Mathematical Foundations¶

This document explains the quantum mechanical and graph theoretical principles underlying QE-KGR.

Quantum Mechanics Fundamentals¶

Hilbert Space Representation¶

In QE-KGR, each node represents a quantum state \(|\psi\rangle\) in a complex Hilbert space \(\mathcal{H}\). The state vector can be expressed as:

\[|\psi\rangle = \sum_{i=0}^{d-1} \alpha_i |i\rangle\]

where:

\(\alpha_i \in \mathbb{C}\) are complex amplitudes
\(|i\rangle\) are orthonormal basis states
\(d\) is the Hilbert space dimension
Normalization: \(\sum_{i=0}^{d-1} |\alpha_i|^2 = 1\)

Density Matrix Formalism¶

For mixed quantum states, we use the density matrix representation:

\[\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|\]

where \(p_i\) are classical probabilities and \(\sum_i p_i = 1\).

Von Neumann Entropy¶

The entanglement entropy of a quantum state is measured using von Neumann entropy:

\[S(\rho) = -\text{Tr}(\rho \log_2 \rho) = -\sum_i \lambda_i \log_2 \lambda_i\]

where \(\lambda_i\) are the eigenvalues of the density matrix \(\rho\).

Quantum Entanglement in Graphs¶

Entanglement Tensors¶

Edges in QE-KGR represent quantum entanglement between nodes through tensor products. For nodes \(A\) and \(B\), the entangled state is:

\[|\Psi_{AB}\rangle = \sum_{i,j} T_{ij} |i\rangle_A \otimes |j\rangle_B\]

where \(T_{ij}\) is the entanglement tensor encoding the correlation structure.

Superposed Relations¶

Unlike classical graphs with single edge types, QE-KGR supports superposed relations:

\[|\text{edge}\rangle = \sum_k \beta_k |\text{relation}_k\rangle\]

where \(\beta_k\) are complex amplitudes for different relation types.

Entanglement Strength¶

The entanglement strength between two nodes is quantified by:

\[E(A,B) = \sqrt{\sum_{k} |\beta_k|^2}\]

Quantum Walks¶

Walk Operator¶

Quantum walks on entangled graphs use a unitary operator \(U = S \cdot C\) where:

Shift operator \(S\): Moves the walker along graph edges
Coin operator \(C\): Determines transition amplitudes

For an entangled graph, the coin operator incorporates quantum correlations:

\[C_{ij} = \sum_k \frac{\beta_k}{\sqrt{d_i}} e^{i\phi_k}\]

where \(d_i\) is the degree of node \(i\) and \(\phi_k\) are phase factors.

Walker State Evolution¶

The quantum walker state evolves according to:

\[|\psi(t+1)\rangle = U |\psi(t)\rangle\]

This enables interference effects that enhance or suppress certain paths.

Quantum Inference Algorithms¶

Grover-Enhanced Search¶

QE-KGR uses quantum amplitude amplification for subgraph discovery. The Grover operator is:

\[G = -H O H^{-1} (2|\psi\rangle\langle\psi| - I)\]

where:

\(H\) is the Hadamard operator creating uniform superposition
\(O\) is the oracle marking target nodes
The optimal number of iterations is \(\sim \frac{\pi}{4}\sqrt{\frac{N}{M}}\) for \(N\) total nodes and \(M\) target nodes

Interference-Based Link Prediction¶

Link prediction uses quantum interference between node states:

\[P(A \leftrightarrow B) = |\langle\psi_A|\psi_B\rangle|^2\]

Constructive interference (\(\text{Re}(\langle\psi_A|\psi_B\rangle) > 0\)) suggests strong correlation, while destructive interference suggests anticorrelation.

Query Processing in Hilbert Space¶

Query Vector Projection¶

Natural language queries are projected into the graph's Hilbert space:

Tokenization: Extract key terms from query text
Embedding: Map terms to quantum state vectors
Superposition: Combine term vectors with learned amplitudes
Normalization: Ensure unit norm in Hilbert space

Context Vector Entanglement¶

Query context is encoded as a quantum state that becomes entangled with node states:

\[|\text{context}\rangle = \sum_i \gamma_i |\text{concept}_i\rangle\]

The query-context entangled state is:

\[|\text{query-context}\rangle = |\text{query}\rangle \otimes |\text{context}\rangle\]

Decoherence and Measurement¶

Decoherence Model¶

Quantum coherence decays over time due to environmental interaction:

\[\rho(t) = e^{-\Gamma t} \rho(0) + (1 - e^{-\Gamma t}) \rho_{\text{mixed}}\]

where \(\Gamma\) is the decoherence rate and \(\rho_{\text{mixed}}\) is the maximally mixed state.

Measurement and Collapse¶

When queries are executed, quantum measurements collapse superposed states to classical outcomes according to the Born rule:

\[P(\text{outcome}_i) = |\langle i|\psi\rangle|^2\]

Complexity Analysis¶

Space Complexity¶

Classical graph: \(O(V + E)\) for \(V\) nodes and \(E\) edges
Quantum graph: \(O(V \cdot d^2 + E \cdot d^4)\) for Hilbert dimension \(d\)

Time Complexity¶

Quantum walk: \(O(\sqrt{V})\) speedup over classical random walk
Grover search: \(O(\sqrt{V})\) vs. \(O(V)\) classical search
Query processing: \(O(d^2 V)\) for Hilbert space projections

Practical Considerations¶

Hilbert Space Dimension¶

Choose Hilbert dimension based on:

Expressivity: Higher dimensions enable richer quantum states
Computational cost: Scales as \(O(d^2)\) for most operations
Memory usage: Density matrices require \(O(d^2)\) storage

Typical choices:

\(d = 2\): Binary quantum states (qubits)
\(d = 4\): Two-qubit states
\(d = 8, 16\): Multi-qubit systems for complex domains

Numerical Stability¶

Use double precision complex arithmetic
Regularize small eigenvalues in entropy calculations
Implement efficient tensor contractions with libraries like opt_einsum

References¶

Nielsen, M.A. & Chuang, I.L. "Quantum Computation and Quantum Information"
Kempe, J. "Quantum random walks: an introductory overview"
Grover, L.K. "A fast quantum mechanical algorithm for database search"
Bonner, E. et al. "Quantum graphs and their applications"