# Landscape for optimal control of quantum-mechanical unitary transformations

## Abstract

The optimal creation of a targeted unitary transformation W is considered under the influence of an external control field. The controlled dynamics produces the unitary transformation U and the goal is to seek a control field that minimizes the cost J= parallel W-U parallel . The optimal control landscape is the cost J as a functional of the control field. For a controllable quantum system with N states and without restrictions placed on the controls, the optimal control landscape is shown to have extrema with N+1 possible distinct values, where the desired transformation at U=W is a minimum and the maximum value is at U=-W. The other distinct N-1 extrema values of J are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.

- Authors:

- Department of Chemistry, Princeton University, Princeton, New Jersey 08544 (United States)
- Department of Chemistry, Drexel University, Philadelphia, Pennsylvania 19104 (United States)

- Publication Date:

- OSTI Identifier:
- 20786489

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.72.052337; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; OPTIMAL CONTROL; QUANTUM COMPUTERS; QUANTUM MECHANICS; TRANSFORMATIONS; UNITARITY

### Citation Formats

```
Rabitz, Herschel, Hsieh, Michael, and Rosenthal, Carey.
```*Landscape for optimal control of quantum-mechanical unitary transformations*. United States: N. p., 2005.
Web. doi:10.1103/PHYSREVA.72.0.

```
Rabitz, Herschel, Hsieh, Michael, & Rosenthal, Carey.
```*Landscape for optimal control of quantum-mechanical unitary transformations*. United States. doi:10.1103/PHYSREVA.72.0.

```
Rabitz, Herschel, Hsieh, Michael, and Rosenthal, Carey. Tue .
"Landscape for optimal control of quantum-mechanical unitary transformations". United States.
doi:10.1103/PHYSREVA.72.0.
```

```
@article{osti_20786489,
```

title = {Landscape for optimal control of quantum-mechanical unitary transformations},

author = {Rabitz, Herschel and Hsieh, Michael and Rosenthal, Carey},

abstractNote = {The optimal creation of a targeted unitary transformation W is considered under the influence of an external control field. The controlled dynamics produces the unitary transformation U and the goal is to seek a control field that minimizes the cost J= parallel W-U parallel . The optimal control landscape is the cost J as a functional of the control field. For a controllable quantum system with N states and without restrictions placed on the controls, the optimal control landscape is shown to have extrema with N+1 possible distinct values, where the desired transformation at U=W is a minimum and the maximum value is at U=-W. The other distinct N-1 extrema values of J are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.},

doi = {10.1103/PHYSREVA.72.0},

journal = {Physical Review. A},

number = 5,

volume = 72,

place = {United States},

year = {Tue Nov 15 00:00:00 EST 2005},

month = {Tue Nov 15 00:00:00 EST 2005}

}