# .qubo format for QUBOs#

`.qubo`

is a format used to model QUBO problems of the form \(x^TQx\),
where \(x\in\{0, 1\}^n\) is the solution vector, and \(Q \in \mathbb{R}^{n \times n}\) is a symmetric matrix.

## Basic Structure#

The file should have no header.

All in-line separators should be spaces.

The first line is a string that sets the optimization sense, which can be

`MINIMIZE`

or`MAXIMIZE`

.The second line needs to be a float, giving the offset: constant offset of the energy.

Next the reader expects the triplets of the upper triangular part of the QUBO matrix. A triplet is given with two integers and a float:

`i`

`j`

`q_{ij}`

:`i`

is the row of the entry;`j`

is the column of the entry;`q_{ij}`

is the value at the position`i`

. Because the matrix \(Q\) is symmetric, only the diagonal and upper-triangular entries of the matrix should be specified. The solver will automatically make the matrix symmetric. Formally, this means that \(i \le j\).Finally, the reader expects a number of variable fixings, i.e., variables whose assignment is set ahead of time. The format is

`f ix val`

where`ix`

is the variable’s index and`val`

sets the value of the variable to either`0`

or`1`

## File syntax#

```
MAXIMIZE <--- optimization sense
# this is a comment
2.0 <--- constant energy offest
0 0 1.0 <-- entries
0 2 1.0
0 4 -1.0
1 1 1.0
2 2 1.0
2 4 1.0
3 3 1.0
4 4 1.0
5 5 1.0
f 0 0 <--- fix variable #0 to value of 0
f 2 1 <--- fix variable #2 to value of 1
```

Here is the same problem from above, but without the comments, in a formally correct syntax:

```
MAXIMIZE
2.0
0 0 1.0
0 2 1.0
0 4 -1.0
1 1 1.0
2 2 1.0
2 4 1.0
3 3 1.0
4 4 1.0
5 5 1.0
f 0 0
f 2 1
```

## Example#

If we take the example from above, the solver would be solving the following problem: