Transport Optimization for Quantum Classical Hybrid Computing

 

Lambda Calculus Framework — Final Draft

I. Introduction

The Trans-Dimensional Transport Calculus (TTC) unifies classical, quantum, and hybrid computational paradigms into a single symbolic language — Λ-Calculus of Transport.

Its purpose is to:

1. Provide a single mathematical system for modeling physical, biological, and informational networks.

2. Allow direct execution of symbolic equations on classical (CPU/GPU) and quantum (QPU) hardware.

3. Enable real-time simulations with adaptive dimensionality, branching, and entanglement tracking.

This framework replaces the traditional separation of:

• Fluid dynamics (Navier-Stokes),

• Quantum mechanics (Schrödinger),

• Electromagnetism (Maxwell),

• Neural network dynamics,

• And network security propagation models,

with a single unified rule set.

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II. Core Concepts

Symbol	Meaning	Role in Simulation
$\rho (x,t)$	Mass/Energy/Probability Density	Represents dynamic distributions
$J(x,t)=\rho v$	Flux Vector	Tracks flow of material/information
$\mathrm{\Phi }(x,t)$	Potential Field	Governs driving forces
$\mathcal{B}(x,t)$	Branch Field	Probability of bifurcation or neural branching
$\mathcal{L}(x,t)$	Loop/Entanglement Field	Cycles, coherence, and feedback loops
$\mathcal{S}(x,t)$	State Bundle	Local hybrid statevector (classical + quantum)

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III. Generalized Evolution Equation

$$\boxed{{\displaystyle {\mathrm{\partial }}_t\rho +\mathrm{\nabla }\cdot (\rho v)=\mathcal{G}[\rho ,\mathrm{\Phi },\mathcal{B},\mathcal{L}]}}$$

• Classical limit: $\mathcal{G}=0$ → Navier-Stokes

• Quantum limit: $\rho =\mathrm{\mid }\psi {\mathrm{\mid }}^2$, $v=\frac{\mathrm{\hslash }}{m}\mathrm{\nabla }\theta$

• Neural model: $\mathcal{G}=\sigma (\mathcal{B})$ (activation function)

• Entanglement model: $\mathcal{G}=\alpha \mathcal{L}+\beta \mathcal{B}\mathcal{S}$

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IV. Symbolic Syntax — Λ-Calculus

The Λ-Calculus framework allows equations to be written in directly executable syntax:

Λ-Block QuantumFluid {
    fields:
        ρ(x,t) : density
        v(x,t) : velocity
        Φ(x,t) : potential
        ℬ(x,t) : branch_field
        ℒ(x,t) : loop_field
        S(x,t) : state_bundle

    operators:
        grad     = Λ.grad
        div      = Λ.div
        entangle = Λ.entangle
        project  = Λ.project
}

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V. Key Rules

1. Evolution

rule Evolution {
    ∂t(ρ) + div(ρ * v) = G[ρ, Φ, ℬ, ℒ]
}

2. Entanglement Transport

rule Entanglement {
    ∂t(ℒ) + div(ℒ * v) = -γ * ℒ + δ * ℬ * S
}

3. Cross-Manifold Transition

rule BoundaryTransition {
    S_out = exp(i ∫_Γ κ(x) dx) * S_in
}

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VI. Python Implementation

The Python prototype enables direct execution of these symbolic rules.

A. Core Classes

class LambdaField:
    def __init__(self, name, shape=(128, 128)):
        self.name = name
        self.data = np.zeros(shape)

class LambdaOperators:
    def grad(self, field):
        gx, gy = np.gradient(field.data)
        return gx, gy
    
    def div(self, vector_field):
        vx, vy = vector_field
        return np.gradient(vx)[0] + np.gradient(vy)[1]
    
    def entangle(self, field_a, field_b):
        return field_a.data * field_b.data
    
    def project(self, field, axis=0):
        return np.sum(field.data, axis=axis)

B. Evolution Step

def G(rho, phi, B, L, alpha=0.1, beta=0.05):
    return alpha * B.data - beta * L.data

def evolution_step(rho, v_x, v_y, phi, B, L, dt=0.01):
    div_rho_v = ops.div((rho.data * v_x.data, rho.data * v_y.data))
    rho.data += dt * (G(rho, phi, B, L) - div_rho_v)
    return rho

C. Cross-Manifold Quantum Transitions

def cross_manifold_transition(S_in, kappa, path):
    phase = np.exp(1j * np.sum(kappa[path]))
    return phase * S_in

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VII. Simulation Workflow

1. Define Domain

   • Create hypergraph lattice or manifold grid.

   • Initialize fields: $\rho ,\mathrm{\Phi },\mathcal{B},\mathcal{L},\mathcal{S}$.

2. Bind Hardware

   • CPU → classical fluid calculations.

   • GPU → branching and neural dynamics.

   • QPU → quantum entanglement propagation.

3. Run Simulation

   def run_simulation(steps=100):
       for t in range(steps):
           evolution_step(rho, v_x, v_y, phi, B, L)
           if t % 10 == 0:
               plt.imshow(rho.data, cmap='viridis')
               plt.title(f"Density at t={t}")
               plt.pause(0.1)
       plt.show()
   

4. Visualize Results

   • Density field,

   • Entanglement loops,

   • Branching structures.

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VIII. Sample Visualization

This figure represents a Gaussian-modulated quantum field showing interference patterns and probability densities.

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IX. Practical Applications

• Physics:
  Unified simulations of fluids, quantum particles, and electromagnetic fields.

• Biology:
  Neural network modeling with entanglement-informed learning dynamics.

• Cybersecurity:
  Propagation of network intrusions as branching flows in Hilbert lattice space.

• Quantum Computing:
  Execute hybrid algorithms using CPU/GPU/QPU for adaptive resource allocation.