QNN PLAYGROUND

Question

Time Evolution

CHALLENGE

PLAYGROUND CHALLENGE

You can use the simple problem presented in the Playground Challenge to learn about Max-Cut problems,

a type of problem that would take a long time to try and solve using a regular computer.

There are five children.

Children connected with red lines get along,

Children connected with blue lines don’t get along.

Children with no line connecting them don’t particularly

interact with each other.

Divide the children between bus A and bus B.

Consider their relationships to each other,

and try and keep children who don’t get along

from being on the same bus.

Level of Harmony on the Bus 0

Level of Harmony on the Bus 0

Click!

Level of Harmony on the Bus 0

COM

YOU

Correct!
Too bad...

Now, what if there were 16 children?

There would be many, many more potential combinations

that need to be tested, making the problem a lot harder.

In mathematics, this sort of problem that involves finding the best answer

out of many possible combinations is called a “combinatorial optimization problem.”

Max-Cut problems are a particularly difficult type of combinatorial optimization problem.

TIn the Playground, you can use the Quatum Neural Network (QNN) simulator

to see what it’s like to solve a difficult Max-Cut problem involving up to 16 people.

TUTORIAL

- Getting started
- Problem setting
- Parameter optimization
- Next step

You can operate a simple simulator to get an idea of how the QNN searches for solutions for combinatorial optimization problems.

The QNN playground consists of three controllers:

**1. simulation controller**to initialize, run, and pause the simulation**2. problem setting controller**to define the Max-Cut problem size and graph structure**3. parameter controller**to input the QNN machine’s various parameters and three displays:

**4. graph display**to show the target Max-Cut problem**5. trajectory display**to report the time dependent amplitudes of constituent OPOs**6. result display**to report the cost function (cut value) and pump rate vs. time

Start the QNN playground by clicking the “Run” buttonin the simulation controller. You will see the three displays change as the simulation proceeds. In the simplest Max-Cut problem with two vertices connected by one edge with a positive weight, the problem mapped to the QNN consists of two OPOs connected by an anti-ferromagnetic coupling. As you can see in the trajectory display, the amplitudes of the two OPOs bifurcate to positive and negative values (anti-ferromagnetic order), meaning the two vertices are separated into two groups, and the cut value is one. This is a fundamental operation principle of the QNN.

In the graph display, a given problem and the states of OPOs are shown according to the following rule:

- □ Blue and red edges correspond to negative-valued (anti-ferromagnetic) and positive-valued (ferromagnetic) couplings.
- □ Blue and red circles represent 0-phase (spin-down) and π-phase (spin-up) states of OPOs.
- □ Circle sizes show the amplitudes of OPOs.

In the results display, you can see the cut value and pump rate vs. normalized time (the number of circulations of OPO pulses inside the ring cavity). When the pump rate is increased to above the threshold of a solitary OPO, the cut value increases rapidly and reaches a steady state value.

Let’s setup a more complicated problem. By using the problem setting controller, you can setup a target graph with up to 16 vertices and various edge structures, including a one-dimensional string, one-dimensional ring, two-dimensional square lattice, complete graph, and random graph. If you click the “random graph” button, you can get a new random graph each time. Each graph structure requires the optimization of QNN parameters to obtain good results. You can simply click the “auto” button ()in the parameter controller to do this pre-requisite automatically.

If you wish to optimize the QNN parameters by yourself, you can adjust the following parameters.

1. Initial pump rate: | Normally set below the OPO network threshold, P_in≲1. |

2.Final pump rate: | Normally set above the OPO network threshold, P_f≳1. |

3.Injection rate: | Determines how strongly or weakly the two OPOs are coupled. |

4.Noise parameter: | Represents quantum zero point fluctuation or squeezed vacuum fluctuation if an external squeezer is assumed. |

We hope you are motivated to understand various quantum features of the QNNs and learn how they are different from classical neural networks (CNNs). You can get quick answers to these questions by clicking “Quantum Features” in Machines-page. Longer but more precise answers are available in the White Paper (Chapter I in particular). If you click the “Real and Virtual Machines” button, MACHINESyou may run the real QNN for experiments and the virtual QNN in supercomputers for simulation.(comming soon)

Enjoy collective computing at criticality of OPO phase transition!

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