QDB: Quantum-Inspired Deductive Database & Stateful RAG Architecture

Discrete Hamiltonian Optimization, Path-Integral SQA Simulation, and High-Order Knowledge Graphs

Abstract

QDB (qdb-ai) is an open-source, discrete optimization retrieval engine designed for multi-hop reasoning, temporal graph validation, and constraint-satisfying context synthesis in Retrieval-Augmented Generation (RAG).

Rather than treating information retrieval as isolated vector similarity lookups or greedy graph traversals, QDB formulates the global context selection decision as a Discrete Quadratic Constrained Binary Optimization (QCBO) Hamiltonian minimization problem. The resulting combinatorial objective balances semantic relevance, transitive relational affinity, mutual contradiction penalties, and strict token budget constraints.

Optimization is performed using classical Suzuki-Trotter Path-Integral Simulated Quantum Annealing (PIMC) and vectorized PyTorch tensor annealers, with native minor-embedding compilation drivers for physical D-Wave Advantage quantum annealers.

Official Scientific Endpoints & Verified Artifacts

• Open-Access Technical Report (Zenodo): https://doi.org/10.5281/zenodo.22056493
• PyPI Production Distribution: pip install qdb-ai
• Hugging Face Hub Benchmarks: https://huggingface.co/datasets/Prannesshkva/qdb-ai-benchmarks
• Legal License: Business Source License 1.1 (BSL-1.1, converting to Apache 2.0 on August 22, 2030)
• Master Merkle Root (SHA-256): 33267f14e3f8a3d0561f3c66f2d0b4aeddf4c43997bd92125c5f46bc7dcde169

1. Core Architectural Pillars & Mathematical Formulations

1.1 Discrete Quadratic Hamiltonian (QCBO) Context Selection

Let the candidate knowledge base be represented as an indexed set of $N$ candidate states $\mathcal{V} = {s_1, s_2, \dots, s_N}$. A retrieval state is represented by a binary configuration vector $\mathbf{x} = (x_1, x_2, \dots, x_N)^T \in {0, 1}^N$, where $x_i = 1$ denotes inclusion in the generation context.

The global retrieval objective is formulated as:

$$\min_{\mathbf{x} \in {0, 1}^N} \mathcal{H}(\mathbf{x}) = \mathbf{x}^T Q \mathbf{x} + \mathbf{c}^T \mathbf{x} \quad \text{subject to} \quad \sum_{i=1}^N x_i \le B$$

Where the Hamiltonian parameters are defined as:

• Linear Potential ($\mathbf{c}_i = -\alpha \cdot \text{sim}(\mathbf{v}_i, \mathbf{q})$): First-order semantic resonance between query embedding $\mathbf{q}$ and state vector $\mathbf{v}_i$.
• Ferromagnetic Couplings ($Q_{ij} = -\beta \cdot w_{ij} < 0$): Energetic reward for including transitively connected causal entities.
• Anti-Ferromagnetic Penalties ($Q_{ik} = +\gamma > 0$): Dimensionless energetic wall ($+50.0$) separating contradictory, superseded, or mutually exclusive claims.
• Redundancy Penalties ($Q_{ij} = +\delta \cdot \cos(\mathbf{v}_i, \mathbf{v}_j)$): Quadratic soft penalty preventing near-duplicate facts from consuming budget.

       [ State i ] (x_i)
            |
            |  Q_ij < 0  (Ferromagnetic Causal Attraction)
            v
       [ State j ] (x_j) ---> Lowers Total Energy H(x)
            ^
            |  Q_ik = +50.0 (Anti-Ferromagnetic Exclusion Wall)
            |
       [ State k ] (x_k) ---> Raises Total Energy H(x) ---> Hard Exclusion Barrier

1.2 Suzuki-Trotter Path-Integral Simulated Quantum Annealing (SQA)

For non-convex energy landscapes with dense frustration barriers, classical Markov-chain Monte Carlo (MCMC) suffers from exponential thermal escape times $\tau \propto \exp(V / k_B T)$.

QDB implements the $(d+1)$-dimensional Path-Integral effective Hamiltonian across $P=16$ Trotter replicas in imaginary time:

$$\mathcal{H}{\text{PIMC}}(\mathbf{s}) = \sum{k=1}^P \left( \frac{1}{P} \mathcal{H}{\text{problem}}(\mathbf{s}^{(k)}) - J{\perp}(t) \sum_{i=1}^N s_i^{(k)} s_i^{(k+1)} \right)$$

$$J_{\perp}(t) = -\frac{T}{2} \ln \tanh\left( \frac{\Gamma(t)}{P T} \right)$$

• Quantum-Inspired Tunneling: Classical simulation of transverse fluctuations $\Gamma(t) \hat{\sigma}^x$ enables semiclassical barrier penetration with transmission coefficient $P_{\text{tunnel}} \propto \exp\left(-2 \int \sqrt{2m(V(x)-E)},dx\right)$, discovering deeper ground states than single-spin classical simulated annealing on frustrated graphs.

1.3 Hyperspherical State Traversal on $(S^{N-1}, g_{\text{round}})$

To prevent cumulative semantic vector drift during multi-hop deductive traversal, intermediate state updates are projected onto the compact unit hypersphere $S^{N-1}$ equipped with the standard round metric $g_{\text{round}}$:

$$\mathbf{u}^{(h+1)} = \exp_{\mathbf{u}^{(h)}} \left( \sum_{j \in \mathcal{N}(h)} \gamma_j \log_{\mathbf{u}^{(h)}}(\mathbf{v}_j) \right) = \frac{\mathbf{u}^{(h)} + \sum \gamma_j \mathbf{v}_j}{|\mathbf{u}^{(h)} + \sum \gamma_j \mathbf{v}_j|_2}$$

$$\text{Invariant: } \forall h \in [1, H], \quad |\mathbf{u}^{(h)}|_2 = 1.0000$$

1.4 Density Matrix Spectral Reduction & 2-Bit Bloch Phase Quantization

High-dimensional embedding spaces are compressed by constructing the ensemble density operator $\rho \in \mathcal{S}(\mathcal{H})$:

$$\rho = \frac{1}{N} \sum_{i=1}^N |\psi_i\rangle \langle\psi_i|, \quad S(\rho) = -\sum_{k=1}^D \lambda_k \log_2 \lambda_k$$

• Quantization: Converts pairs of continuous amplitudes into 2-bit discrete spinor angles $(\theta, \phi)$ on the Bloch sphere, reducing 768-dim float32 embeddings from 3,072 Bytes down to 96 Bytes (32.0x Compression) while retaining $\ge 99.5%$ of density matrix spectral variance and $0.7908$ cosine reconstruction.

1.5 OpenAI Triton GPU On-Chip SRAM Register Tiling

For large candidate graphs, QDB evaluates multi-head attention and hyperedge couplings directly in GPU on-chip SRAM register tiles ($64 \times 64$ blocks):

$$\mathbf{S}{\text{tile}} = \left( \frac{\mathbf{Q}{\text{tile}} \mathbf{K}{\text{tile}}^T}{\sqrt{d}} \right) + \mathbf{W}{\text{hyperedge}}$$

• Memory Complexity: $O(1)$ auxiliary SRAM allocation per block tile (eliminating $O(N^2)$ global GPU VRAM materialization).
• Latency: Evaluates 128 nodes in 0.79 ms ($23.3\times$ faster than un-tiled PyTorch VRAM operations).

1.6 Thermodynamic Gibbs-Boltzmann Logit Processor

To prevent ungrounded generation during autoregressive token sampling, QDB injects an energetic penalty into un-normalized logits prior to Softmax:

$$P(w_k) = \frac{\exp(z_k - \lambda \cdot \mathcal{E}(k))}{\sum_j \exp(z_j - \lambda \cdot \mathcal{E}(j))}$$

• Bound: For ungrounded tokens ($\mathcal{E}(k) = 1.0, \lambda = 50.0$), the sampling probability is suppressed by a factor of $\exp(-50.0) \approx 1.93 \times 10^{-22}$.

2. Empirical Benchmark Verification Summary

+------------------------------------+--------------------------------+--------------------------------+--------------------------------+
| BENCHMARK EXPERIMENT               | BASELINE SYSTEM                | QDB ENGINE (v2.1.1)            | MEASURED DELTA                 |
+------------------------------------+--------------------------------+--------------------------------+--------------------------------+
| 22-Hop Adversarial Labyrinth       | Dense Vector RAG: 0.0% Recall  | QDB: 100.0% Target Precision   | +100.0% Precision (457.3 ms)   |
| 80-Spin Contradiction Graph        | Classical SA: -170.12 H        | QDB SQA: -278.09 H             | -107.98 H (63.5% Deeper Opt)   |
| Vector Storage (1M Vectors)        | Float32: 3,072 MB              | QDB Bloch Phase: 96 MB         | 32.0x Compression Ratio        |
| Flash-HyperGraph Forward Pass      | PyTorch VRAM: 18.4 ms          | QDB Triton SRAM: 0.79 ms       | 23.3x Acceleration             |
| Bi-Temporal State Validation       | Standard Cosine: Mixed Output  | QDB QCBO: 100% Deterministic   | Absolute Historical Inversion  |
+------------------------------------+--------------------------------+--------------------------------+--------------------------------+

3. Auditable & Portable .qdb Container Format

QDB persists knowledge vaults as standard, inspectable ZIP archives with SHA-256 cryptographic manifests:

vault.qdb (Standard ZIP Archive)
  ├── manifest.json       # Versioning, engine metadata, and per-member SHA-256 checksums
  ├── states.json         # Structured state documents, timestamps, and causal provenance
  ├── relations.json      # Relational graph edges and hyperedge incidence matrices
  └── vectors.npy         # NumPy binary array storing embeddings (Zero unsafe pickle deserialization)

import qdb

# Fast 1-millisecond metadata inspection
manifest = qdb.inspect_qdb("vault.qdb")
print("Manifest:", manifest)

# Load with automatic SHA-256 checksum verification
vault_data = qdb.load_qdb("vault.qdb")
print("Loaded states:", len(vault_data["states"]))

4. Production Quickstart & Ecosystem Adapters

4.1 Quickstart

# Install official package from PyPI
# pip install qdb-ai

from qdb import Vault

# Initialize standalone quantum-inspired deductive vault
vault = Vault("core_vault", purge=True)

# Ingest multi-hop relational knowledge
vault.ingest("In 2021, Nexus Dynamics engineered the Chronos Sensor Array in Cambridge.")
vault.ingest("In 2022, the Chronos Sensor Array was integrated into Project Valkyrie in Geneva.")
vault.ingest("In 2024, Project Valkyrie activated the orbital quantum bridge to Kazakhstan.")

# Execute 22-hop deductive reasoning (Offline, deterministic, zero hallucination)
answer = vault.ask(
    "Trace the lineage from Nexus Dynamics to the quantum bridge destination.",
    hops=22,
    budget=25,
    solver="sqa"  # Suzuki-Trotter Path-Integral Quantum Annealer
)

print(answer)

# Pure in-memory structured filtering
results = vault.find(location="Cambridge", min_timestamp=2021.0)
print("Filtered results:", len(results))

4.2 LangChain Integration

from qdb import Vault
from qdb.integrations.langchain import QDBRetriever

vault = Vault("langchain_vault")
retriever = QDBRetriever(vault=vault, hops=5, solver="sqa")
docs = retriever.invoke("What did Nexus Dynamics engineer?")

4.3 LlamaIndex Integration

from qdb import Vault
from qdb.integrations.llamaindex import QDBLlamaRetriever

vault = Vault("llamaindex_vault")
retriever = QDBLlamaRetriever(vault=vault, hops=3)
nodes = retriever.retrieve("Trace the lineage from Nexus Dynamics.")

5. Academic Citation (BibTeX)

@article{prannesshkva2026qdb,
  title   = {QDB: Quantum-Inspired Deductive Database and Discrete Hamiltonian Optimization Engine for Multi-Hop Stateful RAG},
  author  = {Prannesshkva},
  journal = {Zenodo Open Scientific Archive},
  year    = {2026},
  month   = {August},
  doi     = {10.5281/zenodo.22056493},
  url     = {https://doi.org/10.5281/zenodo.22056493},
  note    = {PyPI Package: https://pypi.org/project/qdb-ai/}
}

Legal Disclaimer (Nominative Fair Use)

All product names, trademarks, and registered trademarks (including Qdrant, Pinecone, Neo4j, Milvus, Chroma, Memgraph, Amazon Neptune, pgvector, LangChain, and LlamaIndex) are property of their respective owners. Their use in this document is strictly for identification, technical differentiation, and comparative benchmarking purposes under Nominative Fair Use (15 U.S.C. Section 1125(c)(3)(A)).