# Quantum Strategic Organizational Design: Alignment in Industry 4.0 Complex-Networked Cyber-Physical Lean Management Systems

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## Abstract

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## 1. Introduction

## 2. Background

- Industry 4.0. Industry 4.0 has gained a lot of attention since it was first released [26], claiming the necessity of a new paradigm shift in favour of a less centralized manufacture structure. It is regarded as the fourth industrial revolution, the first three being mechanization through the use of vapor energy, mass production through electricity generation, and ultimately the digital revolution through the integration of electronics and information technology. The industry 4.0 ought to allow for a larger independence of the manufacturing process, since technology is more interrelated and the machines can interact with each other creating a cyber-physical system [27,28,29,30,31,32].
- Cyber-Physical Systems. Cyber-physical system in the context of Industry 4.0 relates to the close bonding and alignment between computing and material resources. A new paradigm of technological systems that is based on embedded collaborative software is impacting the development of such systems [33,34,35].
- Lean Management. Lean management systems in a cyber-physical environment of Industry 4.0 are described as socio-technical structures that are designed to consistently reduce the variability of value creating processes and, therefore, increase their effectiveness and profitability [9,36,37,38,39,40,41,42,43,44,45,46].Variability in this context is understood as any deviation from the desired process state. In quantifiable terms, this work understands the variability of a process, as measured by the systematic reduction of the standard deviation that is associated with the indicators measuring its performance [9,47].
- Complex Networked Organizational Design. According to the network organizational paradigm, modern cyber-physical systems that are oriented to the lean management of Industry 4.0 can be seen as a socio-technical symbiotic ecosystem of human networks [5] interacting with distributed physical sensors interconnected in an increasingly complex network interconnected sensors [48], which readings are modeled as time-dependent signals at the vertices, human, or cyber-physical, respectively. That means that in the structure of the network nodes you can find characteristics that represent them in the form of a certain time series that describes the key performance indicators (KPIs).

- Alignment. This information is typically described by a series of KPIs. Such KPIs are interdependent and they describe certain trajectories in node-related orthonormal bases [2]. These scholars define a node to be in alignment at any given moment in time if the KPI’s trajectory presents asymptotic stability at this point [49]. In other words, the condition for alignment at any given time interval t+$\mathsf{\Delta}t$ is given by Equation (2)$$\forall \mathsf{\Delta}t>0\phantom{\rule{1.em}{0ex}}{D}_{t,t+\mathsf{\Delta}t}<{D}_{t-\mathsf{\Delta}t,t}$$$$\forall \mathsf{\Delta}t>0\phantom{\rule{1.em}{0ex}}{D}_{t,t+\mathsf{\Delta}t}\ge {D}_{t-\mathsf{\Delta}t,t}$$In fact, within this time interval $\mathsf{\Delta}t$, the graph $\mathsf{\Omega}$ that is described in Equation (1) converts into a decision network ${\mathsf{\Omega}}^{\prime}=[{\mathsf{\Gamma}}^{\prime},{E}^{\prime}]$ formed by a set of ${\mathsf{\Gamma}}^{\prime}$ nodes and ${E}^{\prime}$ edges, where ${\mathsf{\Gamma}}^{\prime}=[{\gamma}_{1},{\gamma}_{2},\dots ,{\gamma}_{N}]$ represents the set of all the nodes being part of the network in $\mathsf{\Delta}t$, and the edges are determined by the known probabilistic dependence of alignment occurrence in a node ${\gamma}_{j}$, depending on the alignment occurrence on another ${\gamma}_{i}$. The node ${\gamma}_{i}$ is thus called parent and node ${\gamma}_{j}$ the child. The root nodes are those that do not depend on any other. Subsequently, as described by [50], the joint probability on the nodes can be decomposed into the product of the marginal probabilities that are given by Equation (4):$$P({\gamma}_{1},{\gamma}_{2},\dots ,{\gamma}_{N})=\prod _{i=1}^{N}P\left(\right)open="("\; close=")">{\gamma}_{i}|\prod {\gamma}_{i}$$

- QubitInformation may be represented in many different ways. Quantum computing uses quantum discrete units of information, the qubit (quantum bit) [51]. Qubits represent elementary units of information exchange in quantum computing, similar to the “bits” of classical computing. A bit is always in two basic states, 0 and 1, while a qubit can be in both bases of these states simultaneously. The characteristic is also known as superposition. Quantum computing normally uses the Dirac notation that represents the two bases of computing of these states $|0\rangle $ and $|1\rangle $. The superposition of a $|\mathsf{\Psi}\rangle $ qubit is merely a linear combination of the two basic states $|0\rangle $ and $|1\rangle $, expressed by the Equation (5):$$|\mathsf{\Psi}\rangle ={c}_{1}|0\rangle +{c}_{2}|1\rangle $$$${\left(\right)open="|"\; close="|">{c}_{1}}^{}2=1$$
- Bloch’s sphereBloch’s sphere, as shown in Figure 2A, is commonly used to geometrically represent a qubit [52]. This is a useful and common geometric image of the quantum evolution of a single- or two-level system. On the Bloch sphere, of unitary radius, the Z-axis is the computational axis and its positive direction coincides with the state $|0\rangle $, and the negative with the state $|1\rangle $. A qubit can be represented as a point on the Bloch sphere with the help of two parameters ($\theta $, $\varphi $), as expressed by Equation (7):$$|\mathsf{\Psi}\rangle =cos\left(\right)open="("\; close=")">\frac{\theta}{2}|1\rangle $$When several qubits are utilized, their aggregated state can be determined utilizing the tensorial product of the individual qubits. If the multiple qubit state can be expressed as a linear combination of the $|0\rangle $ and $|1\rangle $ states, then the aggregated state can be represented, as in Equation (8):$$|{\mathsf{\Psi}}_{1}\rangle \otimes |{\mathsf{\Psi}}_{2}\rangle ={c}_{11}{c}_{21}|00\rangle +{c}_{11}{c}_{22}|01\rangle +{c}_{12}{c}_{21}|10\rangle +{c}_{12}{c}_{22}|11\rangle $$The reduced purity ${\kappa}_{j}$ of a qubit ${q}_{j}$ in an $N-qubit$ state $|Psi\rangle $, as given by Equation (9), is a coefficient ${\kappa}_{j}\in [0.5,1]$ that indicates the level of a qubit entanglement in the state [54]. A value of ${\kappa}_{j}=1$ indicates that the qubit is not entangled with the other $N-1$ qubits, and a value of ${\kappa}_{j}=0.5$ indicates that the qubit is maximally entangled with the other qubits in the state.$${\kappa}_{j}=Tr{\left[T{r}_{i\in [0,N-1],i\ne j}\phantom{\rule{0.166667em}{0ex}}|\psi \rangle \langle \psi |\right]}^{2}$$
- Quantum circuitA quantum circuit is a computational sequence that consists of performing a series of coherent quantum operations on qubits. By organizing the qubits into an orderly sequence of quantum gates, measurements, and resets, all of which can be conditioned and use data from the classical calculation in real-time, quantum computing can be simulated. These sequences typically follow a standardized pattern:
- Initialization and reset. First, we begin our quantum calculation with a specified quantum state for each qubit. This is achieved using the initialization operations, typically on the Z-computation axis, and reset. The resets can be done using a single-qubit gate combination that tracks whether we have succeeded in creating the desired state through measurements. Qubit initialization in a desired state $|\mathsf{\Psi}\rangle $ can then continue to apply single-qubit gates.
- Quantum gates. Second, we implement a sequence of quantum gates that manipulate the qubits, as needed by the targeted algorithm following certain quantum circuit design principles.
- Measurement. Third, we measure the qubits. Classical computers translate the measurements of each qubits as classical results (0 and 1) and then store them in either one of the two classical bits. Measurement is understood to be projected into the Z-computational basis unless otherwise stated.

- Quantum gateA quantum gate consists of several mathematical operations applied to the qubits that change the amplitude of their probabilities and, thus, perform the intended computations [54]. The quantum computing basic elements are described in detail:
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- The ${U}_{3}(\theta ,\varphi ,\lambda )$ gate is a single qubit gate that has three parameters $\theta $, $\varphi $ and $\lambda $ which represent a sequence of rotations around the Bloch sphere’s axes such that $[\varphi -\pi /2]$ around the Z axis, $[\pi /2]$ around the X axis, $[\pi -\theta ]$ around the Z axis, $[\pi /2]$ around the X axis, and a $[\lambda -\pi /2]$ around the Z axis. It can be used to obtain any single qubit gate. Equation (10) provides its mathematical representation,$${U}_{3}|\mathsf{\Psi}\rangle =\left[\begin{array}{cc}cos\left(\frac{\theta}{2}\right)& -{e}^{i\lambda}sin\left(\frac{\theta}{2}\right)\\ {e}^{i\varphi}sin\left(\frac{\theta}{2}\right)& {e}^{i(\varphi +\lambda )}cos\left(\frac{\theta}{2}\right)\end{array}\right]|\mathsf{\Psi}\rangle $$
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- The $CNOT$ or conditional NOT gate is a two qubit computation gate with one qubit acting as control $|{\mathsf{\Psi}}_{1}\rangle $ and the other as target$|{\mathsf{\Psi}}_{2}\rangle $. The $CNOT$ gate performs a selective negation of the target qubit. If the control qubit is in superposition, then $CNOT$ creates entanglement. Equation (12) provides its mathematical representation,$$CNOT|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{2}\rangle =|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{1}\oplus \mathsf{\Psi}2\rangle $$
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- The $ccX$ or Toffoli gate is a three qubit computation gate with two qubits $|{\mathsf{\Psi}}_{1}\rangle $ and $|{\mathsf{\Psi}}_{2}\rangle $ acting as controls and one qubit $|{\mathsf{\Psi}}_{3}\rangle $ acting as target. The $ccX$ gate applies an X to the target qubit $|{\mathsf{\Psi}}_{3}\rangle $ only when both controls $|{\mathsf{\Psi}}_{1}\rangle $ and $|{\mathsf{\Psi}}_{2}\rangle $ qubits are in state $|1\rangle $. Equation (14) provides its mathematical represetnation:$$ccX|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{2}\rangle |{\mathsf{\Psi}}_{3}\rangle =|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{1}\oplus \mathsf{\Psi}2\rangle |{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{1}\oplus \mathsf{\Psi}3\rangle $$
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- The Z-measurement of a quantum state—a self-adjoint operator on the Hilbert space—results in the measured object being in an eigenstate of the Z operator or computational basis, with the corresponding eigenvalue being the value measured. The measurement, also called observation, of a quantum state, is a stochastic non-reversible operation and, therefore, cannot be considered as a quantum gate, as it allocates a unique value to the variable observed. In mathematical terms, the probability p of a measurement result m occurring when the state $|\mathsf{\Psi}\rangle $ is measured is given by Equation (16):$$p\left(m\right)=\langle \mathsf{\Psi}|{M}_{m}^{\u2020}{M}_{m}|\mathsf{\Psi}\rangle $$$$|{\mathsf{\Psi}}^{\prime}\rangle =\frac{{M}_{m}|\mathsf{\Psi}\rangle}{\sqrt{p\left(m\right)}}$$Equation (18) shows its quantum circuit equivalent as a symbolic box:

## 3. Quantum Strategic Organizational Design

#### 3.1. QSOD Qubit

#### 3.2. QSOD Design Principles

- Calculation of conditional probabilitiesThe conditional probabilities that will give rise to the quantum circuit are derived from a preliminary analysis of the KPIs that are associated with each node of the complex network in question. This analysis, as indicated above, is based on the method based on genetic algorithms presented in [2]. Specifically, for each node, there are typically three related KPIs. The selected chromosome has subsequently 12 real numbers between 0 and 1, and we have used real value crossover and mutation with probabilities of 60% and 7%, respectively. The population was built over 8000 individuals and it ran over 1000 generations. Once the trajectories associated with each node have been calculated, by applying the Bayes theorem, it is trivial to calculate the relative probability of alignment or non-alignment at each node concerning those to which it is connected.
- Initialization and resetThe initialization and reset of the qubits is typically standardized to the state $|0\rangle $ on the computational Z-axis.
- Rotation angle computationThe conditional probabilities translate into qubit rotation angles depending on its decision network dependencies:
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- For a root node with no parents, the possible states are two $|0\rangle $ and $|1\rangle $. A trivial application of Equation (7), states that a qubit intialized to state $|0\rangle $ and rotated by a gate $U3(\theta ,0,0)$, being $\varphi =0$, transforms it into $|\mathsf{\Psi}\rangle =cos\left(\right)open="("\; close=")">\frac{\theta}{2}|1\rangle $. Therefore, taking Equation (5) into account and the definition of the Bloch sphere’s angles, the rotation angle $\theta $ that is required to calculate the probabilities of being in state $|0\rangle $ and $|1\rangle $ can be expressed by Equation (19):$$\theta =2atan\left(\right)open="["\; close="]">tan\left(\right)open="("\; close=")">\frac{\theta}{2}=2atan\sqrt{\frac{si{n}^{2}\left(\right)open="("\; close=")">\frac{\theta}{2}}{}co{s}^{2}\left(\right)open="("\; close=")">\frac{\theta}{2}}$$
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- In general, for a child node ${\gamma}_{i}$ with m parents, there are ${2}^{m}$ possible states $\prod {\gamma}_{i}^{*}$. Subsequently, taking Equations (4), (5), and (7) into account, as well as the definition of the Bloch sphere’s angles, the rotation angle is given by Equation (20):$${\theta}_{{\gamma}_{i},\prod {\gamma}_{i}^{*}}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(4\right)-Equation\phantom{\rule{3.33333pt}{0ex}}\left(5\right)}{=}2atan\sqrt{\frac{p\left(|1\rangle \right|\prod {\gamma}_{i}=\prod {\gamma}_{i}^{*})}{p\left(|0\rangle \right|\prod {\gamma}_{i}=\prod {\gamma}_{i}^{*})}}$$

- Controlled rotationsControlled rotations are not elementary quantum gates and they need to be deconstructed into elementary operations. As described by Nielsen and Chuang [54], being m the maximum number of parent nodes a child has, the controlled rotation expressing the conditional probabilities needs of the addition of ${a}_{i}$ “dummy” qubits $i=1,\dots ,m-1$ in order to decompose the controlled rotation into $2(m-1)$$CNOT$ gates and one $U3(\theta ,0,0)$. This is exemplified in Equation (21) for $m=5$ qubits, ${a}_{i}$ “dummy” qubits $i=1,\dots ,(m-1=4)$ and a total of $2(m-1)=8$$CNOT$ gates.As first shown in [55], it is important to highlight that $U3(\theta ,0,0)$ is best decomposed as by Equation (22):$$\begin{array}{c}\hfill U3(\theta ,0,0)|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{2}\rangle =U3\left(\right)open="("\; close=")">\frac{\theta}{2},0,0|{\mathsf{\Psi}}_{2}\rangle CNOT|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{2}\rangle U3\left(\right)open="("\; close=")">\frac{-\theta}{2},0,0& |{\mathsf{\Psi}}_{2}\rangle CNOT|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{2}\rangle \end{array}$$Additionally, as a direct application of Equation (13), this equation converts into Equation (23):$$\begin{array}{c}\hfill U3(\theta ,0,0)|{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{2}\rangle \stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(13\right)}{=}U3\left(\right)open="("\; close=")">\frac{\theta}{2},0,0|{\mathsf{\Psi}}_{2}\rangle |{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{1}\oplus \mathsf{\Psi}2\rangle U3\left(\right)open="("\; close=")">\frac{-\theta}{2},0,0& |{\mathsf{\Psi}}_{2}\rangle |{\mathsf{\Psi}}_{1}\rangle |{\mathsf{\Psi}}_{1}\oplus \mathsf{\Psi}2\rangle \end{array}$$This method works, because, if the control qubit $|{\mathsf{\Psi}}_{1}\rangle $ is in state $|0\rangle $, all we have is $U3\left(\right)open="("\; close=")">\frac{\theta}{2},0,0$ followed by a $U3\left(\right)open="("\; close=")">\frac{-\theta}{2},0,0$ and the effect is trivial. If the control qubit $|{\mathsf{\Psi}}_{1}\rangle $ is in state $|1\rangle $, the net effect is a controlled rotation $U3(\theta ,0,0)$ on the $|{\mathsf{\Psi}}_{2}\rangle $ qubit.
- Measurement of the QSOD qubits to obtain the probabilities of alignment states.The measurements are mainly used in the end to extract computational results from the quantum states. This will allow for us to explore the quantum states of the qubits and make an interpretation that allows for improving the management system that is related to the industrial process.

## 4. Case Study. Qsod Circuit

- Section 4.1. Scope establishment.
- Section 4.2. Specification of population and sampling.
- Section 4.3. QSOD circuit design.
- Section 4.4. Analysis of results.

#### 4.1. Scope Establishment

#### 4.2. Specification of Population and Sampling

#### 4.3. QSOD Circuit Design

- Calculation of conditional probabilities.For each node, there are typically three related KPIs. The selected chromosome has subsequently 12 real numbers between 0 and 1, and we have used real value crossover and mutation with probabilities of 60% and 7%, respectively. The population was built over 8000 individuals and it ran over 1000 generations. Once the trajectories associated with each node have been calculated, it is trivial to calculate the relative probability of alignment or non-alignment at each node concerning those to which it is connected. This process is executed for each node in parallel without the loss of performance.
- Initialization and reset.In Step 0, each one of the nodes is assigned to a qubit, ${q}_{A}$, ${q}_{B}$, ${q}_{C}$, ${q}_{D}$, and a “dummy” node ${q}^{*}$ is created. The qubits are initialized and reset to $|0\rangle $ state. This allows for a controlled comparison of the probabilities through qubit rotations.
- Rotation angle computation.Following Equation (19) applied to each root qubit, we obtain the following results given by Equation (25):$$\begin{array}{cc}\hfill {\theta}_{A}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(19\right)}{=}2arctan\sqrt{\frac{0.25327658}{0.74672341}}=1.16479& {\theta}_{C}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(19\right)}{=}2arctan\sqrt{\frac{0.27}{0.73}}=1.0928\\ \end{array}$$Following Equation (20) applied to each child qubit, we obtain following results that are given by Equation (26):$$\begin{array}{cc}\hfill {\theta}_{B,|0\rangle}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(20\right)}{=}2arctan\sqrt{\frac{0.35}{0.65}}=1.2661& {\theta}_{B,|1\rangle}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(20\right)}{=}2arctan\sqrt{\frac{0.57}{0.43}}=1.7113\\ \hfill {\theta}_{D,|00\rangle}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(20\right)}{=}2arctan\sqrt{\frac{0.39}{0.61}}=1.3489& {\theta}_{D,|01\rangle}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(20\right)}{=}2arctan\sqrt{\frac{0.83}{0.17}}=2.29161\\ \hfill {\theta}_{D,|10\rangle}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(20\right)}{=}2arctan\sqrt{\frac{0.63}{0.37}}=1.83382& {\theta}_{D,|11\rangle}\stackrel{Equation\phantom{\rule{3.33333pt}{0ex}}\left(20\right)}{=}2arctan\sqrt{\frac{0.18}{0.82}}=0.87629\\ \end{array}$$
- Controlled rotations.By systematically applying Equation (21) to each qubit, we obtain the QSOD circuit that is shown in Figure 4.Figure 5A visualizes the quantum state of each step of the final state. The Bloch sphere provides a global view of a multi-qubit quantum state in the computational basis. Node size is proportional to state probabilities, and color reflects the phase of each basis state as shown in Figure 5B. The constant color blue that reflects the phase of each basis state does not change throughout the circuit. For the interested reader, in Appendix A, the Bloch sphere states of each step in the circuit are displayed in Figure A1, Figure A2, Figure A3 and Figure A4.The 37 steps that make up the QSOD have been divided into six distinct phases in order to improve the clarity of the explanation. We will now comment on the logic behind each of them:
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- Phase 0. Step 0.Phase 0 performs the initialization and reset, as previously explained in Section 4.3.
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- Phase 1. Step 1–Step 11.This phase is subdivided in three conceptual parts:First. This phase starts in Step 1 by rotating ${q}_{A}$ and ${q}_{C}$ by ${\theta}_{A}=1.16479$ and ${\theta}_{C}=1.0928$ radians respectively as calculated in Equation (25). This is because both ${q}_{A}$ and ${q}_{C}$ are root qubits and Equation (19) applies.Second, two controlled rotations on qubit ${q}_{B}$ in Steps 1–3 are performed, as calculated in Equation (26). As described in Equation (24), since qubit ${q}_{B}$ has a parent qubit${q}_{A}$, we need to perform two controlled rotations properly aligned by CNOT gates. Depending on the state of qubit ${q}_{A}$: $\frac{{\theta}_{B,|1\rangle}}{2}=0.85592$ in Step 1 and $\frac{-{\theta}_{B,|1\rangle}}{2}=-0.85592$ in Step 3, in the case that ${q}_{A}$ is in state $|1\rangle $, and $\frac{{\theta}_{B,|0\rangle}}{2}=0.63305$ in Step 5 and $\frac{-{\theta}_{B,|0\rangle}}{2}=-0.63305$ in the case that ${q}_{A}$ is in state $|0\rangle $. In this case, a ${U}_{3}(\pi ,-\frac{\pi}{2},\frac{\pi}{2})$ is performed, so as to generate proper alignment.Third, on qubit ${q}_{D}$, we need to perform a total of four controlled rotations throughout the circuit. This is because each of its parent qubits ${q}_{B}$ and ${q}_{C}$ can have two states, and we need to represent the rotations corresponding to the states $|00\rangle $, $|10\rangle $, $|01\rangle $, and $|11\rangle $. A controlled rotation ${\theta}_{D,|11\rangle}$ is performed on qubit ${q}_{D}$ conditioned by the state $|11\rangle $ of its parent qubits ${q}_{B}$ and ${q}_{C}$, as calculated in Equation (26). This is done with a controlled rotation $\frac{{\theta}_{D,|11\rangle}}{2}=0.43814$ in Step 1 and a controlled rotation $\frac{-{\theta}_{D,|11\rangle}}{2}=-0.43814$ in Step 11, properly aligned through CNOT and ccX gates, as described in Equation (21) in Steps 9–10 and Steps 31–33.
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- Phase 2. Step 12–Step 17.In phase 2, we perform the second controlled rotation on qubit ${q}_{D}$ as related to the states $|00\rangle $ of qubit ${q}_{B}$ and ${q}_{C}$. A controlled rotation ${\theta}_{D,|00\rangle}$ is performed on qubit ${q}_{D}$ conditioned by the state $|00\rangle $ of its parent qubits ${q}_{B}$ and ${q}_{C}$, as calculated in Equation (26). This is done with a controlled rotation $\frac{{\theta}_{D,|00\rangle}}{2}=0.67449$ in Step 13 and a controlled rotation $\frac{-{\theta}_{D,|00\rangle}}{2}=-0.67449$ in Step 17, properly aligned through CNOT and ccX gates, as described in Equation (21) in Steps 12 and Steps 14–16.
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- Phase 3. Step 18–Step 24.In phase 3, we perform the second controlled rotation on qubit ${q}_{D}$ as related to the states $|10\rangle $ of qubit ${q}_{B}$ and ${q}_{C}$. A controlled rotation ${\theta}_{D,|10\rangle}$ is performed on qubit ${q}_{D}$ conditioned by the state $|10\rangle $ of its parent qubits ${q}_{B}$ and ${q}_{C}$, as calculated in Equation (26). This is done with a controlled rotation $\frac{{\theta}_{D,|10\rangle}}{2}=0.91690$ in Step 19 and a controlled rotation $\frac{-{\theta}_{D,|10\rangle}}{2}=-0.91690$ in Step 24, properly aligned through CNOT and ccX gates, as described in Equation (21) in Steps 18 and Steps 20–23.
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- Phase 4. Step 25–Step 30.In phase 4, we perform the second controlled rotation on qubit ${q}_{D}$ as related to the states $|01\rangle $ of qubit ${q}_{B}$ and ${q}_{C}$. A controlled rotation ${\theta}_{D,|10\rangle}$ is performed on qubit ${q}_{D}$ conditioned by the state $|01\rangle $ of its parent qubits ${q}_{B}$ and ${q}_{C}$, as calculated in Equation (26). This is done with a controlled rotation $\frac{{\theta}_{D,|01\rangle}}{2}=1.1458$ in Step 26 and a controlled rotation $\frac{-{\theta}_{D,|01\rangle}}{2}=-1.1458$ in Step 30, properly aligned through CNOT and ccX gates, as described in Equation (21) in Steps 25 and Steps 27–29.
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- Phase 5. Step 31–Step 33.As mentioned earlier, in phase 5 the controlled rotation of qubit ${q}_{D}$ as related to the states $|00\rangle $ of qubit ${q}_{B}$ and ${q}_{C}$ started in Phase 1, qubits 1 and 9–11, is completed.
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- Phase 5. Step 34–Step 37.Finally, in phase 6 each one of the qubits is measured, as expressed by Equation (17).

#### 4.4. Analysis of Results

## 5. Conclusions, Further Research and Limitations

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

QSOD | Quantum Strategic Organizational Design |

KPI | Key Performance Indicator |

DLT | Distributed Ledger Technology |

## Appendix A

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**Figure 1.**Hoshin Kanri Forest Structure [11].

Qubits | Probability of Alignment ${\mathit{P}}_{\mathit{j}}\left(|0\rangle \right)$ | Reduced Purity ${\mathit{\kappa}}_{\mathit{j}}$ |
---|---|---|

${q}_{D}$ | 49.135% | 0.9178 |

${q}_{B}$ | 58.335% | 0.9026 |

${q}_{C}$ | 73.000% | 0.9178 |

${q}_{A}$ | 69.745% | 0.9794 |

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Villalba-Diez, J.; Zheng, X.
Quantum Strategic Organizational Design: Alignment in Industry 4.0 Complex-Networked Cyber-Physical Lean Management Systems. *Sensors* **2020**, *20*, 5856.
https://doi.org/10.3390/s20205856

**AMA Style**

Villalba-Diez J, Zheng X.
Quantum Strategic Organizational Design: Alignment in Industry 4.0 Complex-Networked Cyber-Physical Lean Management Systems. *Sensors*. 2020; 20(20):5856.
https://doi.org/10.3390/s20205856

**Chicago/Turabian Style**

Villalba-Diez, Javier, and Xiaochen Zheng.
2020. "Quantum Strategic Organizational Design: Alignment in Industry 4.0 Complex-Networked Cyber-Physical Lean Management Systems" *Sensors* 20, no. 20: 5856.
https://doi.org/10.3390/s20205856