MEMS Oscillators‐Network‐Based Ising Machine with Grouping Method

Abstract Combinatorial optimization (CO) has a broad range of applications in various fields, including operations research, computer science, and artificial intelligence. However, many of these problems are classified as nondeterministic polynomial‐time (NP)‐complete or NP‐hard problems, which are known for their computational complexity and cannot be solved in polynomial time on traditional digital computers. To address this challenge, continuous‐time Ising machine solvers have been developed, utilizing different physical principles to map CO problems to ground state finding. However, most Ising machine prototypes operate at speeds comparable to digital hardware and rely on binarizing node states, resulting in increased system complexity and further limiting operating speed. To tackle these issues, a novel device‐algorithm co‐design method is proposed for fast sub‐optimal solution finding with low hardware complexity. On the device side, a piezoelectric lithium niobate (LiNbO3) microelectromechanical system (MEMS) oscillator network‐based Ising machine without second‐harmonic injection locking (SHIL) is devised to solve Max‐cut and graph coloring problems. The LiNbO3 oscillator operates at speeds greater than 9 GHz, making it one of the fastest oscillatory Ising machines. System‐wise, an innovative grouping method is used that achieves a performance guarantee of 0.878 for Max‐cut and 0.658 for graph coloring problems, which is comparable to Ising machines that utilize binarization.

Table S1 summarized the SoA MEMS oscillators operating above 5 GHz.Owing to the high Q value and figure-of-merit (FOM) nature of the LiNbO3 resonator, the proposed design might exhibit a comparatively low phase noise after careful design and optimization.Significant advancements in the field of MEMS oscillators, particularly those employing resonant-fintransistor (RFT) architectures, have led to remarkable achievements.With reported operational frequencies surpassing the 30 GHz benchmark, these developments highlight the burgeoning potential for improving solution speeds in this domain.This progress in academic research underscores the importance of continued exploration and optimization within the context of MEMS oscillator technology.Table S1.Comparison of the SoA MEMS oscillators above 5 GHz [1] [2] [3] simulation [4] simulation [5] simulation [ state is a unit vector whose angle corresponds to its phase.Then, assuming , the continuous Scut can be expressed as follows: .
When adding SHIL, the , so the Scut under SHIL is written as follows: .
And equation ( 5) is the true Scut obtained from the Ising machine with SHIL.Because the oscillator system energy can achieve a lower energy without the constraint of SHIL, we have .(6) As for the oscillators with various phases, we can divide them into two groups randomly using a straight line passing through the center of the circle.The probability of two oscillators being in different groups is , while the probability of two oscillators being in the same group is . The value of is -1 and +1, respectively, when the two oscillators are in different groups and in the same group.Therefore, the expectation value of Ising Hamiltonian under such a grouping method can be expressed as follows: .
From equation (7), the Scut under this grouping method can be written as .( 8) Since the Scut from grouping method cannot exceed the true Scut, combining ( 6) we can get Assuming there is a α making the inequality as follow: , (10) except for the case , we have .(11) In the right side of the inequality (11), there is a minimum value approximately equaling to 0.878 in the domain of definition.As a result, in addition to the case , we can get According to (12), in the case all the weights are positive, namely , we can get the relation associated with ( 4) and (8): .
Combining inequalities (9) and ( 13), finally we can get .( 14) which demonstrates that the SHIL-free Ising machine utilizing a grouping method can achieve a minimum expectation value of 0.878 for the true Max-cut on solving positive-weight Maxcut problems.

Supporting Note 4. Graph coloring problem
The graph coloring problem requires each pair of boarding graph sections to be colored with different colors.The graph to be colored can be transformed as an undirected graph with some vertices and edges.Some graph coloring problems can be solved using 3 colors.Firstly, we regard each vertex si as a unit vector which can be only 3 discrete angles (0, 2π/3 and 4π/3).
The vertices are grouped into 3 groups, V1, V2 and V3 according to their phases.And the edges So, the Scut can be rewritten as follows: . ( 16) From ( 15) and ( 16), we know Scut will be maximized when the Ising Hamiltonian H is minimized.And the largest Scut is when all the neighboring vertices are in different groups.That is, the neighboring group sections are assigned to different colors from the view of graph coloring problems.

Supporting Note 5. Graph coloring problem under grouping method
When using oscillator network to solve graph coloring problems, it can be assumed that each oscillator's state is a unit vector whose angle corresponds to its phase.Then, assuming , there is a continuous Scut that can be expressed as follows: . ( 17) When forcing the phases being 3 discrete values like 0, 2π/3 and 4π/3, the , so the true Scut is written as follows: . ( 18) The true Scut in ( 18) is the same as ( 16).Because the oscillator system energy can achieve a lower energy without constraint, similarly we have: .
As for the oscillators with continuous phases, we can divide them into three groups randomly by a range of 2π/3.When , the probability of two oscillators being in different groups is , while the probability of two oscillators being in the same group is .When , the probability of two oscillators being in different groups is 1, while the probability of two oscillators being in the same group is 0. The value of is -1/2 and +1, respectively, when the two oscillators are in different groups and in the same group.Therefore, the expectation value of Ising Hamiltonian under such grouping method can be expressed as follows: .
From (20), the Scut under this grouping method can be written as associated with a specific angle range, which can be translated to its corresponding probability.
This part will present the calculation process of angle range and probability of one solution.
In the case of n oscillators, if we assign each oscillator a number, there must be a sequence of loop for n oscillators on a unit circle.For example, when n = 5, the sequence of loop for , based on which we can obtain the probability P of this solution.