Optimizing enforcement policies in complex socio-technical systems in Indonesia: a quantum neural network (QNN) simulation of corrupt decision-making

2.1. Qubit Representation of the cognitive state

In the classical Becker framework, an individual faces a binary choice between committing an offense or remaining in the legal sector, typically modeled via deterministic expected utility or classical logit/probit probability distributions. However, classical stochastic models operate under the assumption of epistemic uncertainty, assuming the agent possesses a definite, hidden preference that is merely unknown to the observer. In contrast, the Quantum Becker Model (QBM) conceptualizes corrupt decision-making as an intrinsic state of cognitive superposition, where conflicting psychological motivations (e.g., economic gain vs. moral risk) coexist simultaneously prior to the decision event. This decision state is represented as a complex state vector $\left|\phi \right.〉$ in a two-dimensional Hilbert space ${\mathcal{H}}^2$ spanned by the orthonormal computational basis states $\left|0\right.〉$ and $\left|1\right.〉$:

where $\alpha$ and $\beta$ are complex probability amplitudes ($\alpha ,\beta \in \mathcal{C})$ satisfying the normalization condition:

The term ${\left|\beta \right|}^2$ gives the probability of observing the criminal action upon measurement, directly corresponding to the offense supply function in classical economic theory.

2.2. Feature mapping via parameterizable quantum rotation gates

To encode the classical Beckerian variables into the quantum domain, the QBM employs quantum feature mapping through a parameterized rotation gate. Let $Y$ denote illegal gain, $p$ the probability of apprehension, and $f$ the severity of punishment. The mental rotation angle $\theta$ is defined as a linear function of these inputs:

where $w=[w_1,w_2,w_3{]}^T$ represents the cognitive weights (variational parameters of the QNN) that capture the individual’s risk preferences and sensitivity to enforcement $x=[Y,p,f{]}^T$ and $\varphi$ is the internal moral bias term. The quantum state is evolved from the ground state by applying the rotation operator:

The probability of getting a qubit $\left|1\right.〉$ (illegal/criminal choice) from the output is as follows:

This implies that the cumulative quantum criminal action probability function resulting from an independent individual's conscious decision is generally derived via Born’s Rule as follows:

This formulation yields the individual supply of offense probability function $O_j$ for agent $j$ in a given socio-legal scenario. The cognitive weight vector $w$ scales the subjective elasticities to illegal rewards, apprehension risks, and punishment severity, while the moral bias $\varphi$ accounts for intrinsic ethical resistance. These parameters were numerically mapped, simulated, and optimized across diverse socio-legal profiles using custom hybrid quantum-classical algorithms implemented in MATLAB^®. The model thereby emphasizes that criminal decisions emerge from internal cognitive structures rather than purely automatic responses to external incentives. Modeling human criminality within a complex Hilbert space offers distinct predictive and theoretical advantages over classical stochastic frameworks: 1) Representation of Cognitive Superposition: Rather than assuming a fixed preference, the Hilbert space representation allows the offender’s mind to reside in a continuous, fluid superposition state. Complex probability amplitudes account for internal psychological friction, hesitation, and dynamic cognitive interference prior to action; 2) Capturing Non-linear Phase Transitions: While classical logistic functions model smooth, gradual probability shifts, Born's rule projection introduces periodic non-linearities. This naturally accounts for abrupt behavioral collapses – such as sudden transitions into widespread corruption when enforcement oversight drops below a critical institutional threshold; and 3) Context Sensitivity and Quantum Interference: Complex amplitudes allow the model to incorporate phase shifts $\left(\varphi \right)$, capturing how internal ethical biases and situational context interfere constructively or destructively with external enforcement incentives.

2.3. Formulation of the quantum social loss function

To bridge microscopic quantum cognitive collapses with macroscopic state budgeting, the government is modeled as a centralized optimizing entity seeking to minimize the global expectation value (⟨${\widehat{H}}_{Loss}$⟩) of a social loss Hamiltonian operator (${\widehat{H}}_{Loss}$). While classical economics relies on scalar cost equations, this quantum economic framework translates these costs into an energy matrix where the eigenvalues represent the structural strain or energy of institutional and policy failure. The aggregate population comprising $N$ potential offenders is treated as a multi-qubit grand ensemble with a joint wave function $\left|\phi \right.⟩=\left|{\phi }_1⟩\right.\otimes \cdots \otimes \left|{\phi }_N⟩\right.$. The total Hamiltonian operator is constructed as the summation of individual penalty operators plus the state’s operational enforcement costs $C\left(p,O_j\right)$. The public policy problem is modeled actively as determining enforcement parameters $[p;f$] to minimize the quantum expectation value (⟨${\widehat{H}}_{Loss}$⟩) of a social loss Hamiltonian operator (${\widehat{H}}_{Loss}$).

Step 1: Construction of the Social Loss Hamiltonian.

To scale this framework to a macroscopic system, we treat the overall society containing a population of $N$ potential offenders as a grand ensemble. While localized criminal networks are grouped into entangled blocks governed by density matrices ${\rho }_{AB}=\left|{\phi }_{AB}⟩⟨{\phi }_{AB}|\right.$, the aggregate population is represented by a multi-qubit joint wave function $\left|\phi \right.⟩=\left|{\phi }_1⟩\right.\otimes \cdots \otimes \left|{\phi }_N⟩\right.$. The total Hamiltonian operator is designed as the summation of individual penalty operators plus the operational enforcement costs borne by the state, $C\left(p,O_j\right)$:

where ${\widehat{H}}_{Loss}$ represents the Total Social Loss Hamiltonian Operator of the state system across $N$ distinct socio-legal environments or cases, $\widehat{I}$ represents the identity matrix, and ${\widehat{H}}_j$ integrates real socio-economic damage factors mapped exclusively onto the criminal state via the quantum projection operator $\left|1\right.〉〈\left.1\right|=\left(\begin{array}{c}0\\ 1\end{array}\right)\left(\begin{array}{cc}0& 1\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right).$ The Total Social Loss Hamiltonian Operator, denoted as ${\widehat{H}}_{Loss}$, is the core mathematical operator used to quantify, evaluate, and minimize the total fiscal and behavioral costs inflicted on society by crime and enforcement operations. In classical economics, a policymaker uses a standard scalar cost equation. In a quantum economic framework, this cost is translated into a Hamiltonian matrix ($\widehat{H}$). In physics, a Hamiltonian represents the total energy of a physical system; in quantum economics, it represents the total socioeconomic strain or “loss energy” of the state. Energy of Policy Failure: The eigenvalues of this operator represent the possible levels of social loss. A higher expectation value ⟨${\widehat{H}}_{Loss}$⟩ means the current legal policy ($p,f$) is failing, causing high crime or excessive enforcement costs. ${\widehat{H}}_{Loss}$ acts as the global objective function. The state’s goal during optimization is to find the exact combination of apprehension probability ($p$) and punishment severity ($f$) that minimizes this operator. ${\widehat{H}}_{Loss}$ mathematically fuses quantum behavioral variables (the wave function of criminal intent, embedded in ${\widehat{H}}_j$) with classical economic constraints (the tangible cash spent on hiring police officers, represented by the operational enforcement cost $C\left(p,O_j\right)$):

where, $D$ represents the estimated net damage caused by a criminal offense, $f$ is the severity of the punishment, $p$ is the probability of apprehension, and $b$ defines the punishment friction coefficient ($b=0$ if the punishment is a financial fine, representing a pure wealth transfer without deadweight loss; $b>0$ if the punishment is imprisonment, which triggers budgetary and custodial infrastructure costs for the state). The Localized Crime and Punishment Friction Operator, ${\widehat{H}}_j=\left(D+b.p.f\right)\left|1\right.〉{〈\left.1\right|}_j$, isolates the socioeconomic costs incurred within a specific scenario $j$. The projection operator $\left|1\right.〉{〈\left.1\right|}_j$ acts as a computational switch, penalizing the system only when an agent collapses into the criminal execution state $\left|1\right.〉$. The scalar coefficient combines direct societal damage ($D$) with institutional punishment friction ($b.p.f$). When applied to the agent's wave function, this operator extracts the real-world expected loss via Born's Rule, formalizing the fiscal trade-offs of incarceration infrastructure ($b>0$). In quantum mechanics, a wave function describes all possible physical positions of a particle. However, within the context of Quantum Becker Model (QBM), the agent’s wave function (∣${\phi }_j$⟩) represents the mathematical formalization of an offender's internal psychological state or mental landscape before a final choice is made. Instead of assuming a static, deterministic mind, the wave function captures the fluid human experience of cognitive superposition, the state of mental ambivalence where an individual simultaneously weighs obeying the law versus violating it. Here is the precise breakdown of its structural meaning in your model: 1) The Structure of Mental Superposition: Before a definitive action is executed, the agent's mind does not sit at a hard 0 or 1. It exists as a linear combination of both possibilities within a Hilbert space: $\left|\phi \right.〉=\alpha \left|0\right.〉+\beta \left|1\right.〉$. $\left|0\right.〉$ (The Compliance State): The cognitive baseline of law-abiding behavior. $\left|1\right.〉$ (The Offense State): The cognitive state of executing the crime. $\alpha$ and $\beta$ (Probability Amplitudes): Complex numbers representing the psychological weights pulling the agent toward compliance or defiance, respectively. 2) Dynamic Mental Rotation: The agent’s wave function is highly dynamic. When policymakers adjust external variables, such as scaling penal severity ($f$) or increasing apprehension probability ($p$), these socio-legal shifts act as external forces that actively rotate the wave function by a specific mental angle (${\theta }_j$). Aggressive Enforcement: Rotates the wave function toward the legal compliance vector ($0〉$). High Criminal Payoffs ($Y$) and Degraded Moral Bias (${\varphi }_j$): Rotates the wave function toward the criminal vector (∣1⟩). 3) Wave Function Collapse into Action: When the agent encounters a real-world opportunity to commit an offense, this fluid mental superposition undergoes wave function collapse due to interaction with the environment. The psychological uncertainty instantly collapses into a definitive, observable macro-behavior: either absolute compliance ($\left|0\right.〉$) or active violation (∣1⟩). The statistical likelihood of this collapse occurring is extracted via Born's Rule $\left(O_j\left(w, {\varphi }_j\right)= {\left|\beta \right|}^2=\mathrm{si}{\mathrm{n}}^2\left(\frac{{\theta }_j}{2}\right)\right)$ , which forms the core of our predictive framework.

Step 2: Evaluation of the Quantum Expectation Value.

In quantum economic modeling, Eq. (15) represents the Global Social Loss Expectation Value. It defines how the state calculates its real-world economic and social costs by measuring the “energy” of policy failure embedded within the system's quantum psychological states. While the Hamiltonian ${\widehat{H}}_{Loss}$ is a matrix operator representing potential loss states, this bra-ket projection $\left(⟨\left.\phi \right|\dots \left|\phi ⟩\right.\right)$ collapses those quantum operators into a single, real-world scalar cost (as in currency denomination or utility) that policymakers can actively minimize. The social objective function $L\left(p,f\right)$ is evaluated through standard bra-ket projection:

Due to the orthogonality and normalization properties of quantum wave functions $\left(〈\phi \left|\widehat{I}\right|\phi 〉=1\right)$, the composite multidimensional matrix multiplication simplifies elegantly into single-qubit expectations for each individual $j$:

Step 3: Expansion of the Dirac Inner Product via Born’s Rule.

Referring back to Eq. (4), the inner product bracket is expanded as a projection of state amplitudes:

Taking the absolute square provides the exact probability of the circuit collapsing into a criminal action:

Substituting Eq. (18) into Eq. (16) yields the complete parameterized QNN social loss equation:

Victim Damage Costs represent the actual harm caused by the crime. One obvious instance is when criminals steal money from bank accounts using phishing links, forcing the bank to spend money to reverse the transactions and address the security attack. Punishment Friction Costs are the costs borne by taxpayers to punish imprisoned criminals. A concrete instance is the daily cost of housing an offender in a high-security prison, which includes jail guards, prisoner food, and inmate medical care. Policing Budget Costs are the funds that the government spends directly to arrest criminals and prevent crime. A clear example is purchasing specialized tracking software licenses and funding digital forensics police units. Eq. (14) provides a rigorous bridge between domains: 1) Microscopic Level: Individual decision-making is governed by quantum cognitive rotations (${\theta }_j$) and wave function collapses $\left(\mathrm{si}{\mathrm{n}}^2\left(\frac{{\theta }_j}{2}\right)\right)$; and 2) Macroscopic Level: The state evaluates these expected collapses across the population ensemble, directly summing the statistical victim damage and penal friction costs alongside strictly classical, tangible state budget expenditures (e.g., quadratic policing costs $C\left(p,O_j\right)=1{0p}^2$).

3.1. Modeling optimal anti-corruption strategies via quantum neural networks: Indonesian context

In this study, the Quantum Becker Model (QBM) is initialized with empirical baseline parameters explicitly tied to these Indonesian datasets. The model mapped classical economic metrics into a parameterized quantum landscape using the following baseline values: illegal gain $Y=$ 15 (representing substantial potential rents from corrupt transactions), probability of apprehension $p=$0.10 (directly mirroring Indonesia’s Corruption Perceptions Index score of 34/100, indicating low detection risk), and punishment severity $f=$ 5.0 (aligned with an average custodial sentence of approximately 39 months derived from Indonesia Corruption Watch records [10]). The moral bias parameter $\varphi =$ 0.05 captured limited intrinsic ethical resistance, consistent with societal permissiveness quantified by the 2024 Anti-Corruption Behavior Index (IPAK = 3.85). The Anti-Corruption Behavior Index (IPAK) for Indonesia in 2024 stood at 3.85 on a scale of 0 to 5, reflecting a slight decline from 3.92 recorded in 2023. Higher IPAK values indicate stronger anti-corruption norms and behaviors within society, whereas lower values signal increasing permissiveness toward corrupt practices [8]. The index comprises two core dimensions: Perception and Experience. The Perception Index decreased from 3.82 in 2023 to 3.76 in 2024, while the Experience Index fell from 3.96 to 3.89 over the same period. These trends suggest a modest erosion in both public perception of and direct encounters with corruption. Urban residents exhibited a marginally higher IPAK (3.86) compared to their rural counterparts (3.83). A clear positive association also emerged between educational attainment and anti-corruption behavior: individuals with education below senior high school recorded an IPAK of 3.81, those with senior high school education scored 3.87, and those with higher education achieved 3.97. These patterns underscore the persistent challenge of cultivating robust ethical standards across diverse socioeconomic contexts, while highlighting education’s potential role in strengthening societal resistance to corruption [8]. Complementing these behavioral indicators, Indonesia Corruption Watch documented systemic enforcement weaknesses. Analysis of 1,768 court decisions involving 1,871 defendants revealed an average prison punishment of only 3 years and 3 months and an average fine of Rp180 million. State financial losses totaled Rp330.93 trillion. In this study, the Quantum Becker Model (QBM) is initialized with empirically based parameters obtained from corruption data in Indonesia. The model mapped classical economic metrics into a parameterized quantum landscape using the following baseline values: illegal gain $Y=$15.0 representing substantial potential rents from corrupt transactions), probability of apprehension $p=$0.10 (consistent with Indonesia’s Corruption Perceptions Index score of 34/100, indicating low detection risk), and punishment severity $f=$5.0 (aligned with an average custodial sentence of approximately 39 months). The cognitive weight vector $w=$[$w_1$;$w_2$;$w_3$] $=$[0.25;0.65;0.15] reflected localized risk preferences, with the highest weight assigned to apprehension avoidance. The moral bias parameter $\varphi =$ 0.05 captured limited intrinsic ethical resistance, consistent with societal permissiveness toward petty corruption (IPAK score of 3.85 on a 0-5 scale). Socio-economic net damage per offense was set at 20.0, scaled to reflect the aggregate state financial losses (Rp56 trillion annually from investigated cases) [10]. Substitution of these parameters into the feature mapping function produced a mental rotation angle $\theta =$ 2.9900 radians. Application of the $R_y\left({\theta }_j\right)$ gate evolved the initial state into $\left|\phi \right.⟩=0.0757\left|0\right.⟩+0.9971\left|1\right.⟩$. Application of Born’s rule produced a criminal action probability of 99.43 %, reflecting a near-complete collapse of the cognitive state into the corrupt basis. Evaluation of the macroscopic social loss, which incorporated a quadratic enforcement cost function, revealed notable differences across policy regimes. Under a financial penalty system, the expected Total Social Loss was 19.9853. When institutional frictions associated with imprisonment were factored in, this value increased to 20.2339. In the baseline scenario, the total social loss stood at 20.1221, highlighting the substantial societal burden resulting from inadequate deterrence. These quantum-derived findings closely align with Indonesia’s ongoing enforcement challenges. High potential gains from corruption, persistently low detection probabilities, and relatively moderate sanctions continue to sustain widespread corrupt practices, even in the presence of established anti-corruption institutions. The results emphasize the profound human and societal costs of systemic weaknesses in accountability and justice. The results demonstrate that marginal increases in punishment severity produce limited behavioral shifts when apprehension probability remains depressed. Fig. 1 shows the non-linear response surface of criminal behavior mapped against varying probabilities of apprehension and punishment severities, with behavioral hyperparameters held constant to the case conditions. The surface is bounded dynamically between 0 % and 100 % via Born’s rule, highlighting the distinctive non-linear wave-like phase transitions where increases in legal penalties decouple from traditional deterrence outcomes. Fig. 2 illustrates the direct, steady gradient shift of the agent’s psychological decision state prior to passing through the quantum network’s periodic activation operator, serving as the cognitive foundation for the case analysis.

Fig. 13D criminal action probability model

Fig. 23D cognitive mental rotation angle (θ)

Fig. 3 illustrates the 3D response surface topology of the Global Social Loss Function, denoted as $L(p,f)$, plotted across the complete operational policy domain of the probability of apprehension ($p\in$[0, 1.0]) and punishment severity ($f\in$ [0, 10.0]). The geometric morphology of this continuous non-linear field exposes the critical trade-offs faced by policymakers when distributing public resources between immediate enforcement visibility and subsequent statutory penalty frameworks.

Fig. 33D surface plot of global social loss function

Overall, the findings emphasize that effective anti-corruption strategies must prioritize substantial improvements in detection probability through institutional strengthening and technological monitoring to meaningfully alter cognitive decision landscapes and reduce aggregate social losses.

3.2. Boundary value and topological analysis

A systematic evaluation of the four boundary coordinate vectors of the policy domain matrix across Figure 1 (The Behavioral Response Surface, $O$) and Fig. 3 (The Global Social Loss Field, $L$) reveals the structural mechanics governing the Quantum Becker Model (QBM). Deconstructing these asymptotic limits reveals the non-linear coupling between individual cognitive states, active state expenditure, and aggregate social cost.

– Boundary Coordinate Vector ($p=$ 0, $f=$ 0): The Anarchic Negligence State.

Behavioral reality in Figure 1 shows criminal probability reaches its absolute maximum peak ($O\approx$ 99). Because there is no fear of legal consequences, individual quantum cognitive vectors rotate completely into compliance defiance. Fiscal Dynamics and Social Loss in Fig. 3 shows Total Social Loss rests at a high baseline ($L\approx$ 20.0). Under these coordinates, the state spends absolutely nothing on public safety. The quadratic policing budget cost ($10p^2=0$) is completely empty, and because no one is caught, there are no prisoners to feed or guard, reducing incarceration friction to zero $\left(b.p.f\mathrm{si}{\mathrm{n}}^2\left(\frac{{\theta }_j}{2}\right)=0\right)$. However, Social Loss does not equal the Government Budget. Total social loss is the sum of government spending plus the direct damages felt by citizens $D\mathrm{si}{\mathrm{n}}^2\left(\frac{{\theta }_j}{2}\right)$. Therefore, even though the government's operational bill is zero, society as a whole suffers a severe loss of 20.0 because criminals operate with total freedom, inflicting unmitigated financial and physical damage on the population.

– Boundary Coordinate Vector ($p=$ 0, $f=$10): The Empty Threat Paradox.

Behavioral Reality in Fig. 1 shows criminal probability remains locked at its maximum peak, completely unchanged by the maximum penalty setting. Fiscal Dynamics and Social Loss in Fig. 3 shows Total Social Loss remains stagnated at 20.0. This corner exposes the psychological and mathematical flaw of severe laws without enforcement. In the model’s cognitive equation, punishment severity ($f$) is multiplied directly by the probability of apprehension ($p$). If the probability of being caught is zero, the functional deterrent value drops to zero. Criminals recognize that a severe statutory sentence is an empty threat if there are no police officers to arrest them. Because no arrests occur, the state budget remains zero (no surveillance costs, no active prisoners), while society continues to absorb the maximum victim damage cost (20.0).

– Boundary Coordinate Vector ($p=$ 1, $f=$ 10): The Totalitarian Over-Enforcement Crisis.

Behavioral Reality in Fig. 1 shows Criminal probability drops down to its lowest baseline rate, meaning the population is successfully deterred into behavioral compliance. Fiscal Dynamics & Social Loss in Fig. 3 shows Total Social Loss climbs to its absolute global maximum peak ($L\approx$ 32), covered entirely in the dark red heat zone. At this node, the state achieves its goal of a highly controlled environment, but it triggers an economic collapse. The state is hit by a massive financial burden from two compounding directions: 1) Surveillance Costs: Maintaining a perfect, omnipresent detection apparatus ($p=$ 1.0) maximizes the quadratic policing function and 2) Incarceration Friction: For the small percentage of offenses that still occur, processing them under maximum severity ($f=$ 10) with a heavy friction coefficient ($b=$ 0.5) creates an expensive penal tax. The state budget is completely drained by surveillance hardware, police payroll, and long-term correctional facility overhead. This proves that a crime-free society can be an expensive, unsustainable economic ideal.

– Boundary Coordinate Vector ($p=$ 1, $f=$ 0): The Omnipresent Surveillance Trap.

Behavioral Reality in Fig. 1 shows criminal probability drops to a moderate mid-range baseline. The high visibility of the police causes some quantum cognitive hesitation, but compliance is incomplete. Fiscal Dynamics and Social Loss in Fig. 3 shows Total Social Loss remains high ($L\approx$ 23.5), colored in a cool cyan/yellow transition zone. Here, the state deploys police squads on every corner ($p=$ 1.0), which immediately drains the public ledger by the maximum fixed amount of 10.0. However, because the statutory code mandates zero actual penalties ($f=$ 0), the deterrence is incomplete. Opportunistic offenders quickly realize that even though they will definitely be caught, they face no real punishment or fines. As a result, the state is forced to pay maximum surveillance costs while society continues to suffer from mid-level victim damage, making this position highly inefficient.

3.3. Analysis of the quantum gradient descent optimization code

Section 3 of the Appendix outlines the pure computational quantum gradient descent algorithm built without external libraries. The algorithm evaluates analytical partial shifts across the social loss landscape via the calculus chain rule. The marginal change in criminal response relative to the internal circuit angle is evaluated as:$O_j\left(w, {\varphi }_j\right)= {\left|\beta \right|}^2=\mathrm{si}{\mathrm{n}}^2\left(\frac{{\theta }_j}{2}\right)$:

The algorithm then backpropagates this quantum gradient to the state's policy control instruments $p$ and $f$ using the cognitive system weights: