（SBM）Technologies

A problem for finding the best combination among an exponential number of candidates. With an increase in the problem size, i.e., the number of combinations in total, it is practically impossible to exhaustively test every combination and arrive at a good solution, which is one of the limitations of traditional computing.

**Examples of combinatorial optimization problems**

**Logistic optimization**

Find a route with the shortest travel distance.

**Traffic congestion alleviation**

Determine the route of each vehicle to minimize congestion.

**Financial portfolio optimization**

Find a combination of different stocks with high return and low risk.

**Molecular design for drug discovery**

Identify the molecular makeup of drugs with the desired efficacy.

**Small-scale problems**

Possible to test every single combination and find the solution.

**Large-scale problems**

Practically impossible to every single combination; the remaining option is to rely on experiences, educated guesses, and trial and error.

that utilizes natural phenomenon for problem solving

Applies and simulates the model of magnets known as the Ising model to solve combinatorial optimization problems.

Combinatorial Optimization Problems

Ising Model

Ising Machine

Combinatorial Optimization Problems.

Ising Machine

Ising Model

- ※Model of magnets representing the interaction of atomic spins that are aligned in a lattice system.

A combinatorial optimization algorithm discovered in the process of research into quantum computers we proposed as "quantum bifurcation machines"

### Quantum bifurcation machine

Schrödinger equation

### Classical bifurcation machine

Hamilton's equation

### Quantum bifurcation machine

Schrödinger equation

### Classical bifurcation machine

Hamilton's equation

We have applied such interesting phenomena in classical dynamics as bifurcation phenomena, adiabatic processes, and ergodic processes (chaos) to combinatorial optimization for the first time.Theoretically derived from our unique quantum computers called "quantum bifurcation machines", the theories behind it represent a purely quantum-mechanical discovery that even suggests the existence of unknown theorem in mathematical physics.

**Bifurcation phenomenon**

Changes in system parameters result in a transition from a single stable state to multiple stable states

**Convert continuous variables into discrete variables.**

**Adiabatic process**

The system remains in a low-energy state when the system parameters slowly change

**Track the bottom of potential.**

**Ergodic process**

The system visits all allowable energy states and stays longer in states with low potential energy

**Find a lower point.**

The motion of 2,000 particles as the Simulated Bifurcation Machine solves an optimization problem with 2,000 fully connected variables

Figure 1: Temporal change of particle position x

Figure 2: Motion of particles in phase space (xy plane surface)

RED: x is positive BLUE: x is negative

* Click the Play button to start the video. YouTube is the services provided by another company; please follow the terms of use in YouTube.

Unlike simulated annealing (SA) characterized by sequential updates, our algorithm for computationally solving differential equations allows parallel updates, which in turn makes it possible for conventional parallel computers to achieve acceleration.

Our algorithm thus highly resonates with the technology trends in parallel computing.

### Simulated annealing (SA)

Sequentially updates the sub-optimal solution

→Does not allow parallel updates of variables### Our simulated bifurcation (SB) algorithm

Computationally solves differential equations.

→Allows parallel updates of variables

### Simulated annealing (SA)

Sequentially updates the sub-optimal solution

→Does not allow parallel updates of variables### Our simulated bifurcation (SB) algorithm Computationally solves differential equations.

→Allows parallel updates of variables

H. Goto (2016). Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network, Scientific Reports, 6, 21686

After translating real-world problems into combinatorial optimization problems, mapping to the so-called "Ising model" is required.

### 1.Real-world problems

eg. Portfolio Optimization

Correlation between assets in portfolioFind a combination of assets.

### 2.Combinatorial optimization problems

Modeling

Model the obtained combination as a problem to find those matrices that have desired properties.

### 3.Ising model

Formulating an equation

Formulate an equation which minimizes when a combination represents matrices with optimal properties.

### 4.SBM

Computing

Obtain the optimal solution.

Ising machine

### 1.Real-world problems

eg. Portfolio Optimization

Correlation between assets in portfolioFind a combination of assets.

### 2.Combinatorial optimization problems

Modeling

Model the obtained combination as a problem to find those matrices that have desired properties.

### 3.Ising model

Formulating an equation

Formulate an equation which minimizes when a combination represents matrices with optimal properties.

### 4.SBM

Computing

Obtain the optimal solution.

Ising machine

* The formulas on this page are used to explain the principles of SBM and may differ from the formulas in the SBM PoC version.

* The formulas on this page are used to explain the principles of SBM and may differ from the formulas in the SBM PoC version.

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