2.1. The Ising Model and the QUBO Problem
The Ising model—a simplified version of the Heisenberg model—is a mathematical model of ferromagnetism in statistical mechanics. In the Ising model, spins that can be in either of two states (+1 or −1) are typically located on the lattice points, and there are interactions between adjacent spins. The system has a quadratic Hamiltonian function and the spin status changes such that it minimizes the Hamiltonian spontaneously through the interaction between adjacent spins.
The Ising model is mathematically equivalent to the QUBO problem of finding a binary vector that minimizes the target quadratic polynomial function without any constraints on the variables. Lucas formulated various NP-hard combinatorial optimization problems, such as the maximum cut problem and the traveling salesperson problem, as Ising models [
13]. He embedded the objective function and constraints of the combinatorial optimization problem into a corresponding QUBO problem by adding penalty terms in the Hamiltonian function that raise the energy of a state only when any of the required constraints are violated.
Nowadays, further efforts are underway to formulate real-world application problems as QUBO problems, such as financial portfolio optimization running on a machine from D-Wave [
14] and drug discovery with a Gaussian boson sampling simulator [
15]. Additionally, the quantum annealer, or the hardware architecture harnessed by quantum entanglement dedicated to solving the QUBO problems based on the Ising model, has been actively developed. Examples include the quantum annealing machine from D-Wave Systems [
16] and the coherent Ising machine (CIM) from NTT [
17]. Furthermore, software simulators running on general-purpose computers such as the simulated bifurcation machine (SBM) from Toshiba [
18] and dedicated but classical semiconductor chips such as the Digital Annealer from Fujitsu [
19] have also been developed.
2.2. Hopfield Neural Network for Solving Combinatorial Optimization Problems
In the field of computer science, mathematical models inspired by the Ising model, such as the Hopfield neural network [
10,
11,
12], have been studied as an exotic computing approach to solve QUBO problems. The Hopfield neural network, proposed by Hopfield in 1982 [
10], is similar to the electron spin-glass model and can be used to solve QUBO problems by utilizing the intrinsic characteristics of its quadratic Hamiltonian minimization. In the original model, the output of a neuron, which is a node of the network, had a binary output value of zero or one. In the later model, the output has been extended to have a continuous value from zero to one [
11]. Further, the spin status is computed as continuous values, and a binarization operation is performed after reading out the final status.
The Hopfield neural network is able to solve combinatorial optimization problems by embedding the objective function and constraints into a QUBO formulation and searching for a ground state of Hamiltonian defined for the network [
12]. The procedure merely provides a local minimum of Hamiltonian depending on the initial states and random noise, and thus many trials are required in general to find the ground state. A carefully designed Hamiltonian function that efficiently represents the characteristics of the target problem and provides a more convex energy landscape can significantly increase the probability of reaching the global minimum. An appropriate procedure for updating the system status can also significantly influence the convergence performance for finding the optimal solution in the Hopfield neural network and its derivative algorithms, such as the Gaussian machine [
20]. In this study, we evaluated the performance of the proposed method by using a software simulator that implements a Hopfield neural network with continuous output values.
Considering the Hopfield neural network, the energy ground state is searched for by iteratively updating the neurons’ output based on specific update rules. Regarding the model with continuous output values, each neuron first calculates its internal value
using the following differential equation:
where
is the number of neurons in the network,
is the weight between neurons
and
,
is the output of another neuron
, and
is the bias (i.e., the input to neuron
). In addition,
is the time constant parameter, which determines how well the internal values of the neuron are conserved, considering the time variation
. The Hopfield neural network is an interconnected network; however, it does not contain any self-connections (
), and the weight values are symmetric (
).
Thereafter, the output value
of the neuron
is determined by the following sigmoid function:
where
is a parameter that determines the gradient of the sigmoid function (also known as the mean-field temperature). A higher
makes the sigmoid function closer to a linear function, whereas a lower
makes it closer to a step function.
The energy of the network in the continuous value model is defined as follows:
The discrete value model minimizes the energy
in Equation (4) by repeatedly updating the neurons, whereas the continuous value model minimizes the energy
, which also considers the entropy term
as in Equation (3) [
11]. Here, it is possible to search for a solution to a combinatorial optimization problem using the following steps.
Express the objective function to be minimized in a quadratic form for binary variables.
Compare the quadratic form to be minimized with the energy function in Equation (4) and obtain the values of and , which make them coincide.
Construct a Hopfield neural network with the obtained values of and .
Select one neuron randomly and update it using Equations (1) and (2).
Perform Step 4 several times to minimize the free energy in Equation (3).
Obtain the output values of the neurons and binarize them based on some criteria. The solution is obtained if the output values of the neurons after the binarization operation satisfy the constraint conditions. In this study, the binarization is performed based on whether the output value is greater than or equal to 0.5.
Regarding this case, for the binarized result of the minimum solution to
to be consistent with the minimum solution to
, the value of
must be considerably small at the end. Moreover, there must be no significant difference between the landscape of the basin created by
and that of
. Akiyama et al. proposed a method known as the sharpening technique, in which
is large and the gradient of the sigmoid function is gradual in the initial states and becomes progressively steeper over time [
5]. This method increases the probability of convergence to the global minimum by gradually transforming the space of
from a state where the ground state is easy to find to a state where there is almost no difference from the original landscape of
. We employ this technique, and the details of
scheduling are presented in
Section 3.3.1.