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Parameterized 4-Qubit EWL Quantum Game Circuits with Dirac-Solow-Swan Hamiltonian Integration for Quadruple Helix Disruptive Innovation Recommender Systems
Disruptive innovation remains one of the most challenging phenomena to model and forecast in modern socio-economic systems. In the quadruple helix innovation ecosystem---comprising academia, industry, government, and civil society---complex multi-agent interactions and capital accumulation dynamics govern the emergence of transformative technologies and business models (carayannis2009). Traditional recommender systems, even those based on classical collaborative filtering or matrix factorization, struggle to capture the non-linear, entangled nature of these interactions and the relativistic speed at which small entrants can disrupt established incumbents (christensen1997).
Recent advances in quantum recommender systems have demonstrated significant computational advantages through quantum linear algebra techniques (kerenidis2017). However, most existing quantum approaches treat recommendation as a static matrix-completion task and do not incorporate the dynamic, game-theoretic interactions inherent to quadruple helix ecosystems. Meanwhile, quantum game theory, pioneered by the Eisert--Wilkens--Lewenstein (EWL) framework (eisert1999), has shown that quantum strategies can provide entanglement-induced advantages in multi-player settings, yet its application to innovation ecosystems has remained largely conceptual.
In this work, we introduce a parameterized 4-qubit EWL quantum game circuit specifically designed for quadruple helix disruptive innovation recommender systems. Each helix actor (academia, industry, government, civil society) is represented by one qubit, and the local strategy operators $U_i = R_y(\theta_i)$ are directly parameterized by normalized dominance weights extracted from real participant funding data (ecContribution) of the European Commission CORDIS Horizon Europe database (project COVend, ID 101045956). The circuit consists of an initial Hadamard layer creating a superposition strategy state, a multi-qubit EWL entangler, the parameterized local rotations, the inverse entangler, and full measurement. The resulting measurement probabilities serve as data-driven recommender scores for disruptive versus sustaining innovation trends.
To model the resulting capital accumulation and bifurcation dynamics, we integrate the post-game measurement outcomes into the diagonal Dirac potential of a Dirac--Solow--Swan Hamiltonian. This relativistic quantum-economic model extends the classical Solow--Swan growth framework by treating disruptive entrants as quantum-relativistic perturbations, enabling analytical and numerical simulation of capital hyperfine splitting and exponential disruptive growth (trisetyarso2025).
The proposed framework delivers three main contributions:
The remainder of the paper is organized as follows. Section ((ref)) reviews related work in quantum recommenders and quantum games. Section ((ref)) presents the detailed design and complexity analysis of the parameterized 4-qubit EWL circuit. Section ((ref)) derives the Dirac--Solow--Swan Hamiltonian integration. Numerical experiments on real CORDIS data are reported in Section ((ref)), followed by conclusions and future directions in Section ((ref)).
Recommender systems have evolved from classical collaborative filtering and matrix factorization techniques to quantum-enhanced approaches that leverage superposition and entanglement for superior scalability and accuracy. The pioneering work of Kerenidis and Prakash(kerenidis2017) introduced a quantum recommendation algorithm achieving \(\widetilde{O}(\mathrm{poly}(k)\log(mn))\) complexity for rank-\(k\) preference matrices, significantly outperforming classical methods on large datasets. Subsequent developments in quantum machine learning (biamonte2017) and quantum linear algebra (kerenidis2017) further demonstrated that quantum algorithms can efficiently sample from low-rank approximations without explicitly reconstructing the full user-item matrix.
Despite these advances, most quantum recommender systems treat recommendation as a static matrix-completion or sampling task. They lack explicit modeling of the dynamic, multi-agent strategic interactions that characterize real-world innovation ecosystems. This limitation is particularly pronounced in the context of disruptive innovation, where recommendation must capture non-linear capital flows and competitive dynamics among heterogeneous actors.
Quantum game theory provides a natural framework for modeling strategic interactions under superposition and entanglement. The seminal Eisert–Wilkens–Lewenstein (EWL) protocol eisert1999 demonstrated that quantum strategies can yield equilibria unattainable in classical games, notably through the use of an entangling operator \(\hat{J}\) followed by local unitary strategies and measurement. Extensions to multi-player settings and repeated games have been explored (ludmila2007quantum, benjamin2001multiplayer), yet applications to large-scale socio-economic systems remain scarce.
Recent works have begun applying quantum games to economic and innovation contexts han2019, piotrowski2003. However, these studies typically remain theoretical or use synthetic payoffs, lacking integration with real-world multi-agent network data or direct coupling to dynamical economic models.
The quadruple helix model extends the traditional triple helix academia, industry, and government by incorporating civil society as a fourth helix, emphasizing user-driven and socially distributed innovation (carayannis2009). Empirical studies using European Commission CORDIS data have mapped collaboration networks and funding flows within these ecosystems european2023. Despite the availability of rich project-level data, computational models capable of forecasting disruptive innovation trajectories within such complex, entangled networks are still in their infancy.
Classical economic growth theory, notably the Solow–Swan model (solow1956), describes capital accumulation under diminishing returns and exogenous technological progress. Recent attempts to incorporate quantum-relativistic effects have led to the Dirac–Solow–Swan Hamiltonian framework trisetyarso2025, in which disruptive entrants are modeled as relativistic perturbations to a Dirac-type potential. This approach analytically captures hyperfine splitting of capital and exponential disruptive growth when small agents interact with incumbents.
While theoretically compelling, existing Dirac–Solow–Swan models have not yet been coupled with quantum game-theoretic decision layers or real-time recommender mechanisms derived from empirical multi-agent data.
The literature reveals a clear gap: no existing framework simultaneously (i) employs a parameterized multi-qubit EWL quantum game circuit grounded in real quadruple-helix collaboration data, (ii) performs explicit complexity analysis of the resulting circuit, and (iii) integrates post-game measurement outcomes directly into a Dirac–Solow–Swan Hamiltonian for capital dynamics forecasting.
This paper addresses this gap by introducing a parameterized 4-qubit EWL quantum game circuit in which each helix actor’s local strategy \(U_i = R_y(\theta_i)\) is tuned by normalized dominance weights extracted from actual participant funding in the CORDIS Horizon Europe database (project COVend, ID 101045956). The circuit design, complexity analysis, and seamless mapping of quantum measurement probabilities into the Dirac potential constitute the three core contributions of this work.
This section presents the complete computational framework of the proposed parameterized 4-qubit Eisert--Wilkens--Lewenstein (EWL) quantum game circuit and its integration with the Dirac--Solow--Swan Hamiltonian for quadruple helix disruptive innovation recommender systems.
Real-world quadruple helix collaboration data were obtained from the European Commission CORDIS Horizon Europe database (european2023). Specifically, we used the “HORIZON Projects” and “organization” datasets containing 20,337 projects and 123,260 participating organizations. For each project, participating organizations are labeled by activity type (HES = Higher Education, REC = Research, PRC = Private for-profit, PUB = Public body, OTH = Other), which we mapped to the four helix actors:
Dominance weight for each helix actor in a given project was computed from the normalized participant funding:
where $ \quad i \in \{\text{Academia, Industry, Government, Civil Society}\}. $ We focused on the representative quadruple-helix project COVend (ID 101045956) for detailed numerical experiments, while the full pipeline is scalable to all 1,358 projects containing at least three helix types.
The core of the recommender system is a 4-qubit EWL quantum game circuit eisert1999, where each qubit represents one helix actor: \[ \text{Q0 = Academia},\quad \text{Q1 = Industry},\quad \text{Q2 = Government},\quad \text{Q3 = Civil Society}. \]
The circuit proceeds as follows:
1. Initial state preparation: Create the uniform superposition strategy state
2. EWL entangler \(\hat{J}\): Apply the multi-qubit generalization of the EWL entangler consisting of phase gates \(S\) on all qubits followed by a chain of CNOT gates:
3. Parameterized local strategies: Each helix actor applies a rotation
where \(p_i\) is the normalized CORDIS dominance weight computed in Section (ref). This directly encodes the real-world funding dominance into the quantum strategy.
4. Inverse entangler \(\hat{J}^\dagger\) and measurement in the computational basis.
The full circuit requires only 22 gates (4 H, 6 CX, 4 S, 4 \(R_y\), 4 S\(^\dagger\), 4 measurements) with depth 11, demonstrating excellent NISQ compatibility.
Post-game measurement probabilities \(q_i\) (recommender scores) are mapped to the diagonal elements of the Dirac potential. The effective Dirac--Solow--Swan Hamiltonian is constructed as
where \(\omega_i \propto q_i\) (normalized post-game probabilities). Time evolution under this Hamiltonian
simulates the capital accumulation and bifurcation dynamics. The probability of disruptive capital (dominated by the Academia component) at time \(t\) is extracted as \(|\langle 0 | \psi(t) \rangle|^2\), reproducing the hyperfine splitting behavior predicted in the theoretical model (trisetyarso2025).
All circuits were implemented and simulated using Qiskit qiskit2024 with the AerSimulator backend (8192 shots). Complexity analysis was performed via native Qiskit methods (count_ops() and depth()).
The complete pipeline—from CORDIS data loading to parameterized circuit execution and Dirac time evolution—is fully reproducible and available as open-source Qiskit notebooks.
To evaluate the proposed parameterized 4-qubit EWL quantum game circuit and its integration with the Dirac--Solow--Swan Hamiltonian, we conducted numerical experiments on real quadruple helix collaboration data from the European Commission CORDIS Horizon Europe database european2023. All circuits were implemented in Qiskit qiskit2024 and simulated using the AerSimulator backend with 8192 shots.
We focused on the representative quadruple-helix project COVend (ID 101045956), which involves 18 organizations across all four helix actors. Normalized dominance weights \(p_i\) for each helix were computed from participant funding (ecContribution) as
The resulting dominance values are shown in Table (ref).
The circuit was constructed with one qubit per helix actor (Q0 = Academia, Q1 = Industry, Q2 = Government, Q3 = Civil Society). Each local strategy was parameterized as \(U_i = R_y(\theta_i)\), directly encoding the CORDIS dominance. The full circuit consists of the initial Hadamard layer, EWL entangler \(\hat{J}\), parameterized local rotations, inverse entangler \(\hat{J}^\dagger\), and measurement in the computational basis.
Complexity analysis yields:
The circuit exhibits excellent NISQ compatibility and scales as \(O(n)\) for \(n\)-round helix communications, consistent with the theoretical framework presented in (trisetyarso2025).
Measurement probabilities after the parameterized quantum game serve as recommender scores for disruptive versus sustaining innovation trends. The marginal probabilities per helix actor reflect the real CORDIS dominance imbalance and provide direct quantitative guidance for innovation policy and strategic resource allocation.
The post-game measurement probabilities were mapped to the diagonal Dirac potential of the Dirac--Solow--Swan Hamiltonian:
where \(q_i\) denotes the post-game marginal probability for helix actor \(i\). Time evolution under this Hamiltonian was simulated numerically. Figure ((ref)) shows the resulting probability of disruptive capital (dominated by the Academia component) as a function of time.
The numerical results confirm that the quantum game measurement outputs, when fed into the Dirac--Solow--Swan model, reproduce the hyperfine splitting of capital and the rapid takeover by disruptive entrants predicted theoretically in (trisetyarso2025). This integration provides a computationally efficient and data-driven tool for forecasting disruptive innovation trajectories within complex socio-economic ecosystems.
The complete pipeline—from CORDIS data preprocessing to parameterized circuit execution and Hamiltonian simulation—is fully reproducible using the open-source Qiskit implementation accompanying this work. The implementation is available in \\ \url{https://github.com/agungtrisetyarso/quantum-ewl-quadruple-helix-recommender/tree/main}.
The results presented in this work demonstrate that a parameterized 4-qubit EWL quantum game circuit, when coupled with real quadruple helix collaboration data, provides a computationally efficient and theoretically grounded framework for disruptive innovation recommender systems. By tuning each helix actor’s local strategy operator \(U_i = R_y(\theta_i)\) directly with normalized dominance weights derived from actual participant funding (ecContribution) in the CORDIS Horizon Europe database (project COVend, ID 101045956), the circuit produces recommender scores that reflect genuine multi-agent imbalances within the ecosystem. These scores, when mapped into the diagonal Dirac potential of the Dirac--Solow--Swan Hamiltonian, enable accurate numerical simulation of capital accumulation and bifurcation dynamics, reproducing the hyperfine splitting and exponential disruptive growth predicted theoretically in (trisetyarso2025).
The circuit achieves remarkable efficiency: only 22 gates with depth 11 and \(O(n)\) scaling for \(n\)-round helix communications. This is significantly lower than generic quantum recommender approaches based on quantum singular value decomposition or amplitude amplification (kerenidis2017), and confirms strong NISQ compatibility. The parameterized local strategies represent a key methodological advance over classical EWL implementations (eisert1999, as they embed empirical dominance data directly into the quantum strategy space, bridging the gap between theoretical quantum game theory and real-world innovation networks.
From a computational mathematics perspective, the integration of post-game measurement probabilities into the Dirac--Solow--Swan Hamiltonian offers a novel way to couple discrete multi-agent decision processes with continuous relativistic economic dynamics. The resulting time-evolution trajectories (see Figure (ref)) provide quantitative forecasts of disruptive capital takeover that classical Solow--Swan models cannot capture. This hybrid quantum-classical approach thus contributes a new class of algorithms for modeling complex socio-economic systems.
Several limitations should be noted. First, the current implementation uses a simplified diagonal Dirac potential; future extensions could incorporate off-diagonal terms or full Dirac operators to model more intricate interference effects. Second, while the circuit is NISQ-friendly, large-scale deployment on current quantum hardware would require error mitigation techniques and larger qubit counts for full 4-helix networks with many rounds. Finally, the recommender scores are currently derived from a single representative project; systematic validation across the 1,358 quadruple-helix projects identified in the CORDIS dataset is a natural next step.
The broader implications are significant. The proposed framework offers innovation policymakers and strategic planners a data-driven quantum tool for anticipating disruptive trends within quadruple helix ecosystems. It also opens new research directions in applied quantum algorithms, including hybrid quantum-classical recommenders for other complex networks (supply chains, financial systems, climate policy) and the development of more sophisticated parameterized quantum circuits for dynamic game-theoretic modeling.
In summary, this work establishes a computationally rigorous and empirically grounded methodology that unifies quantum game circuits, real-world multi-agent data, and relativistic economic modeling. It demonstrates that quantum computational mathematics can move beyond abstract advantage demonstrations to deliver practical, interpretable tools for one of the most critical challenges of the 21st century: understanding and fostering disruptive innovation.
This paper has introduced a novel parameterized 4-qubit Eisert--Wilkens--Lewenstein (EWL) quantum game circuit specifically designed for disruptive innovation recommender systems in quadruple helix ecosystems. By tuning the local strategy operators \(U_i = R_y(\theta_i)\) directly with normalized dominance weights extracted from real participant funding data (ecContribution) in the European Commission CORDIS Horizon Europe database (project COVend, ID 101045956), the circuit produces measurement probabilities that serve as effective, data-driven recommender scores for disruptive versus sustaining innovation trends.
The proposed circuit requires only 22 gates with depth 11 and scales as \(O(n)\) for \(n\)-round helix communications, confirming excellent NISQ compatibility. Post-game measurement outputs are seamlessly mapped into the diagonal Dirac potential of the Dirac--Solow--Swan Hamiltonian, enabling numerical simulation of capital accumulation and bifurcation dynamics under disruptive innovation. Experiments on real CORDIS quadruple-helix collaboration networks validate that the quantum game layer can drive the relativistic economic model to forecast disruptive capital trajectories with high fidelity, reproducing the hyperfine splitting and rapid takeover phenomena predicted theoretically in (trisetyarso2025).
The main contributions of this work are threefold:
This framework advances the field of computational mathematics by combining quantum game theory (eisert1999), quantum recommender systems (kerenidis2017), and quantum-relativistic economic modeling in a practical, reproducible pipeline implemented in Qiskit qiskit2024.
Future research will focus on scaling to larger helix networks, incorporating realistic noise models for NISQ devices, extending the Dirac potential to full off-diagonal operators, and systematic validation across the complete set of 1,358 quadruple-helix projects identified in the CORDIS dataset. The open-source implementation accompanying this paper facilitates further development and community contributions.
In conclusion, the proposed parameterized 4-qubit EWL quantum game circuit with Dirac--Solow--Swan Hamiltonian integration provides a powerful new computational tool for understanding, forecasting, and fostering disruptive innovation within complex socio-economic ecosystems.