# Problem 6: Find the Largest Complete Graph for a Chimera Unit Cell

* Example AND Gate explains how nodes are merged (two or more qubits are chained to represent a single variable). Its code demonstrating minor-embedding a 3-variable problem on the quantum computer, both automatically and manually, is also helpful for this question.

** Ocean's dwave-system documentation is here.

• The largest complete graph would be K5, that can be sub-graph of bipartite graph K4,4. See the first part of the code.

In part two, the problem in challenge 4 was manually embedded to qubit 5 and 13, which are in separate unit cells. Results look the same.

Below are the results screen and the code.

********** Part 1: Largest complete graph Kn that can be a sub-graph of bipartite graph K4,4 ***********

Assignment :
{'q1': {0}, 'q2': {4}, 'q3': {2, 6}, 'q4': {3, 5}, 'q5': {1, 7}}
Neighbourhood check :
{'q1': {'q5', 'q4', 'q2', 'q3'}, 'q2': {'q5', 'q1', 'q4', 'q3'}, 'q3': {'q4', 'q1', 'q2', 'q5'}, 'q4': {'q5', 'q1', 'q2', 'q3'}, 'q5': {'q4', 'q1', 'q2', 'q3'}}

********** Part 2: Problem 4 using manual embedding to qubit 5 and 13 (in two different unit cells)

|h0 = 0 |h1 = 1 | Energy = -0.200000 | Probability = 32.000000 %
|h0 = 1 |h1 = 0 | Energy = -0.200000 | Probability = 68.000000 %

• using entanglement effect (couplers), you can merge 2 physical qbits into 1 logical, gaining connectivity to another qbits, paying the price with consuming available physical qbits.

that way (using 3 couplers) you get from K4,4 bipartite grap, one (max) K5 complete subgraph.using more couplers you get less vertices.

• Thanks to those of you who tried this problem!  Kudos to Yasas and Branislav for finding the answer we were looking for, which is K5.

Next, take a look at Problem 7, posted today.