accelerate/src/sampled_exp_val.rs	1	88.71%
crates/accelerate/src/convert_2q_block_matrix.rs	3	93.02%
crates/qasm2/src/lex.rs	7	91.35%
crates/accelerate/src/pauli_exp_val.rs	8	89.68%
crates/accelerate/src/sparse_pauli_op.rs	15	93.27%
qiskit/circuit/quantumcircuit.py	17	95.69%
crates/accelerate/src/euler_one_qubit_decomposer.rs	20	89.68%

Totals
Change from base Build 9680727079:	-0.006%
Covered Lines:	63801
Relevant Lines:	71092

---

💛 - Coveralls

[t-imamichi]

Thank you for fixing the issue. Could you write a release note of bugfix?
Could you also update the title of this PR because it fixes stds of BackendEstimatorV2 in addition to the documentation?

[ihincks]

mathbb or rn for the E's?

[ihincks]

what is \: ? I've never used that

Suggested change

	\frac{\sum_{i=1}^{n}|a_i|\sqrt{\textrm{Var}\big(P_i\big)}}{\sqrt{N}}\:,
	\frac{\sum_{i=1}^{n}|a_i|\sqrt{\textrm{Var}\big(P_i\big)}}{\sqrt{N}},

[ihincks]

what's the link to that paper which suggests this? it's not intuitive to me.

[SamFerracin]

Eq. 5 of https://arxiv.org/abs/1908.06942. I added a reference to it in the docstring

[SamFerracin]

\: leaves a blank space. I normally use it when I have a comma or a break at the end of a full-line equation

[coveralls]

Pull Request Test Coverage Report for Build 9809468048

Warning: This coverage report may be inaccurate.

This pull request's base commit is no longer the HEAD commit of its target branch. This means it includes changes from outside the original pull request, including, potentially, unrelated coverage changes.

• For more information on this, see Tracking coverage changes with pull request builds.
• To avoid this issue with future PRs, see these Recommended CI Configurations.
• For a quick fix, rebase this PR at GitHub. Your next report should be accurate.

Details

• 3 of 3 (100.0%) changed or added relevant lines in 1 file are covered.
• 774 unchanged lines in 35 files lost coverage.
• Overall coverage increased (+0.1%) to 89.854%

---

Files with Coverage Reduction	New Missed Lines	%
qiskit/circuit/library/standard_gates/r.py	1	97.73%
crates/qasm2/src/expr.rs	1	94.02%
qiskit/circuit/library/standard_gates/rxx.py	1	97.56%
qiskit/transpiler/passes/routing/commuting_2q_gate_routing/pauli_2q_evolution_commutation.py	1	97.92%
qiskit/circuit/library/standard_gates/rzx.py	1	97.56%
crates/accelerate/src/isometry.rs	1	99.65%
crates/accelerate/src/sampled_exp_val.rs	1	88.71%
qiskit/quantum_info/operators/symplectic/clifford.py	1	91.87%
qiskit/circuit/library/standard_gates/rz.py	2	95.83%
qiskit/circuit/library/standard_gates/ryy.py	2	95.12%

Totals
Change from base Build 9680727079:	0.1%
Covered Lines:	65202
Relevant Lines:	72564

---

💛 - Coveralls

[coveralls]

Pull Request Test Coverage Report for Build 9810207246

Warning: This coverage report may be inaccurate.

This pull request's base commit is no longer the HEAD commit of its target branch. This means it includes changes from outside the original pull request, including, potentially, unrelated coverage changes.

• For more information on this, see Tracking coverage changes with pull request builds.
• To avoid this issue with future PRs, see these Recommended CI Configurations.
• For a quick fix, rebase this PR at GitHub. Your next report should be accurate.

Details

• 3 of 3 (100.0%) changed or added relevant lines in 1 file are covered.
• 787 unchanged lines in 36 files lost coverage.
• Overall coverage increased (+0.09%) to 89.837%

---

Files with Coverage Reduction	New Missed Lines	%
qiskit/circuit/library/standard_gates/r.py	1	97.73%
crates/qasm2/src/expr.rs	1	94.02%
qiskit/circuit/library/standard_gates/rxx.py	1	97.56%
qiskit/transpiler/passes/routing/commuting_2q_gate_routing/pauli_2q_evolution_commutation.py	1	97.92%
qiskit/circuit/library/standard_gates/rzx.py	1	97.56%
crates/accelerate/src/isometry.rs	1	99.65%
crates/accelerate/src/sampled_exp_val.rs	1	88.71%
qiskit/quantum_info/operators/symplectic/clifford.py	1	91.87%
qiskit/circuit/library/standard_gates/rz.py	2	95.83%
qiskit/circuit/library/standard_gates/ryy.py	2	95.12%

Totals
Change from base Build 9680727079:	0.09%
Covered Lines:	65189
Relevant Lines:	72564

---

💛 - Coveralls

[coveralls]

Pull Request Test Coverage Report for Build 9810752069

Details

• 3 of 3 (100.0%) changed or added relevant lines in 1 file are covered.
• 4 unchanged lines in 1 file lost coverage.
• Overall coverage increased (+0.02%) to 89.835%

---

Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/lex.rs	4	92.37%

Totals
Change from base Build 9807934619:	0.02%
Covered Lines:	65188
Relevant Lines:	72564

---

💛 - Coveralls

[coveralls]

Pull Request Test Coverage Report for Build 9811721107

Details

• 3 of 3 (100.0%) changed or added relevant lines in 1 file are covered.
• 3 unchanged lines in 1 file lost coverage.
• Overall coverage increased (+0.03%) to 89.845%

---

Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/lex.rs	3	92.62%

Totals
Change from base Build 9807934619:	0.03%
Covered Lines:	65195
Relevant Lines:	72564

---

💛 - Coveralls

[jsaroni]

In improving the backend_estimator_v2.py documentation, (unrelated to stds, sorry), there is this comment under _preprocess_pub

# calculate expectation values for each pair of parameter value set and pauli

but the lines following the comment don't actually calculate the expectation values.

        # calculate expectation values for each pair of parameter value set and pauli
        param_obs_map = defaultdict(set)
        for index in np.ndindex(*bc_param_ind.shape):
            param_index = bc_param_ind[index]
            param_obs_map[param_index].update(bc_obs[index])

[t-imamichi]

Good catch, @jsaroni. We will update the comment when we update the method.

[coveralls]

Pull Request Test Coverage Report for Build 9994949577

Details

• 3 of 3 (100.0%) changed or added relevant lines in 1 file are covered.
• 8 unchanged lines in 2 files lost coverage.
• Overall coverage increased (+0.02%) to 89.908%

---

Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/lex.rs	2	92.62%
crates/qasm2/src/parse.rs	6	97.15%

Totals
Change from base Build 9992274403:	0.02%
Covered Lines:	65798
Relevant Lines:	73184

---

💛 - Coveralls

[t-imamichi]

LGTM