# qiskit.opflow.list_ops

## List Operators

`qiskit.opflow.list_ops`

List Operators are classes for storing and manipulating lists of Operators, State functions, or Measurements, and include some rule or `combo_fn`

defining how the Operator functions of the list constituents should be combined to form to cumulative Operator function of the `ListOp`

. For example, a `SummedOp`

has an addition-based `combo_fn`

, so once the Operators in its list are evaluated against some bitstring to produce a list of results, we know to add up those results to produce the final result of the `SummedOp`

’s evaluation. In theory, this `combo_fn`

can be any function over classical complex values, but for convenience we’ve chosen for them to be defined over NumPy arrays and values. This way, large numbers of evaluations, such as after calling `to_matrix()`

on the list constituents, can be efficiently combined. While the combination function is defined over classical values, it should be understood as the operation by which each Operators’ underlying function is combined to form the underlying Operator function of the `ListOp`

. In this way, the `list_ops`

are the basis for constructing large and sophisticated Operators, State Functions, and Measurements.

The base `ListOp`

class is particularly interesting, as its `combo_fn`

is “the identity list Operation”. Meaning, if we understand the `combo_fn`

as a function from a list of complex values to some output, one such function is returning the list as-is. This is powerful for constructing compact hierarchical Operators which return many measurements in multiple dimensional lists. For example, if we want to estimate the gradient of some Observable measurement with respect to some parameters in the State function, we can construct separate evaluation Operators for each parameter’s gradient which we must keep track of ourselves in a list, or we can construct a single `ListOp`

containing the evaluation Operators for each parameter, so the `eval()`

function returns the full gradient vector. Another excellent example of this power is constructing a Quantum kernel matrix:

```
data_sfn_list_op = ListOp(data_circuit_state_fns)
qkernel_op_circuits = ~data_sfn_list_op @ data_sfn_list_op
qkernel_sampled = CircuitSampler(backend).convert(qkernel_op_circuits)
qkernel_sampled.eval()
```

This will return the two dimensional Quantum kernel matrix, where each element is the inner product of some pair of the data State functions, or in other terms, a measurement of one data `CircuitStateFn`

by another.

You’ll encounter the `ListOp`

subclasses (`SummedOp`

, `ComposedOp`

, or `TensoredOp`

) more often as lazy results of Operator construction operations than as something you need to explicitly construct. Any time we don’t know how to efficiently add, compose, or tensor two `primitive_ops`

or `state_fns`

together, they’re returned in a `SummedOp`

, `ComposedOp`

, or `TensoredOp`

, respectively, so we can still work with their combined function and perhaps convert them into an efficiently combine-able format later.

Combination functions do not always behave predictably, and you must understand the conversions you’re making when you working with `list_ops`

. Most notably - sampling a sum of two circuits on Quantum hardware does not incorporate interference between the wavefunctions! In this case, we’re sending our State functions through a depolarizing channel before adding them, rather than adding them directly before the measurement.

### List Operators

`ListOp` (oplist[, combo_fn, coeff, abelian, ...]) | A Class for manipulating List Operators, and parent class to `SummedOp` , `ComposedOp` , and `TensoredOp` . |

`ComposedOp` (oplist[, coeff, abelian]) | A class for lazily representing compositions of Operators. |

`SummedOp` (oplist[, coeff, abelian]) | A class for lazily representing sums of Operators. |

`TensoredOp` (oplist[, coeff, abelian]) | A class for lazily representing tensor products of Operators. |