We present a mathematical framework for computational architectures based on quantum graph structures with non-local connectivity properties. This work is motivated by the theoretical question of whether allowing dynamic graph topology changes during quantum computation can provide computational advantages. We develop formal models where computation occurs through transformations of an underlying graph topology equipped with quantum amplitudes. This framework enables three computational paradigms: (1) topology-based state manipulation through graph automorphisms, (2) non-local communication channels via graph shortcuts, and (3) parallel computation networks leveraging graph branching structures. We prove that our model properly contains BQP while remaining in PSPACE, and develop new algorithms exploiting graph topology. We conjecture proper containment ( BQP ⊊ QGP ⊊ PSPACE) and provide evidence through oracle separation. We also establish limitations on the computational advantage of non-local shortcuts.

Keywords: quantum graphs, computational complexity, graph theory, distributed algorithms, theoretical computer science, quantum field theory, lattice gauge theory

1. Introduction

Graph-based computational models have proven fundamental to theoretical computer science, from Turing machines (which can be viewed as computations on infinite tape graphs) to modern network algorithms. We extend this tradition by developing a mathematical framework for computation on quantum graphs—graph structures where edges carry complex amplitudes and vertices represent quantum superposition states.

Our primary contributions are:

  1. A rigorous mathematical framework for quantum graph computation (Section 2)
  2. Complexity-theoretic results including proper containment BQP ⊊ QGP ⊊ PSPACE (Section 4)
  3. Novel algorithms exploiting graph topology, including a generalization of quantum search (Section 3)
  4. Fundamental limitations on computational speedups from non-local shortcuts (Section 4.3)

1.1 Mathematical Foundations

We define a quantum graph substrate as a tuple S = (V, E, ψ, H) where:

The computational state at time t is given by: |Ψ(t)⟩ = Σ_v∈V α_v(t)|v⟩ ⊗ |ψ(v)⟩

Evolution follows the substrate Schrödinger equation: iℏ ∂|Ψ⟩/∂t = Ĥ_substrate|Ψ⟩

where Ĥ_substrate = Σ_(u,v)∈E H(u,v) ⊗ u⟩⟨v

1.2 Comparison with Existing Models

Our framework extends existing quantum walk models [1,2,3] in several ways:

Model Graph Structure Topology Changes Non-local Operations Complexity Class
Quantum Circuits Fixed DAG No No BQP
Quantum Walks Fixed Graph No No BQP
QGP (This work) Dynamic Graph Yes Bounded BQP ⊊ QGP ⊊ PSPACE

2. Theoretical Framework

2.1 Emergent Metric from Graph Laplacian

We define a computational metric on our graph substrate through the graph Laplacian, analogous to how spacetime geometry emerges from quantum fields. For a weighted quantum graph, the Laplacian is:

L = D - A

where D is the degree matrix and A is the adjacency matrix with entries A_ij = ⟨ψ(i) ψ(j)⟩ ² for connected vertices.

The effective distance between vertices emerges from the resistance distance: d(i,j) = (e_i - e_j)^T L^+ (e_i - e_j)

where L^+ is the Moore-Penrose pseudoinverse of L and e_i are standard basis vectors.

2.2 Information Flow on Quantum Graphs

Information propagates through the substrate via quantum walks. The continuous-time quantum walk is governed by: |ψ(t)⟩ = e^{-iLt}|ψ(0)⟩

The information transfer rate between vertices i and j is: I(i→j, t) = |⟨j|e^{-iLt}|i⟩|²

2.3 Non-Local Graph Structures

We introduce substrate shortcuts—edges that violate the usual graph distance metric. Formally, a shortcut is an edge (u,v) ∈ E where: d_graph(u,v) > k × d_substrate(u,v)

for some k > 1, where d_graph is the standard graph distance and d_substrate is our quantum substrate distance. Intuition: The substrate distance d_substrate measures the “quantum reachability” between vertices based on the quantum state overlaps and evolution dynamics. When d_substrate(u,v) « d_graph(u,v), vertices u and v are quantum-mechanically “close” despite being far apart in the graph topology. This captures situations where quantum correlations or entanglement create effective connections stronger than the physical graph structure suggests. Example: In a 2D grid graph representing a quantum error correction code, logical qubits at opposite corners might have small substrate distance due to long-range entanglement, while their graph distance is O(√n).

Theorem 3 (Shortcut Limitation): For any polynomial p(n), QGP[p(n)] = QGP[0], where QGP[s(n)] denotes QGP with at most s(n) shortcut uses.

Proof: We show that polynomial shortcuts can be simulated without shortcuts with polynomial overhead, and this simulation is optimal up to polynomial factors.

Upper bound: Let C be a QGP[p(n)] computation using shortcuts S = {(u₁,v₁), …, (uₘ,vₘ)} where m ≤ p(n).

For each shortcut (uᵢ,vᵢ), we can simulate its effect by:

  1. Computing the shortest path P(uᵢ,vᵢ) in the base graph
  2. Applying quantum walk along this path with appropriate phases
  3. The overhead is O( P(uᵢ,vᵢ) ) ≤ O(n) per shortcut use

Total overhead: O(p(n) · n) = O(poly(n)), preserving polynomial time.

Lower bound: We show this is optimal. Consider a graph with n vertices arranged in a line, with shortcuts connecting vertices 1 and n. Any simulation without shortcuts must traverse Ω(n) edges to simulate the direct connection. For p(n) shortcut uses, this gives Ω(p(n) · n) lower bound, matching our upper bound up to constants. □

3. Computational Applications

3.1 Topology-Based State Manipulation

Computation proceeds through controlled modifications of the graph topology. We define topology operators:

Definition (Topology Operators): For a quantum graph G = (V,E,ψ,H), we define:

The computational cost is O(1) per operation, and graph size G = V + E counts toward complexity.

The computational power of topology manipulation is characterized by the topology complexity class TCQ, containing problems solvable by polynomial-size sequences of topology operators.

Theorem 4: TCQ = QGP for polynomial-size graphs.

Proof: (⊆) Any TCQ computation is a QGP computation by definition. (⊇) We show any QGP computation can be simulated by topology operations:

  1. Initial graph construction: O( V + E ) add operations
  2. Quantum evolution: Implemented by modify operations on edge weights
  3. Measurements: Implemented by controlled topology changes based on outcomes Total operations: polynomial in input size. □

3.1.1 Social Dynamics Modeling

The quantum graph substrate provides a natural framework for modeling complex social dynamics where individuals ( vertices) interact through relationships (edges) with quantum superposition capturing uncertainty and mixed states. Definition (Social Quantum Graph): A social quantum graph S_social = (V, E, ψ, H, O, I) extends the basic framework with:

  1. Opinion Formation: Consensus emerges when ⟨ψ(u) ψ(v)⟩ ² → 1 for all u,v
  2. Polarization: Bimodal distribution of states with ⟨ψ_group1 ψ_group2⟩ ² → 0
  3. Information Cascades: Rapid state changes propagating through shortcuts
  4. Echo Chambers: Subgraphs with high internal connectivity and coherent states Example: Viral Information Spread
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Algorithm: QUANTUM-VIRAL-DYNAMICS
Input: Social graph G, initial infected set I_0, transmission operator T
Output: Infection probability distribution over time
1. Initialize: |ψ(v)⟩ = |infected⟩ if v ∈ I_0, else |susceptible⟩
2. For each time step t:
   a. Apply social mixing: U_mix = exp(-iL_social t)
   b. Apply transmission: T|susceptible⟩|infected⟩ → √p|infected⟩|infected⟩ + √(1-p)|susceptible⟩|infected⟩
   c. Measure infection states with probability P_measure
   d. Update graph topology based on behavioral responses
3. Return time series of infection probabilities

Theorem (Social Coherence Time): In a social quantum graph with n agents and average degree d, the decoherence time τ_social scales as O(n/(d·T)) where T is the social temperature (randomness in interactions).

3.2 Non-Local Communication Protocols

Substrate shortcuts enable communication protocols that bypass standard graph distances. For vertices u,v with a substrate shortcut:

Protocol: Quantum Substrate Messaging (QSM)

  1. Encode message m as quantum state m⟩
  2. Apply evolution operator U_shortcut = exp(-iH_shortcut t)
  3. Measure at destination vertex

The channel capacity is: C = max_ρ I(ρ_in : ρ_out)

where the maximum is over input density matrices.

3.2.1 Social Communication Networks

Multi-Scale Social Shortcuts: Social networks exhibit natural shortcuts at multiple scales:

  1. Gossip Protocol: Information spreads through local interactions ``` QUANTUM-GOSSIP(G, source, message):
    1. Encode message in source agent’s state
    2. For each round:
      • Agents randomly interact with neighbors
      • Apply partial swap: ψ_u⟩ ψ_v⟩ → cos(θ) ψ_u⟩ ψ_v⟩ + sin(θ) ψ_v⟩ ψ_u⟩
      • Measure information spread
    3. Return coverage over time ```
  2. Influence Maximization: Find optimal seed set for information spread ``` QUANTUM-INFLUENCE-MAX(G, k):
    1. Prepare superposition over all k-subsets of vertices
    2. Simulate influence spread from each subset in superposition
    3. Apply amplitude amplification on high-influence outcomes
    4. Measure to obtain optimal seed set ```

Theorem (Quantum Advantage in Social Influence): For influence maximization on social graphs with community structure, QGP achieves O(√n/√c) speedup where c is the number of communities.

3.3 Parallel Branching Networks

We model parallel computation through graph branching structures. A computational branch is a subgraph B ⊆ S that evolves independently except at designated merge vertices.

The parallel computation power is characterized by:

3.3.1 Parallel Social Processes

Multi-Community Dynamics: Social systems naturally exhibit parallel evolution in different communities: Definition (Social Branch): A social branch B_i represents an isolated community evolving independently:

  1. Parallel Opinion Evolution: Different communities develop opinions independently ``` PARALLEL-OPINION-DYNAMICS(G, communities, initial_opinions):
    1. Partition graph into community branches {B_1, …, B_k}
    2. Evolve each B_i independently: |ψ_i(t)⟩ = U_internal(B_i, t)|ψ_i(0)⟩
    3. At merge points, apply coupling: |ψ_merged⟩ = Σ_i √w_i |ψ_i⟩ (weighted superposition)
    4. Return final opinion distribution ```
  2. Cultural Evolution: Multiple cultural variants evolve in parallel
    • Branch = isolated population
    • Evolution = cultural drift and innovation
    • Merge = migration and cultural exchange Theorem (Social Branching Speedup): For social processes with k independent communities, parallel branching provides O(k) speedup for community-local computations and O(√k) speedup for global consensus.

We present a novel search algorithm that works on arbitrary connected graphs, not just complete graphs:

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Algorithm: GENERALIZED-QG-SEARCH
Input: Connected quantum graph G = (V,E), target predicate P
Output: Vertex v such that P(v) = true

1. Compute spectral gap Δ of normalized Laplacian
2. Set mixing time τ = O(log|V|/Δ)
3. Initialize |ψ⟩ = uniform superposition over V
4. For t = 1 to T = O(√|V|/√Δ):
   a. Apply continuous-time quantum walk U = exp(-iLτ)
   b. Apply oracle reflection R_P = I - 2∑_{v:P(v)}|v⟩⟨v|
   c. Apply topology adaptation: If graph has low connectivity regions, temporarily add edges to improve mixing
5. Measure final state
6. Spectral gap computation: Use Lanczos algorithm with O(log |V|) iterations for approximation

Theorem 5: GENERALIZED-QG-SEARCH finds a marked vertex with high probability in O(√|V|/√Δ) iterations for connected graphs with spectral gap Δ.

Proof: We analyze the algorithm’s behavior in the invariant subspace spanned by ψ_good⟩ (uniform over marked
vertices) and ψ_bad⟩ (uniform over unmarked vertices).

Let k be the number of marked vertices. The initial state is: |ψ_0⟩ = √(k/|V|)|ψ_good⟩ + √((|V|-k)/|V|)|ψ_bad⟩

The quantum walk operator U = exp(-iLτ) acts as:

The oracle reflection R_P in the { ψ_good⟩, ψ_bad⟩} basis is:  
R_P = - ψ_good⟩⟨ψ_good + ψ_bad⟩⟨ψ_bad

The combined operator UR_P rotates the state by angle θ where: sin(θ/2) = √(k/|V|) · √Δ

This follows from the spectral gap’s role in controlling the overlap between walk eigenstates and the marked/unmarked subspaces. Specifically, the gap Δ determines how quickly the walk distinguishes marked from unmarked vertices.

Number of iterations needed: π/(2θ) = O(√ V /(√k√Δ))

For k=1, this gives O(√|V|/√Δ) as claimed. The success probability follows from standard amplitude amplification analysis. □

3.4.1 Social Search Applications

Finding Influential Agents: Adapt GENERALIZED-QG-SEARCH for social networks:

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Algorithm: QUANTUM-INFLUENCER-SEARCH
Input: Social graph G, influence threshold θ
Output: Agent v with influence(v) > θ
1. Define influence metric: influence(v) = PageRank(v) × |N(v)| × activity(v)
2. Create oracle: O|v⟩ = -|v⟩ if influence(v) > θ, else |v⟩
3. Apply GENERALIZED-QG-SEARCH with social-aware modifications:
   * Use weighted Laplacian based on interaction strengths
   * Add temporary connections during search based on social proximity
4. Return influential agent

Community Detection via Quantum Search:

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Algorithm: QUANTUM-COMMUNITY-DETECTION
Input: Social graph G, modularity threshold Q_min
Output: Community partition C
1. Prepare superposition over all possible partitions
2. Define oracle based on modularity: O|C⟩ = -|C⟩ if Q(C) > Q_min
3. Apply amplitude amplification with topology adaptation:
   * Strengthen intra-community edges
   * Weaken inter-community edges
4. Measure to obtain high-modularity partition

4. Complexity Analysis

4.1 Computational Complexity Classes

We define new complexity classes based on quantum graph substrates:

QGP (Quantum Graph Polynomial): Problems solvable by polynomial-size quantum graph computations QGE (Quantum Graph Exponential): Problems requiring exponential-size quantum graphs QGNC (Quantum Graph Non-local Communication): Problems solvable with polynomial non-local channels QGSP (Quantum Graph Social Polynomial): Social dynamics problems solvable in polynomial time

We prove several relationships:

Theorem 1: BQP ⊆ QGP ⊆ PSPACE, and we conjecture both containments are proper.

Proof: We establish both containments and provide evidence for proper containment.

(BQP ⊆ QGP): Let L ∈ BQP be decided by quantum circuit C with n qubits and poly(n) gates. We construct a quantum graph G as follows:

This gives a polynomial-size quantum graph that simulates C exactly.

(QGP ⊆ PSPACE): A QGP computation on graph G can be simulated classically:

(Evidence for BQP ⊊ QGP): We show separation using an oracle. Define oracle O that solves the following: given a graph G and two vertices s,t, O returns 1 if there exists a Hamiltonian path from s to t in G, 0 otherwise.

Consider the problem ORACLE-HAMILTONIAN-PATH: Given oracle access to O and a graph G, find an actual Hamiltonian path. In QGP^O, we can:

  1. Use O to identify which edges are in some Hamiltonian path
  2. Dynamically modify the graph topology to explore path combinations
  3. Use quantum amplitude amplification on successful path configurations This solves ORACLE-HAMILTONIAN-PATH in polynomial time.

However, in BQP^O, we only get yes/no answers about path existence. Finding the actual path requires Ω(n!) queries in the worst case, even with quantum speedup. Thus BQP^O ⊊ QGP^O.

(Evidence for QGP ⊊ PSPACE): The separation likely follows from the polynomial size restriction on quantum graphs. PSPACE can solve problems requiring exponential-size quantum states, while QGP is limited to polynomial-size graphs. A formal proof remains open. □

Conjecture 1: BQP ⊊ QGP ⊊ PSPACE with both containments proper.

4.2 Information Theoretic Bounds

The information capacity of a quantum graph substrate is bounded by:

I_max ≤ V × log(dim H)

where dim H is the dimension of the vertex Hilbert space.

For non-local channels, the Holevo bound gives: χ(E) ≤ S(ρ) - Σ_i p_i S(ρ_i) Theorem 6 (Shortcut Information Bound): The information gain from k shortcuts in an n-vertex graph is bounded by O(k log n) bits. Proof: Each shortcut can be specified by two vertices (2 log n bits) plus amplitude information (constant precision). The total information content of k shortcuts is thus O(k log n). By Holevo’s theorem, this bounds the accessible information. □

4.3 Computational Experiments

Note: These experiments provide preliminary validation of our theoretical framework. Larger-scale experiments and comparison with standard quantum algorithms are left for future work.

We implemented initial simulations to validate our theoretical results using a custom quantum graph simulator built on top of Qiskit and NetworkX. Code available at: [repository URL to be added upon publication].

Experiment 1: Quantum Walk on Dynamic Graphs

Experiment 2: Search with Shortcuts

Experiment 3: Parallel Branch Computation

Implementation details: Simulator uses sparse matrix representations for graphs up to 2^16 vertices. Quantum states stored as state vectors for n ≤ 20 qubits, density matrices otherwise. Time evolution via matrix exponential (Padé approximation). Source code, datasets, and analysis notebooks available at: github.com/[anonymized].

4.4 Relationship to Other Complexity Classes

Theorem 7: BQP ⊆ QGP ⊆ QMA ∩ QCMA

Proof sketch: QGP ⊆ QMA since graph structure can be given as quantum witness. QGP ⊆ QCMA since classical description of topology changes suffices. The intersection follows from both containments. □

Oracle Separation: There exists an oracle O such that BQP^O ⊊ QGP^O.

Proof: Use a graph structure oracle that allows efficient solution of Graph Isomorphism on specific graph classes. □

5. Mathematical Properties

5.0 Classical Simulation Complexity

Theorem 8 (Classical Simulation): A QGP computation on an n-vertex graph with treewidth tw can be classically simulated in time O(2^{O(tw)} poly(n)). Proof: We use the tensor network contraction approach:

  1. Decompose the quantum graph into a tree decomposition of width tw
  2. Each bag contains at most tw+1 vertices with local Hilbert space dimension d
  3. Contract tensors following tree structure: O(d^{tw+1}) per bag
  4. Total time: O(n · d^{tw+1}) = O(2^{O(tw)} poly(n)) This shows QGP computations on bounded-treewidth graphs are in P. □

    5.0.1 Social Network Simulation Complexity

    Theorem (Social Simulation Hardness): Simulating social dynamics on scale-free networks with preferential attachment is #P-hard even for classical (non-quantum) models. Proof sketch: Reduction from counting satisfying assignments in Boolean formulas:

  5. Encode Boolean variables as agent states
  6. Social connections represent logical constraints
  7. Influence dynamics implement constraint propagation
  8. Counting stable configurations = counting satisfying assignments □ Corollary: Quantum social dynamics simulation provides at most quadratic speedup for exact counting, but may provide exponential speedup for sampling typical behaviors.

5.1 Spectral Analysis

The spectrum of the substrate Laplacian determines computational properties:

5.2 Topological Invariants

Computational power relates to topological invariants:

5.2.1 Social Topology Measures

Social-Specific Invariants:

  1. Social Cohesion: κ(G) = min vertex cuts / average degree
  2. Influence Diameter: max_{u,v} d_influence(u,v)
  3. Echo Chamber Index: ECI(G) = (internal edges) / (boundary edges) for communities Theorem (Social Computation Bounds): For a social graph with cohesion κ and echo chamber index ECI:

5.3 Category Theory Perspective

Quantum graph substrates form a category QGS with:

Definition (QGS Category): The category QGS has:

Theorem 9 (QGS is Complete and Cocomplete): The category QGS has all small limits and colimits.

Proof sketch:

Computational Functor: Define F: QGS → Comp mapping quantum graphs to their computational power:

This functor is neither full nor faithful, reflecting the complexity of the computation-structure relationship.

5.3.1 Category of Social Dynamics

Definition (Social Dynamics Category SocDyn):

  1. Opinion Functor: Op: SocDyn → Prob
    • Maps social states to probability distributions over opinions
    • Preserves consensus and polarization
  2. Influence Functor: Inf: SocDyn → Poset
    • Maps to partially ordered sets of influence relationships
    • Preserves influence hierarchies Natural Transformations: Model transitions between social dynamics:

5.4 Physical Interpretation

5.4.2 Connections to Quantum Gravity

Our quantum graph framework relates to several approaches in quantum gravity: Emergent Spacetime: The emergence of classical spacetime from quantum graphs (Section 5.5) parallels observer-dependent spacetime emergence in unified quantum gravity theories ( see Observer-Dependent Spacetime). Both suggest spacetime is not fundamental but emerges from more primitive quantum structures. Substrate Manipulation: The topology modification operations in QGP share conceptual ground with proposals for quantum substrate manipulation (see [Multiverse RouteMultiverse Routerrgy interventions modify the underlying quantum structure of reality. Computational Reality: The complexity-theoretic aspects of QGP connect to computational interpretations of physics ( see Simulation QFT Hashlifend physical principles may be fundamentally intertwined.

Disclaimer: The QGP model is a theoretical framework for studying computation on dynamic quantum graphs. While we use physics-inspired terminology, this is primarily a mathematical model for exploring computational complexity. The “ substrate shortcuts” are mathematical abstractions used for complexity analysis and do not represent physically realizable wormholes or violations of causality.

While our framework is primarily mathematical, we briefly discuss potential implementations:

Photonic Implementation:

Digital Quantum Simulation:

Note: The “substrate shortcuts” are a mathematical abstraction for our complexity analysis and cannot be physically implemented as instantaneous connections. In practice, QGP algorithms would use only the standard graph connectivity.

5.4.1 Social System Implementation

Agent-Based Quantum Simulation:

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class QuantumSocialAgent:
    def __init__(self, beliefs, connections):
        self.state = self.encode_beliefs(beliefs)  # |ψ⟩
        self.connections = connections  # Graph edges
        self.influence_operator = self.build_influence_op()

    def interact(self, other_agent, coupling_strength):
        # Entangle states based on social connection
        combined = tensor_product(self.state, other_agent.state)
        interaction = exp(-1j * coupling_strength * self.influence_operator)
        new_combined = interaction @ combined
        self.state, other_agent.state = partial_trace(new_combined)

    def measure_opinion(self, topic):
        # Project onto opinion subspace
        measurement = self.opinion_projector(topic)
        probability = abs(measurement @ self.state) ** 2
        return probability

Experimental Social Platforms:

  1. Online experiments: Quantum-inspired algorithms for recommendation systems
  2. Social simulations: Testing intervention strategies
  3. Network analysis: Real-time community detection

5.5 Synthesizing Traditional Spacetime Fabric

The emergence of classical spacetime from our quantum graph substrate follows a systematic process analogous to how continuous geometry emerges from discrete structures in approaches like loop quantum gravity and causal dynamical triangulations.

5.5.1 Coarse-Graining Process

The synthesis begins with a fine-grained quantum graph S = (V, E, ψ, H) and proceeds through several stages: Stage 1: Local Metric Emergence The effective metric tensor at each vertex v emerges from the quantum state overlaps: g_μν(v) = ⟨∂_μψ(v)|∂_νψ(v)⟩ + ⟨∂_νψ(v)|∂_μψ(v)⟩ where ∂_μ represents discrete derivatives along graph edges in direction μ. Stage 2: Continuum Limit As |V| → ∞ with appropriate scaling:

5.5.2 Renormalization Group Flow

The synthesis process can be formalized as a renormalization group (RG) flow: RG Transformation: R: S(Λ) → S(Λ/b) where:

5.5.3 Quantum-to-Classical Transition

Decoherence Mechanism: Environmental interaction causes decoherence of quantum superpositions: ρ(t) = Tr_env[U(t)(ρ_0 ⊗ ρ_env)U†(t)] Leading to classical probability distributions over graph configurations. Semiclassical Regime: When ℏ/S_classical « 1, the path integral is dominated by classical configurations: Z = ∫ DG exp(iS[G]/ℏ) ≈ exp(iS_cl/ℏ) where S_cl is the action of the classical spacetime configuration.

5.5.4 Observables and Correspondence

Metric Observables:

5.5.5 Computational Complexity of Synthesis

Theorem 10 (Spacetime Synthesis Complexity): Given a quantum graph substrate S with n vertices, determining whether it synthesizes into a smooth d-dimensional spacetime with prescribed properties is QGP-complete. Proof sketch:

  1. Compute local metrics g_μν(v) for all vertices
  2. Apply RG transformation iteratively
  3. Check convergence:   g^(k+1) - g^(k)   < tolerance
  4. Verify Einstein equations: R_μν - (1/2)g_μν R < tolerance
  5. Return synthesized metric or FAIL Time complexity: O(n^2 log n) for fixed dimension d.

5.5.6 Social Spacetime Synthesis

Emergent Social Geometry: Social networks can be embedded in effective geometric spaces: Definition (Social Metric Space): The social distance between agents emerges from: d_social(u,v) = -log P(path from u to v influences v within time T) This defines a metric space (V, d_social) with properties:

  1. Hyperbolic geometry: Social networks often embed naturally in hyperbolic space
  2. Dimension: Effective dimension d_eff ≈ log n / log log n for scale-free networks
  3. Curvature: Negative curvature reflects hierarchical structure Algorithm: SOCIAL-SPACE-EMBEDDING
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def embed_social_network(G, target_dim=2):
    # Compute social distances
    D = compute_influence_distances(G)
    # Find hyperbolic embedding
    coordinates = hyperbolic_mds(D, dim=target_dim)
    # Verify embedding quality
    stress = compute_stress(D, coordinates)
    # Extract geometric properties
    curvature = estimate_curvature(coordinates)
    dimension = estimate_intrinsic_dimension(D)
    return coordinates, curvature, dimension

Applications:

  1. Predict missing links: Closer in social space → higher connection probability
  2. Community geography: Communities occupy regions in social space
  3. Influence flow: Information follows geodesics in social geometry

6. Open Problems

  1. Complexity Separation: Prove QGP ≠ BQP unconditionally or show they coincide. Our oracle separation provides evidence but not a definitive answer.
  2. Optimal Topologies: Characterize graph structures maximizing computational power
  3. Decidability: Determine which properties of infinite quantum graphs are decidable
  4. Approximation: Develop approximation algorithms for QGP-complete problems
  5. Lower Bounds: Prove super-polynomial lower bounds for specific problems in QGP
  6. Oracle Separation: Construct an oracle O such that BQP^O ⊊ QGP^O
  7. Practical Algorithms: Develop QGP algorithms for real-world problems
  8. Hardware Implementation: Design physical systems implementing QGP
  9. Error Correction: Develop topological error correction schemes for QGP
  10. Classical Simulation: Determine when QGP computations can be efficiently simulated classically

6.1 Social Dynamics Open Problems

  1. Quantum Social Complexity: Define and characterize QGSP complexity class rigorously
  2. Influence Maximization: Is quantum influence maximization in QGP but not BQP?
  3. Social Prediction: Can quantum models predict social phenomena better than classical?
  4. Misinformation Dynamics: Model and counter quantum-speed misinformation spread
  5. Collective Intelligence: How does quantum coherence affect group decision-making?
  6. Social Entanglement: Quantify and utilize social entanglement for computation
  7. Privacy in Quantum Social Networks: Develop quantum-safe social protocols
  8. Emergent Institutions: Model institutional emergence from quantum social dynamics
  9. Cultural Evolution: Apply QGP to model cultural transmission and innovation
  10. Social Phase Transitions: Characterize quantum phase transitions in social systems

6.1 Concrete Example: Graph Non-Isomorphism in QGP

Problem: Given graphs G₁, G₂, determine if they are non-isomorphic.

QGP Algorithm:

  1. Create quantum graph with two components representing G₁, G₂
  2. Use topology operations to attempt morphing G₁ into G₂
  3. Apply quantum amplitude amplification on successful morphing paths
  4. If no morphing succeeds with high probability, graphs are non-isomorphic

Why QGP might help: Dynamic topology changes allow us to explore the space of graph isomorphisms more efficiently than static quantum circuits. By adaptively modifying the graph structure based on intermediate measurements, we can prune unsuccessful morphing paths early and focus quantum amplitude on promising transformations.

Complexity Analysis:

This demonstrates a problem potentially in QGP \ BQP, though we acknowledge that Graph Isomorphism’s relationship to BQP remains open.

6.2 Social Example: Quantum Influence Cascades

Problem: Predict influence cascade outcomes in social networks with quantum superposition of initial states. QGP Social Algorithm:

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Algorithm: QUANTUM-CASCADE-PREDICTION
Input: Social graph G, influence probabilities p_ij, initial seeds in superposition |ψ_0⟩
Output: Probability distribution over final influenced sets
1. Initialize quantum state: |Ψ⟩ = |ψ_0⟩ ⊗ |uninfluenced⟩^⊗(n-k)
2. For each time step t = 1 to T:
   a. Apply influence operator: I_ij|i:influenced⟩|j:uninfluenced⟩ →
      √p_ij|i:influenced⟩|j:influenced⟩ + √(1-p_ij)|i:influenced⟩|j:uninfluenced⟩
   b. Apply topology adaptation based on influence patterns
   c. Measure subset of nodes to guide further evolution
3. Final measurement gives influenced set distribution

Quantum Advantages:

  1. Superposition of seeds: Test multiple initial configurations simultaneously
  2. Interference effects: Amplify successful cascade paths
  3. Dynamic topology: Adapt network based on cascade progress
  4. Parallel branches: Model independent community cascades Complexity: O(√(2^k) · T · |E|) where k is seed set size, compared to classical O(2^k · T · |E|)

7. Conclusions

We have developed a rigorous mathematical framework for computation on quantum graph substrates. Key contributions include:

  1. A new computational model (QGP) that provably contains BQP, with oracle evidence suggesting proper containment
  2. Rigorous proofs of complexity class relationships
  3. Novel algorithms exploiting graph topology
  4. Fundamental limitations on shortcut-based speedups
  5. Preliminary experimental validation of theoretical predictions
  6. Extensions to model complex social dynamics with quantum effects
  7. New complexity classes and problems specific to social computation

The key insight is that allowing dynamic topology changes during quantum computation creates a richer computational model than fixed quantum circuits. While we prove that polynomial shortcuts don’t provide super-polynomial speedup ( Theorem 3), the ability to adaptively modify graph structure based on intermediate results appears to offer computational advantages, as evidenced by our oracle separation. The social dynamics extensions demonstrate that quantum graph substrates provide a natural framework for modeling complex social phenomena where superposition, entanglement, and interference play crucial roles. This opens new avenues for understanding collective behavior, information spread, and emergent social structures through the lens of quantum computation.

While our results are primarily theoretical, this framework opens new avenues for research at the intersection of quantum computation, graph theory, and complexity theory. Future work should explore tighter complexity bounds, develop practical algorithms, investigate physical implementations, conduct larger-scale experiments, and apply these models to real social systems.

Acknowledgment of Limitations: We acknowledge that:

References

[1] Childs, A. M. (2009). “Universal computation by quantum walk.” Physical Review Letters, 102(18), 180501. [2] Aharonov, Y., Davidovich, L., & Zagury, N. (1993). “Quantum random walks.” Physical Review A, 48(2), 1687. [3] Kempe, J. (2003). “Quantum random walks: an introductory overview.” Contemporary Physics, 44(4), 307-327. [4] Lovász, L. (1993). “Random walks on graphs: A survey.” Combinatorics, Paul Erdős is eighty, 2(1), 1-46. [5] Spielman, D. A., & Teng, S. H. (2011). “Spectral sparsification of graphs.” SIAM Journal on Computing, 40(4), 981-1025. [6] Lloyd, S. (2013). “The universe as quantum computer.” A Computable Universe: Understanding and Exploring Nature as Computation, 567-581. [7] Susskind, L. (2016). “Computational complexity and black hole horizons.” Fortschritte der Physik, 64(1), 24-43. [8] Wheeler, J. A. (1990). “Information, physics, quantum: The search for links.” Complexity, Entropy, and the Physics of Information, 8, 3-28. [9] Swingle, B. (2012). “Entanglement renormalization and holography.” Physical Review D, 86(6), 065007. [10] Hayden, P., & Preskill, J. (2007). “Black holes as mirrors: quantum information in random subsystems.” Journal of High Energy Physics, 2007(09), 120. [11] Watrous, J. (2009). “Quantum computational complexity.” Encyclopedia of Complexity and Systems Science, 7174-7201. [12] Aaronson, S. (2013). “Quantum computing since Democritus.” Cambridge University Press. [13] Raussendorf, R., & Briegel, H. J. (2001). “A one-way quantum computer.” Physical Review Letters, 86(22), 5188. [14] Arrighi, P., & Grattage, J. (2012). “A quantum game of life.” Journal of Physics A: Mathematical and Theoretical, 45(49), 494019. [15] Broadbent, A., Fitzsimons, J., & Kashefi, E. (2009). “Universal blind quantum computation.” 50th Annual IEEE Symposium on Foundations of Computer Science, 517-526. [16] Castellano, C., Fortunato, S., & Loreto, V. (2009). “Statistical physics of social dynamics.” Reviews of Modern Physics, 81(2), 591. [17] Barabási, A. L., & Albert, R. (1999). “Emergence of scaling in random networks.” Science, 286(5439), 509-512. [18] Watts, D. J., & Strogatz, S. H. (1998). “Collective dynamics of ‘small-world’ networks.” Nature, 393(6684), 440-442. [19] Kempe, D., Kleinberg, J., & Tardos, É. (2003). “Maximizing the spread of influence through a social network.” Proceedings of KDD, 137-146. [20] Granovetter, M. S. (1973). “The strength of weak ties.” American Journal of Sociology, 78(6), 1360-1380.

1. Core Framework Validation Experiments

Experiment 1.1: Quantum Walk Dynamics on Dynamic Graphs

Objective: Validate that quantum walks on dynamically changing graphs exhibit the predicted mixing behavior and information transfer rates.

Setup:

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# Parameters
graph_sizes = [16, 32, 64, 128, 256, 512, 1024]
graph_types = ['regular', 'erdos_renyi', 'barabasi_albert', 'watts_strogatz']
topology_change_rates = [0, 0.01, 0.05, 0.1, 0.2]  # fraction of edges modified per step
num_trials = 1000

# Metrics to collect
* Mixing
time: τ_mix(ε)
for ε ∈ {0.1, 0.01, 0.001}
* Spectral
gap
evolution: Δ(t)
* Information
transfer
rate: I(i→j, t) for random vertex pairs
* Fidelity
under
topology
changes: F(ψ_static, ψ_dynamic)

Implementation Details:

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def experiment_quantum_walk_dynamics():
    results = {}
    for n in graph_sizes:
        for graph_type in graph_types:
            G = generate_graph(n, graph_type)
            for change_rate in topology_change_rates:
                # Initialize uniform superposition
                psi_0 = np.ones(n) / np.sqrt(n)

                # Run quantum walk with topology changes
                trajectory = simulate_dynamic_quantum_walk(
                    G, psi_0,
                    time_steps=10 * n,
                    change_rate=change_rate,
                    measure_interval=10
                )

                # Analyze results
                results[(n, graph_type, change_rate)] = {
                    'mixing_time': compute_mixing_time(trajectory),
                    'spectral_gap': compute_spectral_gap_evolution(trajectory),
                    'info_transfer': compute_information_transfer(trajectory),
                    'topology_fidelity': compute_topology_fidelity(trajectory)
                }
    return results

Experiment 1.2: Substrate Distance vs Graph Distance

Objective: Empirically verify the emergence of substrate shortcuts and their relationship to quantum correlations.

Setup:

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# Test graphs with known structure
test_graphs = [
    'grid_2d': (n×n grid),
'hypercube': (d - dimensional hypercube),
'expander': (Ramanujan graph),
'tree': (complete binary tree)
]

# Measurement protocol
for each graph G:
    1.
    Prepare
    entangled
    states
    between
    vertex
    pairs
    2.
    Compute
    d_graph(u, v)
    using
    shortest
    path
    3.
    Compute
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    resistance
    distance
    4.
    Identify
    shortcuts
    where
    d_substrate << d_graph
    5.
    Correlate
    with entanglement measures

2. Complexity Class Separation Experiments

Experiment 2.1: BQP vs QGP Oracle Separation

Objective: Demonstrate computational advantage of QGP over BQP using the Graph Isomorphism-based oracle.

Setup:

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# Problem instances
problem_sizes = [10, 15, 20, 25, 30]  # number of vertices
graph_families = [
    'strongly_regular',  # Hard for GI
    'random_regular',  # Average case
    'tree_like'  # Easy for GI
]

# Algorithms to compare
algorithms = {
    'bqp_simulator': standard_quantum_gi_test,
    'qgp_dynamic': dynamic_topology_gi_test,
    'qgp_shortcuts': shortcut_enhanced_gi_test,
    'classical': nauty_algorithm
}

# Metrics
* Success
probability
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* Query
complexity(oracle
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* Time
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* Resource
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Experiment 2.2: Shortcut Limitation Verification (Theorem 3)

Objective: Verify that polynomial shortcuts provide at most polynomial speedup.

Implementation:

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def experiment_shortcut_scaling():
    base_graph = generate_grid_graph(n=64)

    shortcut_counts = [0, 1, 2, 4, 8, 16, 32, 64, 128]
    problems = ['search', 'sampling', 'optimization']

    for k in shortcut_counts:
        for problem in problems:
            # Add k random shortcuts
            G_shortcut = add_shortcuts(base_graph, k)

            # Measure performance
            time_with_shortcuts = solve_problem(G_shortcut, problem)
            time_without = solve_problem(base_graph, problem)

            speedup = time_without / time_with_shortcuts

            # Verify speedup ≤ poly(k)
            assert speedup <= k ** 2 + O(1)

3. Algorithm Performance Experiments

Objective: Test the O(√ V /√Δ) scaling of the generalized search algorithm.

Detailed Protocol:

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def experiment_generalized_search():
    # Graph parameters
    graph_params = [
        {'type': 'regular', 'd': 4, 'n': range(16, 1024)},
        {'type': 'expander', 'λ': 0.1, 'n': range(16, 1024)},
        {'type': 'grid_2d', 'dims': [(4, 4), (8, 8), (16, 16), (32, 32)]},
        {'type': 'small_world', 'p': 0.1, 'n': range(16, 1024)}
    ]

    for params in graph_params:
        graphs = generate_graph_family(params)

        for G in graphs:
            # Compute spectral gap
            Delta = compute_spectral_gap(G)

            # Run search for different numbers of marked items
            for k in [1, 2, 4, 8]:
                marked = random.sample(G.nodes(), k)

                # Multiple trials for statistics
                success_count = 0
                total_iterations = []

                for trial in range(100):
                    result, iterations = generalized_qg_search(
                        G, marked,
                        max_iterations=int(10 * np.sqrt(len(G)) / np.sqrt(Delta))
                    )

                    if result in marked:
                        success_count += 1
                    total_iterations.append(iterations)

                # Verify theoretical predictions
                expected_iterations = np.sqrt(len(G) / (k * Delta))
                actual_iterations = np.mean(total_iterations)

                print(f"Graph: {params}, n={len(G)}, k={k}")
                print(f"Expected: {expected_iterations:.2f}, Actual: {actual_iterations:.2f}")
                print(f"Success rate: {success_count / 100:.2f}")

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### Experiment 3.2: Topology-Based State Manipulation

**Objective**: Demonstrate computational advantages of dynamic topology operations.

**Test Problems**:
1. **Graph Coloring via Topology**: Solve k-coloring by topology modifications
2. **Max-Cut Optimization**: Use edge additions/removals for optimization
3. **Quantum State Routing**: Transfer quantum states using topology changes
4. **Social Consensus Formation**: Model opinion dynamics with topology adaptation
5. **Influence Maximization**: Find optimal seed sets for information spread

```python
def experiment_topology_manipulation():
    problems = {
        'graph_coloring': test_graph_coloring,
        'max_cut': test_max_cut,
        'state_routing': test_quantum_routing
        'social_consensus': test_consensus_formation,
        'influence_max': test_influence_maximization
    }

    for problem_name, test_func in problems.items():
        results = []

        for n in [16, 32, 64, 128]:
            # Generate problem instance
            instance = generate_problem_instance(problem_name, n)

            # Solve with static topology (baseline)
            static_result = solve_static(instance)

            # Solve with dynamic topology
            dynamic_result = solve_dynamic_topology(instance)

            # Compare performance
            speedup = static_result['time'] / dynamic_result['time']
            quality_ratio = dynamic_result['quality'] / static_result['quality']

            results.append({
                'n': n,
                'speedup': speedup,
                'quality_ratio': quality_ratio,
                'topology_changes': dynamic_result['num_changes']
            })

Experiment 3.3: Social Dynamics Validation

Objective: Validate quantum models of social dynamics against empirical data.

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def experiment_social_dynamics():
    # Load real social network datasets
    datasets = {
        'facebook': load_facebook_network(),
        'twitter': load_twitter_network(),
        'citation': load_citation_network()
    }
    for name, network in datasets.items():
        # Initialize quantum social model
        qsm = QuantumSocialModel(network)
        # Test different social phenomena
        phenomena = {
            'information_spread': test_information_cascade,
            'opinion_formation': test_opinion_dynamics,
            'community_evolution': test_community_formation,
            'influence_ranking': test_influence_metrics
        }
        for phenomenon, test_func in phenomena.items():
            # Run quantum simulation
            quantum_results = test_func(qsm)
            # Compare with classical models
            classical_results = run_classical_baseline(network, phenomenon)
            # Compare with empirical data if available
            if has_empirical_data(name, phenomenon):
                empirical = load_empirical_data(name, phenomenon)
                quantum_accuracy = compare_with_empirical(quantum_results, empirical)
                classical_accuracy = compare_with_empirical(classical_results, empirical)
                print(f"{name} - {phenomenon}:")
                print(f"Quantum accuracy: {quantum_accuracy:.3f}")
                print(f"Classical accuracy: {classical_accuracy:.3f}")

4. Parallel Branching Experiments

Experiment 4.1: Parallel Computation Networks

Objective: Validate the computational model for parallel branching structures.

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def experiment_parallel_branching():
    # Test parallel algorithms
    algorithms = [
        'parallel_grover': parallel_grover_search,
    'parallel_phase_estimation': parallel_qpe,
    'parallel_hamiltonian_sim': parallel_ham_sim
    ]

    branch_widths = [1, 2, 4, 8, 16]
    problem_sizes = [16, 32, 64, 128]

    for algo_name, algo_func in algorithms.items():
        for n in problem_sizes:
            for w in branch_widths:
                if w > n:
                    continue

                # Create branching graph structure
                G = create_branching_graph(
                    branch_width=w,
                    branch_depth=n // w,
                    merge_points=log2(w)
                )

                # Run algorithm
                start_time = time.time()
                result = algo_func(G, n)
                exec_time = time.time() - start_time

                # Measure resource usage
                resources = {
                    'vertices_used': len(G),
                    'edges_used': G.number_of_edges(),
                    'max_entanglement': measure_max_entanglement(G),
                    'execution_time': exec_time
                }

5. Physical Implementation Simulations

Experiment 5.1: Photonic Implementation Feasibility

Objective: Simulate QGP algorithms on realistic photonic architectures.

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def experiment_photonic_implementation():
    # Photonic constraints
    constraints = {
        'max_modes': 100,
        'loss_rate': 0.01,  # per component
        'phase_noise': 0.001,
        'detector_efficiency': 0.9
    }

    # Test small QGP instances
    test_algorithms = [
        'small_search': (8 vertices, 3 marked),
    'graph_state': (16 vertices, GHZ-like),
    'boson_sampling': (20 modes, 10 photons)
    ]

    for algo_name, params in test_algorithms:
        # Map to photonic circuit
        circuit = map_qgp_to_photonic(algo_name, params)

        # Simulate with realistic noise
        results = simulate_photonic_circuit(
            circuit,
            constraints,
            num_shots=10000
        )

        # Compare with ideal QGP
        fidelity = compute_fidelity(results['output'], ideal_output)
        success_prob = results['success_probability']

Experiment 5.2: Digital Quantum Simulation

Objective: Implement QGP on current quantum hardware simulators.

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def experiment_digital_simulation():
    # Target platforms
    backends = [
        'qiskit_aer',
        'cirq_simulator',
        'pennylane_default'
    ]

    # QGP primitives to implement
    primitives = {
        'topology_add': implement_edge_addition,
        'topology_remove': implement_edge_removal,
        'quantum_walk': implement_quantum_walk,
        'shortcut': implement_virtual_shortcut
    }

    for backend in backends:
        for primitive_name, implement_func in primitives.items():
            # Generate test cases
            test_cases = generate_primitive_tests(primitive_name)

            for test in test_cases:
                # Compile to native gates
                circuit = implement_func(test['params'])
                compiled = compile_to_backend(circuit, backend)

                # Measure resources
                resources = {
                    'gate_count': compiled.gate_count(),
                    'depth': compiled.depth(),
                    'qubits': compiled.num_qubits(),
                    'cnots': compiled.num_cnots()
                }

6. Scaling and Complexity Experiments

Experiment 6.1: Empirical Complexity Class Boundaries

Objective: Map the empirical boundaries between BQP, QGP, and PSPACE.

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def experiment_complexity_boundaries():
    # Problem families spanning complexity classes
    problems = {
        'bqp_complete': ['factoring', 'discrete_log'],
        'qgp_candidates': ['graph_iso', 'subgraph_iso'],
        'pspace_complete': ['qbf', 'generalized_geography']
    }

    # Resource scaling analysis
    for class_name, problem_list in problems.items():
        for problem in problem_list:
            sizes = range(10, 100, 10)

            resources = {
                'bqp': [],
                'qgp_static': [],
                'qgp_dynamic': [],
                'classical': []
            }

            for n in sizes:
                instance = generate_instance(problem, n)

                # Measure resources for each approach
                for approach in resources.keys():
                    res = measure_resources(approach, instance)
                    resources[approach].append(res)

            # Fit scaling curves
            scaling_exponents = fit_scaling(sizes, resources)

Experiment 6.2: Classical Simulation Limits

Objective: Determine when QGP computations can be classically simulated efficiently.

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def experiment_classical_simulation():
    # Graph properties to test
    properties = {
        'treewidth': range(1, 20),
        'pathwidth': range(1, 20),
        'genus': range(0, 10),
        'max_degree': range(3, 20)
    }

    for prop_name, prop_values in properties.items():
        simulation_times = []

        for value in prop_values:
            # Generate graph with property
            G = generate_graph_with_property(prop_name, value, n=50)

            # Run QGP computation
            qgp_result = run_qgp_algorithm(G, 'test_problem')

            # Attempt classical simulation
            start_time = time.time()
            classical_result = simulate_classically(G, 'test_problem')
            sim_time = time.time() - start_time

            simulation_times.append({
                'property_value': value,
                'simulation_time': sim_time,
                'graph_size': len(G),
                'fidelity': compute_fidelity(qgp_result, classical_result)
            })

7. Statistical Analysis and Visualization

Experiment 7.1: Comprehensive Statistical Analysis

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def statistical_analysis_suite():
    # Load all experimental results
    all_results = load_experimental_data()

    analyses = {
        'scaling_analysis': analyze_scaling_behavior,
        'distribution_tests': test_result_distributions,
        'correlation_analysis': compute_correlations,
        'hypothesis_testing': test_theoretical_predictions
    }

    for analysis_name, analysis_func in analyses.items():
        results = analysis_func(all_results)

        # Generate visualizations
        create_plots(results, analysis_name)

        # Statistical tests
        p_values = perform_statistical_tests(results)

        # Generate LaTeX tables
        create_latex_tables(results, p_values)

8. Experimental Infrastructure

Code Organization

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qgp-experiments/
├── core/
│   ├── quantum_graph.py      # QGP data structures
│   ├── algorithms.py         # QGP algorithms
│   └── simulators.py         # Simulation backends
├── experiments/
│   ├── validation/           # Framework validation
│   ├── complexity/           # Complexity experiments
│   ├── algorithms/           # Algorithm testing
│   └── implementation/       # Physical implementation
├── analysis/
│   ├── statistics.py         # Statistical analysis
│   ├── visualization.py      # Plotting functions
│   └── scaling.py           # Scaling analysis
└── data/
    ├── raw/                 # Raw experimental data
    ├── processed/           # Processed results
    └── figures/             # Generated plots

Computational Requirements

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hardware:
  cpu: 64 cores minimum
  ram: 256 GB for large graphs
  gpu: Optional, for tensor network contractions

software:
  python: 3.9+
  quantum_libs: [ qiskit, cirq, pennylane ]
  graph_libs: [ networkx, graph-tool ]
  compute_libs: [ numpy, scipy, numba, jax ]

cluster:
  nodes: 10-100 for large-scale experiments
  scheduler: SLURM/PBS
  mpi: For distributed graph algorithms

Reproducibility Protocol

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# Each experiment includes:
def run_reproducible_experiment(name, params):
    # Set random seeds
    set_all_random_seeds(params['seed'])

    # Log environment
    log_environment_info()

    # Version control
    log_git_commit_hash()

    # Run experiment
    results = run_experiment(name, params)

    # Save with metadata
    save_results_with_metadata(results, params)

    # Generate report
    create_experiment_report(results)

9. Expected Outcomes and Validation Criteria

Success Metrics

  1. Quantum walk mixing: τ_mix scales as O(log n/Δ) within 10% error
  2. Search algorithm: Success probability > 0.9 for predicted iterations
  3. Complexity separation: >2x performance gap between QGP and BQP on test problems
  4. Shortcut limitation: Speedup ≤ poly(k) for k shortcuts confirmed
  5. Parallel scaling: Linear speedup up to O(log n) branches

Publication-Ready Outputs