List Operators (
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
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.
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 (
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
state_fns together, they’re returned in
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.
A Class for manipulating List Operators, and parent class to
A class for lazily representing compositions of Operators.
A class for lazily representing sums of Operators.
A class for lazily representing tensor products of Operators.