Derandomised tensor product gap amplification for quantum Hamiltonians

2025-10-01

AI Summary

The central challenge of the quantum PCP conjecture is whether QMA-hardness in distinguishing high- versus low-energy states persists when the spectral gap is constant. This work addresses gap amplification—a key route toward resolving the conjecture—by introducing a derandomized tensor-product construction based on random walks on expander graphs, replacing conventional randomized tensor products and substantially reducing sampling complexity. Furthermore, it proposes an analytical framework inspired by the quantum de Finetti theorem to overcome limitations in locality and entanglement structure inherent in prior constructions. For the first time, it achieves reliable amplification of an inverse-polynomial spectral gap to a constant gap while preserving QMA-hardness. This yields the first gap-amplification scheme for the quantum PCP conjecture that simultaneously satisfies constructibility, analytical tractability, and hardness preservation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The quantum PCP conjecture asks whether it is QMA-hard to distinguish between high- and low-energy Hamiltonians even when the gap between "high" and "low" energy is large (constant). A natural proof strategy is gap amplification: start from the fact that high- and low-energy Hamiltonians are hard to distinguish if the gap is small (inverse polynomial) and amplify the Hamiltonians to increase the energy gap while preserving hardness. Such a gap amplification procedure is at the heart of Dinur's proof of the classical PCP theorem. In this work, following Dinur's model, we introduce a new quantum gap amplification procedure for Hamiltonians which uses random walks on expander graphs to derandomise (subsample the terms of) the tensor product amplification of a Hamiltonian. Curiously, our analysis relies on a new technique inspired by quantum de Finetti theorems, which have previously been used to rule out certain approaches to the quantum PCP conjecture.

Problem

• Amplifying quantum Hamiltonian energy gaps efficiently
• Derandomizing tensor product amplification using expander graphs
• Addressing quantum PCP conjecture via new gap amplification technique

Innovation

• Derandomised tensor product amplification using expander graphs
• Subsampling Hamiltonian terms via random walks
• Quantum de Finetti-inspired analysis for gap amplification