• Simulated Bifurcation Machine (SBM)
Simulated Bifurcation Machine (SBM)

# Simulated Bifurcation Machine（SBM）Technologies

## What is a combinatorial optimization problem?

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.

## Ising machine that utilizes natural phenomenon for problem solving

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

## What is the Simulated Bifurcation algorithm?

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

## Theories behind the SBM

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

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## Reasons for high speed

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.

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

## Steps for applying the SBM to real-world problems

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

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