# Quantum Max-flow/Min-cut

## Abstract

The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network and, more specifically, as a linear map from the input space to the output space. The quantum max-flow is defined to be the maximal rank of this linear map over all choices of tensors. The quantum min-cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network. We show that unlike the classical case, the quantum max-flow=min-cut conjecture is not true in general. Under certain conditions, e.g., when the capacity on each edge is some power of a fixed integer, the quantum max-flow is proved to equal the quantum min-cut. However, concrete examples are also provided where the equality does not hold. We also found connections of quantum max-flow/min-cut with entropy of entanglement and the quantum satisfiability problem. We speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity.

- Authors:

- Department of Mathematics, University of California, Santa Barbara, California 93106 (United States)
- (United States)
- Computer Science Division, University of California, Berkeley, California 94720 (United States)
- Center for Communications Research, La Jolla, California 92121 (United States)

- Publication Date:

- OSTI Identifier:
- 22596848

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ENTROPY; INTEGRALS; QUANTUM ENTANGLEMENT; QUANTUM GRAVITY; TENSORS

### Citation Formats

```
Cui, Shawn X., E-mail: xingshan@math.ucsb.edu, Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052, Freedman, Michael H., E-mail: michaelf@microsoft.com, Microsoft Research, Station Q, University of California, Santa Barbara, California 93106, Sattath, Or, E-mail: sattath@gmail.com, Stong, Richard, E-mail: stong@ccrwest.org, and Minton, Greg, E-mail: gtminto@ccrwest.org.
```*Quantum Max-flow/Min-cut*. United States: N. p., 2016.
Web. doi:10.1063/1.4954231.

```
Cui, Shawn X., E-mail: xingshan@math.ucsb.edu, Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052, Freedman, Michael H., E-mail: michaelf@microsoft.com, Microsoft Research, Station Q, University of California, Santa Barbara, California 93106, Sattath, Or, E-mail: sattath@gmail.com, Stong, Richard, E-mail: stong@ccrwest.org, & Minton, Greg, E-mail: gtminto@ccrwest.org.
```*Quantum Max-flow/Min-cut*. United States. doi:10.1063/1.4954231.

```
Cui, Shawn X., E-mail: xingshan@math.ucsb.edu, Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052, Freedman, Michael H., E-mail: michaelf@microsoft.com, Microsoft Research, Station Q, University of California, Santa Barbara, California 93106, Sattath, Or, E-mail: sattath@gmail.com, Stong, Richard, E-mail: stong@ccrwest.org, and Minton, Greg, E-mail: gtminto@ccrwest.org. Wed .
"Quantum Max-flow/Min-cut". United States.
doi:10.1063/1.4954231.
```

```
@article{osti_22596848,
```

title = {Quantum Max-flow/Min-cut},

author = {Cui, Shawn X., E-mail: xingshan@math.ucsb.edu and Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052 and Freedman, Michael H., E-mail: michaelf@microsoft.com and Microsoft Research, Station Q, University of California, Santa Barbara, California 93106 and Sattath, Or, E-mail: sattath@gmail.com and Stong, Richard, E-mail: stong@ccrwest.org and Minton, Greg, E-mail: gtminto@ccrwest.org},

abstractNote = {The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network and, more specifically, as a linear map from the input space to the output space. The quantum max-flow is defined to be the maximal rank of this linear map over all choices of tensors. The quantum min-cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network. We show that unlike the classical case, the quantum max-flow=min-cut conjecture is not true in general. Under certain conditions, e.g., when the capacity on each edge is some power of a fixed integer, the quantum max-flow is proved to equal the quantum min-cut. However, concrete examples are also provided where the equality does not hold. We also found connections of quantum max-flow/min-cut with entropy of entanglement and the quantum satisfiability problem. We speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity.},

doi = {10.1063/1.4954231},

journal = {Journal of Mathematical Physics},

number = 6,

volume = 57,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}