Perform gradient-based deterministic optimization of generic real-valued functions.
Use this function to determine a choice of variables that minimizes the
value of a scalar real-valued cost function of those variables. You
express that cost function as a graph describing how the input variables
are transformed into the cost value.
Note that this function will pick a different initial guess for each optimization run.
You can provide initial values (or a list of them of the same length) for the
optimization variables in the graph, and the optimizer will start with those values.
If the optimization_count is larger than the number of provided initial
values, the optimizer will use random values after consuming all available initial
values.
- Parameters
graph (qctrl.graphs.Graph) – The graph describing the cost \(C(\mathbf v)\) and outputs
\(\{F_j(\mathbf v)\}\) as functions of the optimizable input
variables \(\mathbf v\). The graph must contain nodes with names
\(s\) (giving the cost function) and \(\{s_j\}\) (giving the
output functions).
cost_node_name (str) – The name \(s\) of the real-valued scalar graph node that defines
the cost function \(C(\mathbf v)\) to be minimized.
output_node_names (List[str]) – The names \(\{s_j\}\) of the graph nodes that define the output
functions \(\{F_j(\mathbf v)\}\). The function evaluates these
using the optimized variables and returns them in the output.
max_iteration_count (int, optional) – The maximum number of cost evaluations to perform. You can set this as
an early stop condition for the optimizer. Defaults to None, which
means that the optimizer runs until it converges.
target_cost (float, optional) – A target value of the cost that you can set as an early stop condition
for the optimizer. If the cost becomes equal or smaller than this
value, the optimization halts. Defaults to None, which means that the
optimizer runs until it converges.
cost_tolerance (float, optional) – The tolerance \(\mathrm{tol}\) to check whether the cost function
has converged. You can set this as an early stop condition for the
optimizer. The optimizer stops when the improvement of the cost
function over the iterations is below the tolerance. That is, it stops
when \((C_j - C_{j+1}) / \max(|C_j|, |C_{j+1}|, 1) <=
\mathrm{tol}\). Defaults to None, meaning the optimizer handles the
tolerance automatically. You can try different values depending on your
optimization problem. A recommended starting value is about 1e-9.
optimization_count (int, optional) – The number \(N\) of independent randomly seeded optimizations to
perform. The function returns the results from the best optimization
(the one with the lowest cost). Defaults to 20.
cost_history_scope (qctrl.dynamic.types.HistoryScope, optional) – Configuration for the scope of the returned cost history data. Use this
to select if you want the history data to be returned. Defaults to no
cost history data returned.
seed (int, optional) – Seed for the random number generator used in the optimizer. Use this
option to generate deterministic results from the optimizer. Note that
if your graph contains nodes generating
random samples, you need to also set seeds for those nodes to ensure a
reproducible result.
- Returns
The result of an optimization. It includes the minimized cost
\(C(\mathbf v_\mathrm{optimized})\), and the outputs \(\{s_j:
F_j(\mathbf v_\mathrm{optimized})\}\) corresponding to the variables
that achieve that minimum cost.
- Return type
qctrl.dynamic.types.optimization.Result
Notes
Given a cost function \(C(\mathbf v)\) of variables \(\mathbf v\), this
function computes an estimate \(\mathbf v_\mathrm{optimized}\) of
\(\mathrm{argmin}_{\mathbf v} C(\mathbf v)\), namely the choice of
variables \(\mathbf v\) that minimizes \(C(\mathbf v)\). The function
then calculates the values of arbitrary output functions
\(\{F_j(\mathbf v_\mathrm{optimized})\}\) with that choice of variables.
This function represents the cost and output functions as nodes of a
graph. This graph defines the input optimization variables \(\mathbf v\),
and how these variables are transformed into the corresponding cost and
output quantities. You build the graph from primitive nodes defined in
the
graphs of the Q-CTRL Python package.
Each such node, which can be identified by a name, represents a function
of the previous nodes in the graph (and thus, transitively, a function of
the input variables). You can use any named scalar real-valued node \(s\)
as the cost function, and any named nodes \(\{s_j\}\) as outputs.
After you provide a cost function \(C(\mathbf v)\) (via a graph), this
function runs an optimization process \(N\) times, each with random
initial variables, to identify \(N\) local minima of the cost function,
and then takes the variables corresponding to the best such minimum as
\(\mathbf v_\mathrm{optimized}\).
A common use-case for this function is to determine controls for a
quantum system that yield an optimal gate: the variables \(\mathbf v\)
parameterize the controls to be optimized, and the cost function
\(C(\mathbf v)\) is the operational infidelity describing the quality of
the resulting gate relative to a target gate. When combined with the node
definitions in the Q-CTRL Python package, which make it convenient to
define such cost functions, this function provides a highly configurable
framework for quantum control that encapsulates other common tools such
as gradient ascent pulse engineering and chopped random basis (CRAB)
optimization .
The optimizer will carry out the gradient-based optimization until it
converges to a minimum. You can set up early stop conditions such as a
target cost or a maximum number of cost evaluations via the target_cost
and max_iteration_count parameters. If both are provided, the optimizer
will terminate when either of the two conditions is satisfied.
References
Examples
See the How to optimize controls in D-dimensional systems using graphs
user guide.