# Category Theory Approach to Solution Searching Based on Photoexcitation Transfer Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Solution Searching Based on Bounceback Principles

#### 2.1. Spatiotemporal Amoeba Dynamics

#### 2.2. Photoexcitation Transfer Modeling

_{S}, located in the center, surrounded by six larger quantum dots denoted by QD

_{Li}$(i=1,\dots ,6)$. The optical excitation generated at QD

_{S}, or more precisely the energy level (denoted by S) of QD

_{S}in Figure 1b, is transferred to the energy level marked ${L}^{(U)}$ in QD

_{Li}via optical near-field interaction ${U}_{SL}$. The subsequent energy dissipation $\mathrm{\Gamma}$ induced in QD

_{L}transfers the excitation to lower energy level ${L}^{(L)}$, prohibiting the excitation from being transferred back to the original QD

_{S}. The dynamics are described by a density matrix formalism in the Lindbrad form, given by

_{S}to each QD

_{Li}differs depending on the state filling effects induced in the destination QD

_{Li}[9]. Figure 2a(0–3) shows evolutions of populations with respect to the lower energy levels of the six larger dots (QD

_{L1},…,QD

_{L6}), where zero, one, two, or three QD

_{Li}quantum dots experience state fillings. In the numerical calculations, we assume inter-dot near-field interactions ${U}_{\mathrm{SL}i}^{-1}=100$ ps, sublevel energy dissipation ${\mathrm{\Gamma}}_{i}^{-1}=10$ ps, optical radiation from a larger dot ${\gamma}_{\mathrm{L}i}^{-1}=1$ ns, and that from a smaller dot ${\gamma}_{\mathrm{S}}^{-1}=2.92$ ns as a typical parameter set. When there is no state filling, an optical excitation sitting initially at QD

_{S}can be transferred to any one of QD

_{L1}to QD

_{L6}with the same probability (Figure 2a(0)). If one of the QD

_{Li}quantum dots, for example QD

_{L1}, suffers from state filling, the initial exciton in QD

_{S}is more likely to be transferred to the non-state-filled QD

_{Li}quantum dots (QD

_{L2}QD

_{L6}), as shown in Figure 2a(1). Likewise, the energy transfer probability differs depending on the occupation of the destination QD

_{Li}quantum dots, as shown in Figure 2a(2,3), where two and three QD

_{Li}quantum dots, respectively, are subjected to state filling. The insets of Figure 2a show the energy transfer probabilities to QD

_{Li}($i=1,\dots ,6$) derived on the basis of the numerical integral of the calculated populations. There are 2

^{6}inherently different energy transfer patterns in total.

_{L1}, QD

_{L2}, QD

_{L3}, QD

_{L5}, and QD

_{L6}experience state fillings because ${x}_{1}={x}_{2}={x}_{6}=1$. The system is transferred to $({x}_{i},\dots ,{x}_{6})=(0,0,0,1,0,1)$ at $t=2$. Based on the updated state fillings, the system evolves to $({x}_{i},\dots ,{x}_{6})=(0,1,0,1,0,1)$, which is one of the correct solutions of the given CSP. Figure 2c shows the evolution of the correct-selection rate from the initial state of $({x}_{i},\dots ,{x}_{6})=(0,0,0,0,0,0)$ for 10,000 trials, in which the correct selection rate stabilized at approximately 0.65 after the $t=50$ cycle.

## 3. Category Theoretic Picture and Analysis

#### 3.1. Product, Coproduct, and Short Exact Sequence

**Definition**

**(product and coproduct)**

**state of the problem**, which is decomposed into the

**optical emission from the system**(P) and the

**bounceback rule**(Q) to be applied to the system, where all the environmental processes are taken into account. In other words, the total system (X) is a composite system comprising P and Q. Therefore, we assume that the total system that is under study forms a monoidal category. Hence, the theorems, lemmas, and axioms of the Abelian category apply to the system.

**bounceback controller**. Here, the sequence $C\to X\to Q$ is a short exact sequence.

**Definition**

**(short exact sequence)**

_{i}(where $i=1,2,3,4,6$) according to Equation (1).

**state of energy transfer**. Here, we pay attention to the morphism $v:Q\to Y$. We can naturally introduce the cokernel of v, which is defined as the quotient $Y/\mathrm{Im}(v)$. According to category theory, the cokernel can also be regarded as a set comprising the subsequent object and morphism [29]. Thus, an object D is placed after $Q\to Y$ as shown in Figure 3d. Physically, the object D corresponds to an

**optical environmental condition**in which all the unobservable environmental conditions for the light emission are included implicitly. For example, suppose that no bounceback rules are active in Q. Hence, no control lights are supplied to inhibit energy transfer. Therefore, the optical environmental conditions allow a variety of energy transfer patterns that could potentially be induced in the QD system. In this manner, we can comprehend the cokernel as implying the potential for growth.

#### 3.2. Short-Exact-Sequence-Based Time

**Definition**

**(complex):**

**Lemma**

**(long exact sequence of cohomology: snake lemma)**

#### 3.3. Braid Structure of Solution Searching

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Solution searching based on stochastic spatiotemporal dynamics. (

**a**) amoeba-plus-optical-stimulus system; (

**b**) optical excitation transfer dynamics in a multiple quantum-dot system.

**Figure 2.**Solution searching based on photoexcitation transfer. (

**a**) evolutions of populations that differ from each other depending on state fillings induced in the larger quantum dots; (

**b**) example of evolution of the excitation transfer based on the given bounceback rule; (

**c**) correct-solution rate increases with time.

**Figure 3.**Category theoretic picture of solution searching. (

**a**) rudimentary picture of solution searching; (

**b**) hyper-dimensional view of constraint satisfaction problem; (

**c**) introducing product and coproduct for the light emission (P) and bounceback rule (Q); (

**d**) introducing bounceback controller (C) and optical environment (D) as the kernel and cokernel of the morphisms $X\to Q$ and $Q\to Y$, respectively; (

**e**) schematic diagram of chain-wise short exact sequences and exact long sequences of homology; and (

**f**) derivation of four triangulated sequences known in triangulated categories.

**Figure 4.**Short-exact-sequence-based time. (

**a**) schematic diagram of the unit of time based on the satisfaction of short exact sequence; (

**b**) correct-selection rate exhibits poor performances when the integration time of the populations is below approximately 3000 ps; that is, the unit of the short-exact-sequence-based time should be larger than 3000 ps for the particular system under study.

**Figure 5.**(

**a**) octahedral structure of solution searching; (

**b**,

**c**) braid structures of the solution searching.

**Figure 6.**Solution searching as unfolding of knots of the braids. (

**a**) introduction of the consecutive unfolded knots (CUKs) of the braids; (

**b**,

**c**) evolution of the CUK statistics. Longer-length CUKs emerge as time elapses, which is the manifestation of the decreased homology dimension known in triangulated categories.

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**MDPI and ACS Style**

Naruse, M.; Aono, M.; Kim, S.-J.; Saigo, H.; Ojima, I.; Okamura, K.; Hori, H.
Category Theory Approach to Solution Searching Based on Photoexcitation Transfer Dynamics. *Philosophies* **2017**, *2*, 16.
https://doi.org/10.3390/philosophies2030016

**AMA Style**

Naruse M, Aono M, Kim S-J, Saigo H, Ojima I, Okamura K, Hori H.
Category Theory Approach to Solution Searching Based on Photoexcitation Transfer Dynamics. *Philosophies*. 2017; 2(3):16.
https://doi.org/10.3390/philosophies2030016

**Chicago/Turabian Style**

Naruse, Makoto, Masashi Aono, Song-Ju Kim, Hayato Saigo, Izumi Ojima, Kazuya Okamura, and Hirokazu Hori.
2017. "Category Theory Approach to Solution Searching Based on Photoexcitation Transfer Dynamics" *Philosophies* 2, no. 3: 16.
https://doi.org/10.3390/philosophies2030016