quantum-cut-sparsification

name	quantum-cut-sparsification
description	Hamiltonian sparsification methodology for Quantum Cut (QC) Hamiltonians. Achieves O(n) term sparsification in n-qubit systems while preserving energy of every state. Uses invariant subspace decomposition and expander graph techniques.
tags	["quantum","sparsification","hamiltonian","graph-algorithms","expander-decomposition","spectral-approximation"]
related_skills	["quantum-algorithm-framework-designer","quantum-hamiltonian-learning-long-times"]

Quantum Cut Sparsification

Context

Hamiltonian sparsification is a key technique for reducing the complexity of quantum simulations. The Quantum Cut (QC) Hamiltonians are widely studied but suffer from exponential term counts that make simulation intractable.

Core Methodology

1. Main Result

In an n-qubit system, any n-qubit QC Hamiltonian can be sparsified to O(n) many terms while preserving the energy of every state up to a multiplicative factor.

2. Importance Sampling Scheme

The result provides an importance sampling scheme for graph edges such that:

• The Kikuchi graph at level ℓ of the sampled graph is a spectral approximation to the Kikuchi graph of the original graph G
• The same sampling scheme works simultaneously for all levels ℓ

3. Why Leverage Score Sampling Fails

Standard leverage score sampling (analyzed via matrix concentration inequalities) yields polynomially worse bounds because the underlying matrices have dimension 2^n — exponential in qubit count.

4. Invariant Subspace Decomposition

The approach decomposes the action of these matrices into invariant subspaces, avoiding the exponential dimension problem.

5. Expander Graph Extension

Using an operator-valued inequality of Alon and Kozma (building on the octopus inequality of Caputo, Liggett, and Richthammer):

• Extends sparsification technique to all expander graphs
• Invokes expander decomposition to extend sparsifier to all graphs

Implementation Steps

1. Identify the QC Hamiltonian structure in your problem
2. Apply importance sampling to Hamiltonian terms
3. Decompose action into invariant subspaces
4. Apply operator-valued inequalities for spectral approximation bounds
5. For non-expander graphs: use expander decomposition as preprocessing
6. Verify energy preservation on test states

Pitfalls

• Leverage score trap: Standard matrix concentration fails due to 2^n dimension — must use invariant subspace approach
• Octopus inequality requirement: The Alon-Kozma inequality requires specific graph properties
• Expander decomposition overhead: Converting general graphs to expanders adds preprocessing cost
• Energy preservation factor: The sparsified Hamiltonian only approximately preserves energies — the factor depends on sampling quality

Verification

1. Test on small n-qubit QC Hamiltonians (n ≤ 10)
2. Compare ground state energies: original vs sparsified
3. Verify spectral approximation quality on Kikuchi graphs
4. Measure term count reduction: should achieve O(n) scaling

Activation

quantum cut sparsification, Hamiltonian sparsification, Kikuchi graph, expander decomposition, spectral approximation, quantum simulation, invariant subspaces, arxiv:2606.09728