Fixing and documenting how the Estimator calculates stds (#12670)

* riverlane paper

* docs

* improvement

* empty

* fix linting and add abs

* Update qiskit/primitives/backend_estimator_v2.py

Co-authored-by: Takashi Imamichi <31178928+t-imamichi@users.noreply.github.com>

* CR

* Update qiskit/primitives/backend_estimator_v2.py

Co-authored-by: Ian Hincks <ian.hincks@gmail.com>

* CR

* indent

* Update releasenotes/notes/backend-estimator-v2-variance-905c953415ad0e29.yaml

Co-authored-by: Takashi Imamichi <31178928+t-imamichi@users.noreply.github.com>

---------

Co-authored-by: Takashi Imamichi <31178928+t-imamichi@users.noreply.github.com>
Co-authored-by: Ian Hincks <ian.hincks@gmail.com>

qiskit/primitives/backend_estimator_v2.py

@@ -72,7 +72,7 @@ class _PreprocessedData:
 
 
 class BackendEstimatorV2(BaseEstimatorV2):
-    """Evaluates expectation values for provided quantum circuit and observable combinations
+    r"""Evaluates expectation values for provided quantum circuit and observable combinations.
 
     The :class:`~.BackendEstimatorV2` class is a generic implementation of the
     :class:`~.BaseEstimatorV2` interface that is used to wrap a :class:`~.BackendV2`
@@ -87,7 +87,19 @@ class BackendEstimatorV2(BaseEstimatorV2):
     precludes doing any provider- or backend-specific optimizations.
 
     This class does not perform any measurement or gate mitigation, and, presently, is only
-    compatible with Pauli-based observables.
+    compatible with Pauli-based observables. More formally, given an observable of the type
+    :math:`O=\sum_{i=1}^Na_iP_i`, where :math:`a_i` is a complex number and :math:`P_i` is a
+    Pauli operator, the estimator calculates the expectation :math:`\mathbb{E}(P_i)` of each
+    :math:`P_i` and finally calculates the expectation value of :math:`O` as
+    :math:`\mathbb{E}(O)=\sum_{i=1}^Na_i\mathbb{E}(P_i)`. The reported ``std`` is calculated
+    as
+
+    .. math::
+
+        \frac{\sum_{i=1}^{n}|a_i|\sqrt{\textrm{Var}\big(P_i\big)}}{\sqrt{N}}\:,
+
+    where :math:`\textrm{Var}(P_i)` is the variance of :math:`P_i`, :math:`N=O(\epsilon^{-2})` is
+    the number of shots, and :math:`\epsilon` is the target precision [1].
 
     Each tuple of ``(circuit, observables, <optional> parameter values, <optional> precision)``,
     called an estimator primitive unified bloc (PUB), produces its own array-based result. The
@@ -104,6 +116,12 @@ class BackendEstimatorV2(BaseEstimatorV2):
 
     * ``seed_simulator``: The seed to use in the simulator. If None, a random seed will be used.
       Default: None.
+
+    **Reference:**
+
+    [1] O. Crawford, B. van Straaten, D. Wang, T. Parks, E. Campbell, St. Brierley,
+    Efficient quantum measurement of Pauli operators in the presence of finite sampling error.
+    `Quantum 5, 385 <https://doi.org/10.22331/q-2021-01-20-385>`_
     """
 
     def __init__(
@@ -254,8 +272,8 @@ def _postprocess_pub(
             for pauli, coeff in bc_obs[index].items():
                 expval, variance = expval_map[param_index, pauli]
                 evs[index] += expval * coeff
-                variances[index] += variance * coeff**2
-        stds = np.sqrt(variances / shots)
+                variances[index] += np.abs(coeff) * variance**0.5
+        stds = variances / np.sqrt(shots)
         data_bin = DataBin(evs=evs, stds=stds, shape=evs.shape)
         return PubResult(data_bin, metadata={"target_precision": pub.precision})
 

releasenotes/notes/backend-estimator-v2-variance-905c953415ad0e29.yaml

@@ -0,0 +1,5 @@
+---
+fixes:
+  - |
+    Changes the way in which the :class:`.BackendEstimatorV2` class calculates the ``std`` to ensure that
+    it matches the correct formula.