# 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.