Gap Amplification for Local Hamiltonians with Combinatorial Soundness

2026-10-04

AI Summary

This work addresses the long-standing absence of locality-preserving gap amplification methods for the quantum PCP conjecture, a limitation imposed by the quantum no-cloning theorem. We propose the first viable framework for quantum locality-preserving gap amplification. Rather than compromising locality, our approach enlarges the combinatorial gap by increasing qudit dimensions. By introducing fault-tolerant computation theory to analyze this process, we successfully transplant core classical PCP mechanisms into the quantum domain. The framework further integrates high-dimensional expanders, circuit-to-Hamiltonian constructions, and state-of-the-art quantum coding techniques. We rigorously prove that this framework effectively amplifies the combinatorial gap of local Hamiltonians, thereby opening a new pathway toward resolving the quantum PCP conjecture.

Abstract

The quantum PCP conjecture is one of the major open problems in quantum complexity theory. It has resisted attack in part because many primitives used in the proof of the classical PCP theorem, such as locality-preserving gap amplification and alphabet reduction, have no obvious quantum analogues due to quantum no-cloning. Locality-preserving gap amplification is a procedure that takes as input a local Hamiltonian problem instance and produces a new instance with a larger promise gap, without increasing the locality of the Hamiltonian, and instead moderately increasing its local qudit dimension. Obtaining this kind of control over the locality during gap amplification is critical to the success of many known strategies for proving the classical PCP theorem. In this work, we put forth the first known viable template for quantum locality-preserving gap amplification, and we prove that our procedure amplifies the combinatorial gap of local Hamiltonians. Our work introduces a new framework for reasoning about quantum gap amplification in terms of fault-tolerant computation, and illuminates a route toward importing one of the central ingredients in classical PCPs into the quantum setting. In particular, we build upon ideas from the recent classical PCP of Bafna--Minzer--Vyas based on high-dimensional expanders, and the circuit-to-Hamiltonian construction of Anshu--Breuckmann--Nguyen, in addition to several recent advances in quantum coding theory.

Problem

• Quantum PCP conjecture
• Local Hamiltonians
• Gap amplification
• Locality-preserving
• Combinatorial soundness

Innovation

• Quantum PCP conjecture
• Locality-preserving gap amplification
• Local Hamiltonians
• Fault-tolerant computation
• High-dimensional expanders