The aforementioned “base” Ising Model is a 1-D open model and is its energy, H, is commonly denoted by the following mathematical expression

H = − J 2 ∑ i = 1 N − 1     σ i σ i + 1 + h ∑ j = 1 N     σ j (1)

where N is the number of particles, σ i is the spin (between −1 and 1) of the particle with positional index i, and J and h are the interaction coefficient of the particle and the magnitude of any existing perturbations, respectively [15] . The left summation describes the interactions between electrons, and the right summation factors in any perturbations or external magnetic fields acting on the system.

Since quantum systems are time-dependent, we must time-evolve H. Instead of a scalar value σ representing a particle’s spin, we now use the Z basis, where Z = [ 1 0 0 − 1 ] since spin can have any value between −1 and 1 [16] . Additionally, we represent perturbations as the X basis, where X = [ 0 1 1 0 ] [16] . We could choose a different matrix to represent perturbations so long as it differs from the Z basis. After making these changes, our mathematical model is now equivalent to Equation (2):

H ^ = − J 2 ∑ i = 1 N − 1     Z i Z i + 1 + h ∑ j = 1 N     X j (2)

However, in order to adequately implement an Ising Model on a quantum circuit, we must break down the Hamiltonian H ^ to predefined rotation operators on a quantum circuit. Let the following be true:

H ^ = − J 2 ∑ i = 1 N − 1     Z i Z i + 1 + h ∑ j = 1 N     X j = A ^ + B ^ . (3)

The terms A ^ and B ^ are intelligible by a quantum circuit because the expressions can be simplified to R Z Z ( θ ) gates (composed of CNOT, R Z ( θ ) , and CNOT logical operators) for interactions between neighboring particles and R X ( θ ) gates for perturbations [13] [17] . For clarification, an R Z Z ( θ ) gate performs a rotation about the z-axis across 2 qubits whereas an R X ( θ ) gate performs a rotation about the x-axis across 1 qubit.

We then time-evolve the Hamiltonian by implementing the following expression: ψ ( t ) = e − i H ^ t / ℏ ψ ( 0 ) .

Recall that H ^ = − J 2 ∑ i = 1 N − 1     Z i Z i + 1 + h ∑ j = 1 N     X j = A ^ + B ^ . Since Z i Z i + 1 does not commute with X j , in other words, [ Z i Z i + 1 , X j ] ≠ 0 :

e − i H ^ t ≠ e − i A ^ t e − i B ^ t (4)

Therefore, we cannot directly represent H ^ on a quantum circuit, so we must apply Trotter Decomposition [17] [18] . We decompose e − i H ^ t to a particular combination of e − i A ^ t and e − i B ^ t with minimal error [17] [18] . This particular version of Trotter Decomposition is known as the Suzuki-Trotter approximation, a direct improvement from the traditional Lie-Trotter approximation. The Suzuki-Trotter decomposition is a corollary from the Baker-Campbell-Hausdorff Zassenhaus formula, which states the following [19] :

e t ( X + Y ) = e t X e t Y e − t 2 2 [ X , Y ] e t 3 6 ( 2 [ Y , [ X , Y ] ] + [ X , [ X , Y ] ] ⋯ (5)

Hence, for H ^ = A ^ + B ^ , Suzuki-Trotter decomposition states the following:

e − i H ^ t = e − i A ^ / 2 t e − i B ^ t e − i A ^ / 2 t + O ( t 3 ) (6)

The error O ( t 3 ) is negligible for our purposes. Shown below is the final mathematical expression for the open 1-D Ising Model Hamiltonian

H ^ = A ^ / 2 + B ^ + A ^ / 2 (7)

in that specific order. We used this expression as the base model for our quantum circuits in both the magnetic material and neural network applications.

For the magnetic materials application, we simulate the effect of perturbations on a chain of electrons. Recall Equation (2). To do this, we alter the values of J, the interaction coefficient between electrons, and h, the perturbations coefficient. We want to determine if the values of J and h have implications on the alignment probabilities of electrons in a given system, which influences whether the system is ferromagnetic or paramagnetic. We also model a chain of electrons that wraps around in a circle, known as a closed model. The closed model is more realistic because it closely represents a real system, but its glaring drawback is the increased complexity compared to the open model. Overall, in this paper, we show the feasibility of accurately modeling ferromagnetic and paramagnetic systems using the open 1-D Ising Model. See Figure 1 for an image showing the differences between an open and closed Ising Model.

Mathematically, for a linear chain of electrons, the Hamiltonian remains the same as that of the base Ising Model (refer to Figure 1(a)).

H ^ = − J 2 ∑ i = 1 N − 1     Z i Z i + 1 + h ∑ j = 1 N     X j (8)

For the closed model, since the first and last electrons interact (refer to Figure 1(b))

H ^ = − J 2 ∑ i = 1 N − 1     Z i Z i + 1 + Z N Z 1 + h ∑ j = 1 N     X j (9)

The additional term, Z N Z 1 , quantifies the spin interaction between the Nth electron, the last in the chain, and the 1st electron at the beginning of the chain.

In addition to magnetic materials, we applied the Ising Model to more complex geometries such as biological neural networks. We modified the coefficient of interaction J and coefficient of perturbation h to develop a high-level simulation of a human BNN. In this geometry, we included a main branch and 4 different auxiliary branches which stems from the main branch.

In order to differentiate the main branch from the auxiliary branches, we increased J on the main branch. In this paper, we evaluate how changing the ratio of J_main and J_aux impacted the information retention of the system, which is quantified by entropy. We also showed how removing or damaging certain neurons affect the overall entropy of the neural network, which provides useful information regarding the relative importance of each neuron and therefore which to avoid during neurosurgery. See Figure 2 for an image showing a particular neural network with four neurons per branch. Note the Main Branch, Central Qubit of Contact, Central Nervous System, Peripheral Nervous System, and Auxiliary Branches.

Since the BNN is a 2-D tree-like structure, the Hamiltonian of the entire system is the summation of the Hamiltonian for each branch.

H ^ = H ^ m a i n + Z n Z 1 A + Z n Z 1 B + Z n Z 1 C + Z n Z 1 D + H ^ A + H ^ B + H ^ C + H ^ D (10)

where n is the last neuron in the Main Branch. The Hamiltonian for each branch, denoted by H ^ m a i n , H ^ A , H ^ B , H ^ C , and H ^ D , are evaluated using the base Ising Model discussed in Section 2.1. Note that because we do not consider perturbations for the biological neural network model, the X basis summation in the base Ising Model equates to 0. The Suzuki-Trotter decomposition is therefore not necessary for our purposes. Hence, H ^ m a i n , H ^ A , H ^ B , H ^ C , and H ^ D simply represent series of the following:

H ^ = − J 2 ∑ i = 1 N − 1     Z i Z i + 1 (11)

where N is the length of said branch.

In order to create a quantum circuit that maps to an Ising Model, we develop an algorithm so that variations e.g. the number of neurons in the biological neural network application are automatically or can be easily accounted for with minimal changes.

Recall that the number of qubits in the quantum circuit is equivalent to the number of particles in the system. After each qubit is initialized and prepared for rotation, we begin processing the algorithm.

The algorithm depends on the specific application. Since the Ising Model is 1-D for the magnetic materials application, we use a basic for-loop to generate the quantum circuit. However, the BNN application is far more complex due to its 2-D tree-like structure. The algorithm for biological neural network involves first generating the branches for the tree, which is easily done with for-loops with boundaries based on the length of each branch (or the number of neurons or particles on that branch). If the length of the main branch differs, we simply modify the algorithm to append the main branch separately. While generating the tree, each neuron is simultaneously numbered as R Z Z ( θ ) gates are inputted in their respective locations to the quantum circuit. See the pattern for the enumerated neurons in Figure 2. To join each branch of the tree, we use the Breadth First Search (BFS) algorithm that connects the child of the main branch to the parent of each auxiliary branch [21] . At the same time, R Z Z ( θ ) gates are inputted into the quantum circuit.

See Figure 3 for a visual diagram describing the quantum circuit generation process as a whole.

Figure 4 shows a diagram of a typical magnetic material system with 2 electrons. Each purple circle represents an electron, and the blue arrow represents internal interactions between those electrons that affects their spins. Figure 5 shows the system’s corresponding quantum circuit.

In Figure 5, the light red squares with “H” represent the Hamiltonian operators that initialize each qubit, the red vertical lines represent the R Z Z ( θ ) operators, the dark red squares with “RX” represents the R X ( θ ) operators, and the two gray squares measures the spin probabilistic results. Recall that R Z Z ( θ ) operators quantify interactions between electrons and R X ( θ ) operators quantify

any perturbations or external magnetic field acting on the system. Also recall that in Section 2.1 Equation (5), after applying Trotter Decomposition, H ^ = A ^ / 2 + B ^ + A ^ / 2 . That is why there are two R Z Z ( θ ) operators with phase angles half of that of the R X ( θ ) gates.

In this application, the results of the quantum circuit are the probabilities of measuring each electron spin configuration and therefore the probability of each electron being spin up or down. An example of an electron configuration is 01, where 0 means the left electron was measured spin-up and 1 means the right electron was measured spin-down.

The expression under the R Z Z ( θ ) operator is its phase angle θ R Z Z where θ R Z Z = 2 π J . The value of J in this particular circuit is 2. Similarly, the expression under the R X ( θ ) operator is also its phase angle, where θ R X = 2 π h . The value of h in this particular configuration is 1.

Figure 6 shows a diagram of a biological neural network with 2 neurons per branch. The Main Branch is black and the two Auxiliary Branches are red and blue. Each neuron is indexed from 0 to N where N is the number of neurons in the neural network. Figure 7 shows the system’s corresponding quantum circuit.

In Figure 7, the light blue squares represent the initialized Hamiltonian operators, the purple vertical lines represent the R Z Z ( θ ) operators, and each black square measures the spin probabilistic results.

In this application, the results of the quantum circuit are the probabilities of measuring each neuron configuration and therefore the probabilities of each neuron being active or inactive. If a neuron is active, that means it is able to send and receive signals through neurotransmitters. If a neuron is inactive, that means

it lacks the ability to send or receive signals. An active neuron is represented by a spin-up qubit denoted by 0, and an inactive neuron is represented by a spin-down qubit denoted by 1. An example of a neuron configuration (quantum state) is 010011, which shows the neurons that are measured as active or inactive and the neuron index increases from left to right.

Furthermore, notice that the qubit in Figure 7 denoted q₁ has the most number of connections or R Z Z ( θ ) interactions with other qubits. This means that this qubit is the Central Qubit of Interaction and the child of the Main Branch. The Central Qubit of Interaction is labeled in Figure 6.

Note that for this paper, we use the BNN depicted in Figure 2. The BNN shown in Figure 6 is merely for example purposes.

In order to classify a magnetic material as ferromagnetic (same spin direction in the presence of external magnetic field) or paramagnetic (equal number of spin-up and spin-down electrons in the presence of external magnetic field), we need to utilize the values of J and h as metrics. Recall that J represents the coefficient of interaction between particles, and h represents the coefficient of perturbation from external magnetic fields. We determine the relationship between J and h by inputting different values of J and h (phase angles) in a particular system’s quantum circuit before running it. We analyze data from a total of 10,000 shots per run.

Figure 8 measures the probability of alignment, or the probability that every electron is spin-up or spin-down, for a hypothetical magnetic material with 4 electrons. When the electrons are all aligned, the material is ferromagnetic and therefore produces the strongest electric field towards the source of the magnetic field [14] . We notice a pattern when J and h have the same sign, for instance, if the value of J is positive and the value of h is also positive, the probability of alignment is greatest. On the other hand, when J and h have the opposite sign, the probability of alignment is the lowest, and therefore the particular material would produce the weakest electric field [14] .

We confirm this proposed phenomenon by analyzing bar charts that depict the number of shots for each possible spin state when J and h have the same sign or opposite sign. We ensure that the magnitudes of J and h are the same to reduce variability from extraneous factors.

Figure 9 shows that, when J and h are both positive, the spin configurations (quantum states) 0000 and 1111 have the highest number of shots out of 10,000. Likewise, when they were both negative, a similar pattern emerged. Hence, the probability of the electrons being aligned and therefore the material being ferromagnetic are likely higher when J and h share the same sign.

Figure 10 shows that, when the signs of J and h were different, in this case when J is negative and h is positive, the spin configurations 0101 and 1010 have the highest number of shots out of 10,000. Likewise, when J is positive and h is

negative, 0101 and 1010 still had the highest number of shots. Therefore, the probability of the magnetic material being paramagnetic is likely higher when J and h have opposite signs.

The brain is composed of approximately 10 billion neurons [22] . Each neuron can be seen as a single processing unit that receives and sends electrochemical signals from one neuron to another in a chain-like process. These electrochemical signals represent the “information” that constantly transfers across all the neurons in a body known as the biological neural network [22] .

One reason why neurons are important is that they allow for motor functions [23] . For example, when neurotransmitters send signals from the brain to the peripheral nervous system (other areas of the body), that area is given a command to move. However, if neurons in that path are damaged, the electrochemical signals never travel down to the area, which can lead to paralysis. In other cases, damage to neurons may cause phenomena such as the loss of feeling and sensations. Without properly functioning neurons, daily tasks such as eating, thinking, and walking are impossible [23] .

Neurosurgery is commonly performed on patients who exhibit issues with existing neurons. However, surgeons are not immune to making mistakes, around 4000 surgical errors or “never events” occur in the United States annually [24] .

Consequently, prior to neurosurgery, it is essential for experts to know which neurons are more valuable to the overall neural network to avoid damaging it. This section introduces a method to achieve that information with quantum computing and the Ising Model.

The value or importance of a neuron can be quantified by the neural network’s information retention if that neuron is damaged or outright removed. In a BNN, we think of information retention as the amount of information remaining when an electrical signal passed in the first neuron reaches the dendrite of the final neuron [22] . When neurons are damaged, for instance during neurosurgery, neurotransmitters fail to pass from the synapse of one neuron to the dendrite of the neighboring neuron [22] . Therefore, the total information retained is reduced.

Total system information retention is quantified by entropy, which is the level of disorder in a system and therefore the information stored. Entropy can be found by evaluating the following expression:

S ( ρ ) = − t r a c e ( ρ log ρ ) (12)

where ρ is a density matrix depicting the probabilities corresponding to each of the 2^N possible qubit spin configurations. There are 2^N possible spin configurations (or quantum states) because there are N qubits, and each qubit can be measured as spin-up (0) or spin-down (1). Hence, 2 ^N possible permutations [25] .

Figure 11 shows the effect of removing an arbitrary number of neurons from a hypothetical neural network with 4 neurons. We see an exponential decline in total system information retention as more neurons are removed. Refer back to Figure 2 for a detailed image of the biological neural network model we use.

To gauge the relative importance of each neuron in a biological neural network, we compare the fractions of initial entropy retained when the neurons are removed. Figure 12 plots the percent information retained after each neuron is removed based on the model shown in Figure 2. The largest decrease in entropy occurs after removing Neuron 3, therefore, Neuron 3 is the most important to the neural network. This makes sense because Neuron 3 maps to the Central Qubit of Interaction. If this was a real system, experts should be aware that

Neuron 3 is the most important and should avoid surgery around it.

Different neural networks may vary in terms of how information is transferred from neuron to neuron, branch to branch. One possible difference between neural networks is the ratio of the length of the main branch versus the length of each auxiliary branch. Figure 13 shows the effects of changing this ratio on the entropy of the neural network. As the ratio increases, so does the entropy, albeit minimally. Occasionally, the entropy decreases, for example, when the ratio is 3.5 or 4.5. We aren’t sure why entropy decreases at those values, but we are looking into possible causes. This is potentially an area for future work and analysis.