Quantum Gravity Lab

Variational Quantum Eigensolver (VQE) simulations for four fundamental quantum gravity models.

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What This Is

A computational framework that encodes four quantum gravity Hamiltonians as qubit operators and solves them using the Variational Quantum Eigensolver (VQE). Ground truth is established via exact diagonalization (numpy.linalg.eigvalsh). All VQE results are validated against these exact eigenvalues.

This is not a claim of new physics — it is a systematic demonstration that discrete quantum gravity models can be faithfully encoded and solved on near-term quantum hardware.

The Four Models

#	Model	Qubits	Hamiltonian	Exact E₀	VQE E₀	Deviation
1	Spin Network Area Spectrum	6	Heisenberg intertwiner on cyclic graph	-2.662637	-2.420351	9.10%
2	LQC Big Bounce	4	FRW mini-superspace finite-diff Laplacian	-23.692806	-21.385583	9.74%
3	Regge / CDT Simplicial Gravity	6	Transverse-field Ising + exchange on ring	-3.434249	-3.405818	0.83%
4	Wheeler-DeWitt Equation	4	WDW 1D mini-superspace Laplacian	-11.155048	-11.130582	0.22%

Physics Background

1. Spin Network Area Spectrum (Loop Quantum Gravity)

The area operator in Loop Quantum Gravity has a discrete spectrum: $A = 8\pi\gamma\ell_P^2 \sum_i \sqrt{j_i(j_i+1)}$ (Rovelli & Smolin 1995). We encode the intertwiner Hilbert space as a Heisenberg model on a cyclic graph:

$$H = \beta \sum_{\langle i,j \rangle} \vec{\sigma}_i \cdot \vec{\sigma}_j + \frac{\beta^2}{2} \sum_i Z_i$$

where $\beta = 0.2375$ (Immirzi parameter). The ground state encodes the lowest area eigenvalue of the quantum geometry.

References:

• Rovelli, C. & Smolin, L. (1995). Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442, 593-622.
• Mielczarek, J. (2018). Spin Foam Vertex Amplitudes on Quantum Computer. arXiv:1810.07100
• Czelusta, G. & Mielczarek, J. (2021). Quantum Simulations of a Qubit of Space. Phys. Rev. D 103, 046001. arXiv:2003.13124

2. LQC Big Bounce (Loop Quantum Cosmology)

The quantum bounce replaces the Big Bang singularity at $\rho_c \sim 0.41\rho_{\text{Planck}}$. We discretize the FRW mini-superspace Wheeler-DeWitt equation on a grid with domain $[0, 2\sqrt{K/2\Lambda}]$:

$$H = -K \nabla^2 + \Lambda a^2$$

where $K$ = kinetic coupling and $\Lambda$ = cosmological constant. The ground-state wavefunction peaks away from $a=0$, confirming singularity avoidance.

References:

• Ashtekar, A., Pawlowski, T. & Singh, P. (2006). Quantum nature of the Big Bang: Improved dynamics. Phys. Rev. D 74, 084003.
• Ganguly, A., Behera, B.K. & Panigrahi, P.K. (2019). Demonstration of Minisuperspace Quantum Cosmology Using Quantum Computational Algorithms on IBM Quantum Computer. arXiv:1912.00298
• Bojowald, M. (2001). Absence of a singularity in loop quantum cosmology. Phys. Rev. Lett. 86, 5227.

3. Regge / CDT Simplicial Gravity

The Regge action discretizes gravity on a simplicial lattice. We encode it as a transverse-field Ising model:

$$H = -J \sum_{\langle i,j \rangle} Z_i Z_j - h \sum_i X_i + \kappa \sum_{\langle i,j \rangle} (X_i X_j + Y_i Y_j)$$

The deficit angle interactions (ZZ coupling) compete with quantum geometry fluctuations (X field). The ground state represents the most probable discrete spacetime configuration.

References:

• Regge, T. (1961). General relativity without coordinates. Nuovo Cimento 19, 558-571.
• Ambjørn, J., Jurkiewicz, J. & Loll, R. (2004). Emergence of a 4D world from causal quantum gravity. Phys. Rev. Lett. 93, 131301.
• Ferguson, C., Nasiri, S. & Sheridan, L. (2025). Dynamics of Discrete Spacetimes with Quantum-enhanced MCMC. arXiv:2506.19538

4. Wheeler-DeWitt Equation

The Wheeler-DeWitt equation $\hat{H}|\Psi\rangle = 0$ is the fundamental quantum constraint on spacetime geometry. We discretize the mini-superspace WDW Hamiltonian for a closed FRW universe with cosmological constant:

$$H = -k\nabla^2 + \Lambda a^2$$

with domain $[0, 2\sqrt{k/2\Lambda}]$. The Hartle-Hawking no-boundary solution is encoded in the ground state.

References:

• DeWitt, B.S. (1967). Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160, 1113-1148.
• Hartle, J.B. & Hawking, S.W. (1983). Wave function of the Universe. Phys. Rev. D 28, 2960.
• McGuigan, M. (2021). Quantum Computing for Inflationary, Dark Energy and Dark Matter Cosmology. arXiv:2105.13849

Technical Architecture

Hamiltonian Construction:
  Physics equations → Dense NumPy matrix → SparsePauliOp.from_operator()
  (No hardcoded Pauli strings — all verified Hermitian, roundtrip error < 1e-15)

Exact Ground Truth:
  numpy.linalg.eigvalsh(H_matrix) → exact eigenspectrum

VQE Strategy:
  ├── Aer Simulator
  │   ├── Statevector evaluation (exact ⟨ψ|H|ψ⟩, ~0.01s/eval)
  │   ├── COBYLA optimizer (Heisenberg models, multi-start)
  │   └── L-BFGS-B + spectral normalization (grid models, informed init)
  │
  └── IBM Hardware (EstimatorV2)
      ├── SPSA optimizer (noise-resilient)
      ├── hamiltonian.apply_layout() for qubit mapping
      └── 10-min timeout per job, 10 iteration cap

Ansatz: EfficientSU2 with 'ry','rz' rotations, 'circular' entanglement

Key Technical Findings

1. Spectral normalization is critical for grid-based Hamiltonians. The kinetic/potential scale imbalance (500×) prevents optimizer convergence without $H_{\text{norm}} = (H - E_{\text{shift}}) / E_{\text{scale}}$.

2. Informed initialization (Ry ≈ π/2, near-Hadamard states) dramatically outperforms random starts for Laplacian ground states — the ground state of $-\nabla^2 + V$ is close to uniform superposition.

3. L-BFGS-B converges 15× faster than COBYLA for normalized grid models (10s vs 165s), but COBYLA with multi-start is more robust for un-normalized Heisenberg-type Hamiltonians.

Quick Start

# Install dependencies
pip install qiskit qiskit-aer numpy scipy

# Run validation of all 4 models
python validate.py

# Run a single model
python -c "
from quantum_gravity_sim import run_gravity_simulation
result = run_gravity_simulation('wheeler_dewitt', 'aer_simulator',
                                 max_iterations=200, depth=3)
print(f'VQE E₀ = {result[\"ground_energy\"]:.6f}')
print(f'Exact E₀ = {result[\"exact_E0\"]:.6f}')
print(f'Deviation = {result[\"deviation_from_exact\"]:.2f}%')
"

IBM Hardware (requires IBM Quantum account)

from quantum_gravity_sim import run_gravity_simulation
result = run_gravity_simulation('spin_network_area', 'ibm_torino',
                                 shots=4096, max_iterations=10, depth=3)

Results

Pre-computed results are in results/. Each JSON file contains:

• ground_energy: VQE-optimized ground state energy
• exact_E0: Exact diagonalization ground truth
• deviation_from_exact: Percentage deviation
• convergence_history: Energy at each function evaluation
• comparison.spectrum: Lowest 4 eigenvalues
• comparison.spectral_gap: E₁ - E₀
• comparison.references: Published literature references
• physical_interpretation: Physics meaning of the result

Honest Caveats

• These are toy models (4-6 qubits). Real quantum gravity requires thousands of logical qubits with error correction.
• The "ground energies" are effective Hamiltonian eigenvalues, not direct physical observables of spacetime geometry.
• Agreement with exact diagonalization confirms the VQE framework works, not that new physics was discovered.
• No quantum advantage — classical exact diagonalization solves these in microseconds. The VQE demonstrates feasibility at small scale.
• IBM hardware results show 10-30% additional deviation due to decoherence and gate errors.

Where This Sits in the Literature

Work	Year	Qubits	Model	Our Extension
Ganguly et al. (arXiv:1912.00298)	2019	2-3	WDW mini-superspace	Extended to 4 qubits with L-BFGS-B
Mielczarek (arXiv:1810.07100)	2018	5	Spin foam vertex	Intertwiner ground state on 6 qubits
McGuigan (arXiv:2105.13849)	2021	2-4	WDW dark energy	Systematic exact-diag validation
Asaduzzaman et al. (arXiv:2311.17991)	2023	—	SYK on IBM	Different model, similar methodology
Ferguson et al. (arXiv:2506.19538)	2025	—	Causal set MCMC	We do Regge/CDT as Ising model

Our contribution: A systematic 4-model VQE comparison framework with honest exact-diagonalization validation — no published work combines all four quantum gravity directions (LQG, LQC, CDT, WDW) in a single benchmarking framework.

License

MIT

Part Of

QubitPage OS — Full quantum computing operating system at quantumos.qubitpage.com