# An exact formulation of the time-ordered exponential using path-sums

## Abstract

We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs, and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.

- Authors:

- Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU (United Kingdom)
- Colby College, 4000 Mayflower Hill Dr., Waterville, Maine 04901 (United States)
- Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 Munich (Germany)
- (Singapore)

- Publication Date:

- OSTI Identifier:
- 22403149

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONTINUED FRACTIONS; DIAGRAMS; GRAPH THEORY; MATRICES; TIME DEPENDENCE

### Citation Formats

```
Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk, Lui, K., Thwaite, S. J., Jaksch, D., and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543.
```*An exact formulation of the time-ordered exponential using path-sums*. United States: N. p., 2015.
Web. doi:10.1063/1.4920925.

```
Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk, Lui, K., Thwaite, S. J., Jaksch, D., & Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543.
```*An exact formulation of the time-ordered exponential using path-sums*. United States. doi:10.1063/1.4920925.

```
Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk, Lui, K., Thwaite, S. J., Jaksch, D., and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543. Fri .
"An exact formulation of the time-ordered exponential using path-sums". United States. doi:10.1063/1.4920925.
```

```
@article{osti_22403149,
```

title = {An exact formulation of the time-ordered exponential using path-sums},

author = {Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk and Lui, K. and Thwaite, S. J. and Jaksch, D. and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543},

abstractNote = {We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs, and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.},

doi = {10.1063/1.4920925},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 5,

volume = 56,

place = {United States},

year = {2015},

month = {5}

}