Online Stochastic Combinatorial Optimization
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Ising spin chip 7 based on CMOS architecture and coherent Ising machine 8 based on laser networks are newer additions to this family. The difference in their mechanism of operation can be illustrated based on the three main components of each unit. Manipulation of these states is induced by the current flow through the spin Hall materials.
Online Stochastic Combinatorial Optimization (English) 1st Edition (Paperback)
This poses challenge to the scalability of the system as the problem size increases. Read: Reading the magnetization state in both methods are performed by a current flow through an MTJ, where the free layer is coupled to the nanomagnet of the write unit through dipolar interaction. However, the output parameter in previous works is current proportional to the resistance of the MTJ , which is passed on to the inputs of interacting spin units. But, in our device, the MTJ current is converted to an output voltage by using a voltage divider circuit.
This voltage later provides inputs to coupled units. Interconnect: The spin-spin interaction is emulated by sending the output from one spin unit to the other. The current-based input-outputs in previous works cause current flow from one interacting unit to the other, whereas, in our device, there is no current flow between connected spin units due to voltage-based coupling. This is an important aspect of scalability of the scheme. Also previous works suggest additional amplification 9 or CMOS circuits 10 in between coupled spin units in order to drive the fan-out, whereas in our device, direct cascading is possible by proper device engineering.
It is to be noted that the network of spins in our model does not entirely resemble a Boltzmann machine The focus of our scheme is to make the lower energy states to be more probable with no restriction on the overall energy landscape. Therefore, although the probability of observing a state with energy E gradually increases with decreasing E , it may or may not follow the Boltzmann distribution.
In this work, we have presented the structure and switching characteristics of these voltage-controlled spin units. Additionally, we have showed a way of direct configuration of the spin-spin and spin-external source coupling network from the Ising Hamiltonian, taking TSP as an example. Ising model, although originally introduced as a model for ferromagnetic materials, is widely applied in molecular biology, chemistry and other areas due to its combinatorial interpretation.
According to this model, the spin dynamics in a ferromagnetic lattice, consisting of N lattice sites, is governed by the following Hamiltonian 2 :. J i , j and h i correspond to the energies due to the interactions with the nearest neighbors and external fields, respectively. The spins interact in such a way that they tend to eventually line up in the configuration producing the lowest value of H , thus transitioning from a high energy random state to a low energy ordered state at or below a critical temperature.
Finding the spin configuration which minimizes H is itself an NP-hard problem. Hence, the elements of a combinatorial optimization problem can be thought of as a collection of spins, x i , where the Ising energy function, H represents the parameter to be optimized. For example, the traveling salesman problem, which asks for the ordering of cities to visit so that the total distance travelled is the minimum, can be formulated as an Ising energy function in the following way 13 :.
W uv is the distance between cities u and v. At the ground state of the system, H A equals 0 and H B denotes the minimum distance travelled. Hence, at this state, the collection of x u , i gives us the order of cities to visit for achieving minimized H. Each spin, x u , i denotes whether city u should be visited at order i , where the rows represent the cities and the columns stand for the order of visit. This conversion from bipolar to binary variables is shown in the table. The magnetization moment vs time plots are shown in the insets at positive C , negative A and zero B input voltages.
The stochastic switching characteristic of nanomagnets, arising from the inherent thermal noise, enables its use as RNGs. Nanomagnet-based RNGs have been demonstrated previously, like spin transfer torque based spin-dice 14 , spin-orbit torque based spin-dice 15 , voltage controlled spin-dice 16 etc.
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However, high current requirement in spin-torque-based devices and unipolar switching 17 of VCMA-based devices have stirred the quest for new mechanisms, like the magnetoelectric ME effect. Exchange bias interaction occurs at the interface between the ME oxide and the ferromagnet, while the spin and charge polarization of the multiferroic material BFO are coupled to each other. In our model, we have used an ultrathin CoFeB film thickness of 0. Each spin x u , i is represented by the magnetization moment of CoFeB.
However, since the crucial aspects of these fields, like coupling mechanisms, switching dynamics etc. The details of the simulation methodology is explained in the next section. The read unit Fig. However, dipolar interaction couples the free layer of the MTJ to the magnetization x u , i of the CoFeB layer in the write unit.
Here, V REF denotes the supply voltage.
At the same time, it should be sufficient to let V OUT reach the desired values to drive subsequent units. Here x v , j acts as the input to x u , i and they lie either on the same row or same column. Bottom panel: A plot of the Hamiltonian cycle, H A with time. The input voltage V IN consists of two components, V neighbor and V external which model the interaction with neighboring units and external sources, respectively.
H A imposes the restriction that each city should be visited only once. It can be reformulated into the following form:. Each term in H A contains pairs of spins lying on the same row or same column of the N by N system. Note that, the rows indicate individual cities and the columns represent their order of travel. In that case, x u , i is decided by the other part of the Hamiltonian associated with the weight matrix. The truth table in Fig. The top panel in Fig. The corresponding switching probability is demonstrated in the bottom panel of this figure.
We have performed a sample simulation of two bits to validate this model. The result is shown in the top panel of Fig. As expected, H A is concentrated at 0 during the course of time Fig. The other component of the input voltage V external depends on H B which can be re-written in the following form:. Each term in H B dictates whether city u and v should be visited consecutively depending on the distance between them, W u , v.
Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems
At the ground state of the system, H B refers to the minimum distance travelled. With this restriction in mind, the truth table in Fig.
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The value of V u , v is set by the W u , v vs V u , v curve shown in the bottom panel of Fig. Note that the first letters in the subscript of the variable names x A ,1 etc.
In addition, all units are being injected with V neighbor , as described in the previous paragraph, to prevent simultaneous selection of spins in the same column. Therefore, Fig. Top panel: The switching probability curve with applied voltage.
Combinatorial Optimization — Eindhoven University of Technology research portal
City A is predetermined to be visited first. H B is concentrated near the minimum distance W AB , as expected. In our numerical simulation, we have used a time step of 0. After each time step, the magnetization moment of each bit is updated, and these updated values are used to generate the input voltages for the next time cycle Fig.
A behavioral model is used to generate the input voltages, taking corresponding spin values as inputs. The details of the micromagnetic simulation and the behavioral model are described in this section. A schematic showing the flow-diagram of our coupled simulation framework for two sample spin units x 1 and x 2. The LLG equation with monodomain approximations is described as follows 22 ,.
Note that there is no spin transfer torque due to negligible current flow through the oxide. Note that the easy axis of the PMA magnet is along z-direction. The experiment in ref. However, for the voltage range in our model, we have used an average value of 0.