Quantum Walks on Hypergraphs

What is a Quantum Walk?

A quantum walk is the quantum analog of a classical random walk. Instead of a particle randomly hopping between vertices, a quantum walker exists in a superposition of all positions simultaneously. Interference effects cause probability amplitudes to cancel on suboptimal paths and reinforce on optimal ones.

For search problems, this yields a quadratic speedup: finding a marked vertex among n vertices takes O(√n) steps instead of O(n). This is Grover's algorithm generalized to graphs.

Why Hypergraphs?

Standard graphs connect pairs of vertices. Hypergraphs connect any number of vertices in a single edge. This matters for finance:

Pairwise (Graph)

"AAPL correlates with MSFT"

Higher-Order (Hypergraph)

"When Fed raises rates, tech stocks, bonds, and gold move together"

The Eidos hypergraph captures these multi-way relationships between assets, issuers, venues, sectors, and macro indicators. Quantum walks on this structure can discover non-obvious correlations and optimal portfolio allocations.

The Algorithm

Our implementation combines continuous-time quantum walks with Grover's amplitude amplification:

1. Initialize uniform superposition over all vertices
2. Oracle flips phase of marked (target) vertices
3. Diffusion inverts amplitudes about the mean
4. Repeat steps 2-3 for O(√(N/M)) iterations
5. Measure to find marked vertices with high probability

For M marked vertices among N total, the algorithm finds a marked vertex with probability approaching 100% after approximately π/4 × √(N/M) iterations.

Implementation

Python Simulation

We classically simulate the quantum walk using matrix operations. The Hamiltonian is derived from the hypergraph's normalized Laplacian:

Python

from eidos.hypergraph.quantum_walk import quantum_walk_search

# Define oracle: find all INDICATOR vertices
def oracle(vertex):
    return vertex.type == VertexType.INDICATOR

# Run Grover search
result = quantum_walk_search(hypergraph, oracle)

print(f"Found {len(result.marked_vertices)} targets")
print(f"Success probability: {result.success_probability:.1%}")

Lean 4 Proofs

Core theorems are formalized in Lean 4 with Mathlib, providing mathematical guarantees:

Lean 4

-- Grover convergence: O(√(N/M)) iterations suffice
theorem grover_convergence
    (ψ₀ : QuantumState n) (hψ : ψ₀ = uniformSuperposition hn)
    (hM : 0 < G.numMarked) :
    ∃ k : ℕ,
      k ≤ ⌈π / 4 * √(n / G.numMarked)⌉₊ ∧
      ∑ i ∈ G.marked, (G.iterateN ψ₀ k hn).probability i ≥ 1/2

Practical Implications

1

Today: Verified Algorithm

Classical simulation proves correctness. Formal proofs guarantee the math works.

2

Near-term: Structure Discovery

Use quantum-inspired heuristics for portfolio optimization and risk clustering on classical hardware.

3

Future: Quantum Advantage

When fault-tolerant quantum computers arrive, this code ports directly for genuine √n speedup on large financial graphs.

References

Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. STOC. arXiv
Childs, A. M., et al. (2003). Exponential algorithmic speedup by a quantum walk. STOC. arXiv
Szegedy, M. (2004). Quantum speed-up of Markov chain based algorithms. FOCS. IEEE
Magniez, F., et al. (2011). Search via quantum walk. SIAM J. Comput. arXiv