QuantEvolve Framework: Unified Evolutionary Paradigm

• QuantEvolve Framework is a unified, multi-domain paradigm that integrates stochastic quantization and evolutionary optimization to model diverse complex systems.
• It employs rigorous mathematical techniques, including spectral decomposition and natural gradient descent, to ensure convergence and control error propagation.
• Algorithmic instantiations such as EVQE and MQNE demonstrate hardware efficiency and robustness across applications in quantum computing, machine learning, and quantitative finance.

QuantEvolve Framework is a unified, multi-domain paradigm for modeling and automating complex evolutions—ranging from biological systems, quantum circuits, time-dependent physical processes, to diverse algorithmic architectures—using stochastic, evolutionary, and optimization methodologies. This framework emerges from foundational links between stochastic quantization and quasi-species theory (Bianconi et al., 2010), and subsequently integrates quantum algorithmic primitives, neuroevolutionary strategies, and multi-agent optimization to address problems in quantum computing, machine learning, and quantitative finance.

1. Stochastic Evolution Equation and Universal Dynamics

At its mathematical core, the QuantEvolve Framework generalizes biological evolution equations through operator methods grounded in statistical physics:

$$\frac{dP(x,t)}{dt} = \frac{M_{xx'}\left[e^{-\beta U(x')} P(x',t)\right]}{Z_t} - P(x,t)$$

where $P(x,t)$ is the probability to find a system (or an individual particle) at state $x$ and time $t$; $U(x)$ maps to the fitness function (or potential energy); $M_{xx'}$ is a mutation operator realized as Gaussian noise; and $\beta$ governs the influence of selection (or temperature, in analogy to thermodynamical systems). This equation encapsulates both the out-of-equilibrium migration toward optimal states and the steady state formation characterized by Bose–Einstein statistics under the right scattering processes.

Via spectral decomposition, the evolution equation admits solutions:

$$P(x, t) = \sum_n c_n(t) \pi_n(x), \quad M_{xx'}[e^{-\beta U(x')} \pi_n(x')] = \lambda_n \pi_n(x)$$

with discrete eigenvalue spectrum $\{\lambda_n\}$, driving system relaxation toward the dominant (fundamental) state—a phenomenon formally analogous to the Fisher theorem in evolutionary biology.

2. Eigenfunction Analysis and Generalized Fisher Theorem

The Eigenfunction expansion centralizes the quantification of long-time behavior and relaxation to equilibrium. Under projection onto the eigenbasis, the temporal dynamics obey:

$$c_n(t) = \exp[\lambda_n G(t) - t]\, c_n(0), \quad G(t) = \int_0^t dt' \frac{1}{Z_{t'}}$$

Generalization of Fisher's theorem is elegantly expressed as:

$$\frac{1}{2} \frac{d \langle \lambda \rangle^2}{dt} = \langle \lambda^2 \rangle - \langle \lambda \rangle^2$$

This implies that the rate of increase of (squared) mean fitness is the variance of the reproductive rates, controlling the deterministic drift toward the optimal genetic or solution state. In QuantEvolve, such a formalism underpins the analytic tracking of convergence, variance reduction, and mode selection throughout the evolutionary cycle.

3. Algorithmic Instantiations: Evolutionary Quantum Solvers and Neuroevolution

The framework operationalizes evolutionary principles in several quantum algorithmic strategies. In the Evolutionary Variational Quantum Eigensolver (EVQE) (Rattew et al., 2019), circuit topologies are encoded as genomes, and mutations drive adaptive ansatz generation for general-purpose optimization. EVQE demonstrates hardware efficiency (up to $18.6\times$ shallower circuits and $12\times$ fewer CX gates than fixed-form VQE) and notable noise resistance (at least $3.6\times$ lower error on tested NISQ devices). The fitness function:

$$f_i = \langle \psi_i | H | \psi_i \rangle + \alpha \cdot (\text{circuit depth}) + \beta \cdot (\text{number\,of\,CX})$$

is minimized through asexual reproduction, speciation, and three distinct mutation operators (topological, parameter, removal), favoring shallow, robust quantum algorithms applicable across chemistry, optimization, and control domains.

Similarly, Markovian Quantum Neuroevolution (MQNE) (Lu et al., 2020) models quantum architecture search as directed graph path selection, with gate blocks as nodes and connection rules as edges. Evolution proceeds as a Markovian process, extending circuits by sampling permissible transitions. This results in efficient quantum neural networks that achieve high performance in classification and recognition tasks with minimal circuit depth, leveraging the reduced susceptibility to barren plateaus and maximizing suitability for NISQ hardware.

4. Information-Theoretic Geometry: Fisher–Bures, Wigner–Yanase, and Kubo–Mori Metrics

QuantEvolve incorporates sophisticated gradient and geometry-aware optimization. For evolved Quantum Boltzmann Machines (Minervini et al., 6 Jan 2025), the natural gradients of an evolved ansatz,

$$\omega(\theta,\phi) = e^{-i H(\phi)}\, \rho(\theta)\, e^{i H(\phi)}, \quad \rho(\theta) = \frac{e^{-G(\theta)}}{Z(\theta)}$$

are estimated via quantum algorithms (random time sampling, Hamiltonian simulation, Hadamard test), and the information matrices—Fisher–Bures, Wigner–Yanase, Kubo–Mori—quantify curvature and sensitivity:

$$I_{ij}^{FB(\theta)} = \frac{1}{2} \langle \{ \Phi_\theta(G_i),\, \Phi_\theta(G_j) \} \rangle_{\rho(\theta)} - \langle G_i \rangle_{\rho(\theta)} \langle G_j \rangle_{\rho(\theta)}$$

Theoretical results (generalizing Luo's 2004 theorem) establish that Fisher–Bures and Wigner–Yanase matrices differ by at most a factor of two, confirming their essential interchangeability in natural gradient descent.

5. Quantum Walks and Resource Trade-offs

Efficient search and simulation in QuantEvolve are enhanced by the unified quantum walk search framework (Apers et al., 2019), which interpolates between hitting time, electric network, and MNRS approaches, balancing walk steps and oracle checks:

$$\text{Complexity:}\quad S(\sigma) + \sqrt{C_{\sigma,M}(P^t)} \cdot (\sqrt{t}\, U(\sigma) + C)$$

Fast-forwarding and absorbing chain interpolation facilitate adaptive walk lengths and resource-efficient implementation, critical for scalability in quantum simulation and optimization.

6. Time Evolution and Error Control

TimeEvolver (Michel et al., 2022) extends QuantEvolve's reach to large-scale quantum dynamics by leveraging Krylov subspace methods for exponential matrix computation:

$\vec{v}(t) \approx V_m \exp(-i H_m t) e_1$

Crucially, rigorous a posteriori error bounds,

$$||\text{err}(t)|| \leq \int_0^t |h_{m+1,m}\, (e_m^\top \exp(-i H_m \tau) e_1)|\, d\tau$$

are afforded by the Hermitian property of $H$. This ensures precise control over simulation accuracy and supports robust time-dependent analyses in quantum physics.

7. Multi-Agent Evolution in Quantitative Finance

QuantEvolve's application in finance is formalized by a multi-agent framework for strategy evolution (Yun et al., 21 Oct 2025). A feature map organizes strategies by multidimensional investor-centric attributes (strategy category, trading frequency, risk, and return metrics), and an island-model multi-agent process generates and refines hypotheses. Parent and cousin selection balance exploration and exploitation:

$$P(s_p=s) = \begin{cases} \alpha/|M_I| & s \in M_I\ (1-\alpha)/|I| & s \in I \end{cases}$$

Empirical results show superior Sharpe and Information Ratios and cumulative returns when backtested against conventional baselines. The public dataset release further enables reproducible research and meta-analysis of strategy evolution.

8. Synthesis and Domain-Spanning Implications

The QuantEvolve Framework, by subsuming stochastic quantization, evolutionary quantum optimization, neuroevolution, quantum channel transformation, advanced gradient geometries, and multi-agent evolutionary search into a single rigorous architecture, offers a coherent foundation for modeling, analyzing, and automating evolution in biosystems, quantum devices, learning architectures, and financial strategies. The explicit connection of spectral theory, evolution equations, stochastic processes, and optimization underpins theoretical advances and guides practical implementations that adaptively maximize diversity, robustness, and efficiency in dynamically changing environments. This framework is extensible to future domains where evolution—in its stochastic, quantum, and adaptive forms—remains central.