# calculate_optimization¶

static FunctionNamespace.calculate_optimization(*, graph, cost_node_name, target_cost=None, output_node_names, optimization_count=20, **kwargs)

Performs 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.

Parameters
• graph (qctrlcommons.graph.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.

• 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.

• 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.

• 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.

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 operations namespace 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 1 and chopped random basis (CRAB) optimization 2.

References

1

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Journal of Magnetic Resonance 172, 2 (2005).

2

P. Doria, T. Calarco, and S. Montangero, Physical Review Letters 106, 190501 (2011).

Examples

See the Optimization user guide.