Estimator inputs and outputs

qiskit[all]~=2.3.0
qiskit-ibm-runtime~=0.43.1

This page gives an overview of the inputs and outputs of the Qiskit Runtime primitives that execute workloads on IBM Quantum® compute resources. These primitives provide you with the ability to efficiently define vectorized workloads by using a data structure known as a Primitive Unified Bloc (PUB). These PUBs are the fundamental unit of work that a QPU needs in order to execute these workloads. They are used as inputs to the run() method for the Estimator primitive, which execute the defined workload as a job. Then, after the job has completed, the results are returned in a format that is dependent on both the PUBs used as well as the runtime options specified from the Estimator primitives.

Inputs

The inputs to the Estimator primitive are PUBs. For the Estimator primitive, the format of the PUB should contain at most four values:

• A single QuantumCircuit, which may contain one or more Parameter objects
• A list of one or more observables, which specify the expectation values to estimate, arranged into an array (for example, a single observable represented as a 0-d array, a list of observables as a 1-d array, and so on). The data can be in any one of the ObservablesArrayLike format such as Pauli, SparsePauliOp, PauliList, or str.
  Note

  If you have two commuting observables in different PUBs but with the same circuit, they will not be estimated using the same measurement. Each PUB represents a different basis for measurement, and therefore, separate measurements are required for each PUB. To ensure that commuting observables are estimated using the same measurement, they must be grouped within the same PUB.

• A collection of parameter values to bind the circuit against. This can be specified as a single array-like object where the last index is over circuit Parameter objects, or omitted (or equivalently, set to None) if the circuit has no Parameter objects.
• (Optionally) a target precision for expectation values to estimate

The following code demonstrates an example set of vectorized inputs to the Estimator primitive and executes them on an IBM® backend as a single RuntimeJobV2 object.

from qiskit.circuit import (
    Parameter,
    QuantumCircuit,
)
from qiskit.transpiler import generate_preset_pass_manager
from qiskit.quantum_info import SparsePauliOp
 
from qiskit_ibm_runtime import (
    QiskitRuntimeService,
    EstimatorV2 as Estimator,
)
 
import numpy as np
 
# Instantiate runtime service and get
# the least busy backend
service = QiskitRuntimeService()
backend = service.least_busy(operational=True, simulator=False)
 
# Define a circuit with two parameters.
circuit = QuantumCircuit(2)
circuit.h(0)
circuit.cx(0, 1)
circuit.ry(Parameter("a"), 0)
circuit.rz(Parameter("b"), 0)
circuit.cx(0, 1)
circuit.h(0)
 
# Transpile the circuit
pm = generate_preset_pass_manager(optimization_level=1, backend=backend)
transpiled_circuit = pm.run(circuit)
layout = transpiled_circuit.layout
 
# Now define a sweep over parameter values, the last axis of dimension 2 is
# for the two parameters "a" and "b"
params = np.vstack(
    [
        np.linspace(-np.pi, np.pi, 100),
        np.linspace(-4 * np.pi, 4 * np.pi, 100),
    ]
).T
 
# Define three observables. The inner length-1 lists cause this array of
# observables to have shape (3, 1), rather than shape (3,) if they were
# omitted.
observables = [
    [SparsePauliOp(["XX", "IY"], [0.5, 0.5])],
    [SparsePauliOp("XX")],
    [SparsePauliOp("IY")],
]
# Apply the same layout as the transpiled circuit.
observables = [
    [observable.apply_layout(layout) for observable in observable_set]
    for observable_set in observables
]
 
# Estimate the expectation value for all 300 combinations of observables
# and parameter values, where the pub result will have shape (3, 100).
#
# This shape is due to our array of parameter bindings having shape
# (100, 2), combined with our array of observables having shape (3, 1).
estimator_pub = (transpiled_circuit, observables, params)
 
# Instantiate the new estimator object, then run the transpiled circuit
# using the set of parameters and observables.
estimator = Estimator(mode=backend)
job = estimator.run([estimator_pub])
result = job.result()

Output

Once one or more PUBs are sent to a QPU for execution and a job successfully completes, the data is returned as a PrimitiveResult container object accessed by calling the RuntimeJobV2.result() method. The PrimitiveResult contains an iterable list of PubResult objects that contain the execution results for each PUB. Depending on the primitive used, these data will be either expectation values and their error bars in the case of the Estimator.

Each element of this list corresponds to each PUB submitted to the primitive's run() method (for example, a job submitted with 20 PUBs will return a PrimitiveResult object that contains a list of 20 PubResults, one corresponding to each PUB).

Each of these PubResult objects possess both a data and a metadata attribute. The data attribute is a customized DataBin that contains the actual measurement values, standard deviations, and so forth. This DataBin has various attributes depending on the shape or structure of the associated PUB as well as the error mitigation options specified by the primitive used to submit the job (for example, ZNE or PEC). Meanwhile, the metadata attribute contains information about the runtime and error mitigation options used (explained later in the Result metadata section of this page).

The following is a visual outline of the PrimitiveResult data structure for the Estimator output:

└── PrimitiveResult
    ├── PubResult[0]
    │   ├── metadata
    │   └── data  ## In the form of a DataBin object
    │       ├── evs
    │       │   └── List of estimated expectation values in the shape
    |       |         specified by the first pub
    │       └── stds
    │           └── List of calculated standard deviations in the
    |                 same shape as above
    ├── PubResult[1]
    |   ├── metadata
    |   └── data  ## In the form of a DataBin object
    |       ├── evs
    |       │   └── List of estimated expectation values in the shape
    |       |        specified by the second pub
    |       └── stds
    |           └── List of calculated standard deviations in the
    |                same shape as above
    ├── ...
    ├── ...
    └── ...

Put simply, a single job returns a PrimitiveResult object and contains a list of one or more PubResult objects. These PubResult objects then store the measurement data for each PUB that was submitted to the job.

Each PubResult for the Estimator primitive contains at least an array of expectation values (PubResult.data.evs) and associated standard deviations (either PubResult.data.stds or PubResult.data.ensemble_standard_error depending on the resilience_level used), but can contain more data depending on the error mitigation options that were specified.

The below code snippet describes the PrimitiveResult (and associated PubResult) format for the job created above.

print(
    f"The result of the submitted job had {len(result)} PUB and has a value:\n {result}\n"
)
print(
    f"The associated PubResult of this job has the following data bins:\n {result[0].data}\n"
)
print(f"And this DataBin has attributes: {result[0].data.keys()}")
print(
    "Recall that this shape is due to our array of parameter binding sets having shape (100, 2) -- where 2 is the\n\
         number of parameters in the circuit -- combined with our array of observables having shape (3, 1). \n"
)
print(
    f"The expectation values measured from this PUB are: \n{result[0].data.evs}"
)

Output:

The result of the submitted job had 1 PUB and has a value:
 PrimitiveResult([PubResult(data=DataBin(evs=np.ndarray(<shape=(3, 100), dtype=float64>), stds=np.ndarray(<shape=(3, 100), dtype=float64>), ensemble_standard_error=np.ndarray(<shape=(3, 100), dtype=float64>), shape=(3, 100)), metadata={'shots': 4096, 'target_precision': 0.015625, 'circuit_metadata': {}, 'resilience': {}, 'num_randomizations': 32})], metadata={'dynamical_decoupling': {'enable': False, 'sequence_type': 'XX', 'extra_slack_distribution': 'middle', 'scheduling_method': 'alap'}, 'twirling': {'enable_gates': False, 'enable_measure': True, 'num_randomizations': 'auto', 'shots_per_randomization': 'auto', 'interleave_randomizations': True, 'strategy': 'active-accum'}, 'resilience': {'measure_mitigation': True, 'zne_mitigation': False, 'pec_mitigation': False}, 'version': 2})

The associated PubResult of this job has the following data bins:
 DataBin(evs=np.ndarray(<shape=(3, 100), dtype=float64>), stds=np.ndarray(<shape=(3, 100), dtype=float64>), ensemble_standard_error=np.ndarray(<shape=(3, 100), dtype=float64>), shape=(3, 100))

And this DataBin has attributes: dict_keys(['evs', 'stds', 'ensemble_standard_error'])
Recall that this shape is due to our array of parameter binding sets having shape (100, 2) -- where 2 is the
         number of parameters in the circuit -- combined with our array of observables having shape (3, 1). 

The expectation values measured from this PUB are: 
[[ 0.00948597  0.12163221  0.29100944  0.40535344  0.46625814  0.54716103
   0.57690846  0.59809047  0.5784682   0.50924868  0.4579837   0.40035644
   0.37174056  0.32887613  0.25850853  0.26396412  0.25852429  0.26074166
   0.29282485  0.34388535  0.37368314  0.43562138  0.46912323  0.51955146
   0.54430185  0.55467261  0.5162183   0.52744696  0.47261781  0.42613541
   0.35400013  0.33217125  0.29600426  0.27561903  0.25307754  0.25672088
   0.28783701  0.36612701  0.40433263  0.44428286  0.51028376  0.55034507
   0.55979913  0.57160124  0.54127534  0.49753533  0.42942659  0.32552331
   0.20215918  0.04303087 -0.08115732 -0.18473659 -0.34015892 -0.44489319
  -0.49112115 -0.54588034 -0.60601287 -0.55869218 -0.53353861 -0.51628053
  -0.44978534 -0.38090252 -0.32481576 -0.28832245 -0.27057547 -0.26542929
  -0.27054473 -0.29367389 -0.31531828 -0.38462352 -0.40276794 -0.47168997
  -0.48548191 -0.5382924  -0.52716406 -0.53277032 -0.50776933 -0.48512907
  -0.44335198 -0.38756463 -0.34438156 -0.29199194 -0.2729216  -0.24602918
  -0.23527174 -0.3019153  -0.35159518 -0.38303379 -0.42434541 -0.47743033
  -0.54652609 -0.5877912  -0.59175701 -0.57386895 -0.56416812 -0.48022381
  -0.3853372  -0.2639702  -0.12030502  0.02081148]
 [ 0.00581765  0.0552677   0.15998546  0.20725389  0.25452232  0.34178711
   0.39196437  0.47050268  0.50031815  0.527952    0.57231161  0.64066903
   0.72429779  0.77011181  0.78174711  0.86610308  0.88646487  0.91337151
   0.94245978  0.98100173  0.97372966  1.00936279  1.01881647  1.0544496
   1.01954368  1.03699664  0.99845469  1.03845105  1.00936279  1.00354513
   0.95409508  0.95264067  0.91264431  0.91846196  0.8355604   0.80283611
   0.77956549  0.74102354  0.69520953  0.64575948  0.58976457  0.53231524
   0.43996     0.3956004   0.32069812  0.27706572  0.22470684  0.16653032
   0.07272066 -0.00218162 -0.05817653 -0.06253977 -0.15853104 -0.25015908
  -0.28506499 -0.34251432 -0.44359604 -0.44432324 -0.53158804 -0.60285429
  -0.637033   -0.67630215 -0.71266249 -0.76793019 -0.81519862 -0.86464867
  -0.90173621 -0.93155168 -0.9337333  -0.98245614 -0.99627307 -1.01518044
  -1.01590764 -1.04863194 -1.00499955 -1.02827016 -1.01663485 -1.0108172
  -1.02317971 -0.97518407 -0.96500318 -0.94682302 -0.901009   -0.87846559
  -0.79556404 -0.84937733 -0.78101991 -0.73811472 -0.65521316 -0.57667485
  -0.59921825 -0.49813653 -0.44577766 -0.36505772 -0.33524225 -0.25888556
  -0.21161713 -0.12289792 -0.03781474  0.00654486]
 [ 0.01315429  0.18799671  0.42203343  0.603453    0.67799397  0.75253494
   0.76185256  0.72567827  0.65661825  0.49054535  0.3436558   0.16004385
   0.01918334 -0.11235955 -0.26473006 -0.33817484 -0.36941628 -0.39188819
  -0.35681008 -0.29323102 -0.22636339 -0.13812003 -0.08057002 -0.01534667
   0.06906002  0.07234859  0.03398191  0.01644286 -0.06412716 -0.15127432
  -0.24609482 -0.28829816 -0.32063579 -0.3672239  -0.32940532 -0.28939435
  -0.20389148 -0.00876953  0.11345574  0.24280625  0.43080296  0.5683749
   0.67963826  0.74760208  0.76185256  0.71800493  0.63414634  0.48451631
   0.3315977   0.08824335 -0.10413812 -0.30693341 -0.52178679 -0.6396273
  -0.69717731 -0.74924637 -0.76842971 -0.67306111 -0.53548918 -0.42970677
  -0.26253768 -0.08550288  0.06303097  0.19128528  0.27404768  0.33379008
   0.36064675  0.34420389  0.30309674  0.2132091   0.19073719  0.07180049
   0.04494382 -0.02795286 -0.04932858 -0.03727049  0.00109619  0.04055906
   0.13647575  0.20005481  0.27624007  0.36283913  0.3551658   0.38640723
   0.32502055  0.24554673  0.07782954 -0.02795286 -0.19347767 -0.3781858
  -0.49383393 -0.67744588 -0.73773637 -0.78268019 -0.793094   -0.70156207
  -0.55905728 -0.40504248 -0.20279529  0.0350781 ]]

How the Estimator calculates error

In addition to the estimate of the mean of the observables passed in the input PUBs (the evs field of the DataBin), the Estimator also attempts to deliver an estimate of the error associated with those expectation values. All estimator queries will populate the stds field with a quantity like the standard error of the mean for each expectation value, but some error mitigation options produce additional information, such as ensemble_standard_error.

Consider a single observable $\mathcal{O}$. In the absence of ZNE, you can think of each shot of the Estimator execution as providing a point estimate of the expectation value $\langle \mathcal{O} \rangle$. If the pointwise estimates are in a vector Os, then the value returned in ensemble_standard_error is equivalent to the following (in which $\sigma_{\mathcal{O}}$ is the standard deviation of the expectation value estimate and $N_{shots}$ is the number of shots):

$\frac{ \sigma_{\mathcal{O}} }{ \sqrt{N_{shots}} },$

which treats all shots as part of a single ensemble. If you requested gate twirling (twirling.enable_gates = True), you can sort the pointwise estimates of $\langle \mathcal{O} \rangle$ into sets that share a common twirl. Call these sets of estimates O_twirls, and there are num_randomizations (number of twirls) of them. Then stds is the standard error of the mean of O_twirls, as in

$\frac{ \sigma_{\mathcal{O}} }{ \sqrt{N_{twirls}} },$

where $\sigma_{\mathcal{O}}$ is the standard deviation of O_twirls and $N_{twirls}$ is the number of twirls. When you do not enable twirling, stds and ensemble_standard_error are equal.

If you enable ZNE, then the stds described above become weights in a non-linear regression to an extrapolator model. What finally gets returned in the stds in this case is the uncertainty of the fit model evaluated at a noise factor of zero. When there is a poor fit, or large uncertainty in the fit, the reported stds can become very large. When ZNE is enabled, pub_result.data.evs_noise_factors and pub_result.data.stds_noise_factors are also populated, so that you can do your own extrapolation.

Result metadata

In addition to the execution results, both the PrimitiveResult and PubResult objects contain a metadata attribute about the job that was submitted. The metadata containing information for all submitted PUBs (such as the various runtime options available) can be found in the PrimitiveResult.metatada, while the metadata specific to each PUB is found in PubResult.metadata.

# Print out the results metadata
print("The metadata of the PrimitiveResult is:")
for key, val in result.metadata.items():
    print(f"'{key}' : {val},")
 
print("\nThe metadata of the PubResult result is:")
for key, val in result[0].metadata.items():
    print(f"'{key}' : {val},")

Output:

The metadata of the PrimitiveResult is:
'execution' : {'execution_spans': ExecutionSpans([DoubleSliceSpan(<start='2026-01-15 08:07:33', stop='2026-01-15 08:07:36', size=4096>)])},
'version' : 2,

The metadata of the PubResult result is:
'circuit_metadata' : {},