Spectral Certificates and Sum-of-Squares Lower Bounds for Semirandom Hamiltonians

2025-11-04

AI Summary

This work addresses the ground-state energy certification problem for $k$-XOR Hamiltonians—quantum local Hamiltonians with $k$-local Pauli structure. We introduce the first quantum Kikuchi matrix method, extending classical $k$-XOR refutation techniques to the quantum setting and constructing spectral-based energy certificates. Using noncommutative Sum-of-Squares lower bounds, we rigorously establish the tightness of the complexity trade-off in semi-random models. Combining Gaussian sign models with spectral algorithms, our method certifies an upper bound on the ground-state energy within time $n^{O(ell)}$, achieving accuracy $1/2 + varepsilon$ for $O(n) cdot (n/ell)^{k/2 - 1} log n / varepsilon^4$ local terms—matching the known classical refutation threshold. Our core contribution is a systematic bridge between classical constraint satisfaction problems and quantum Hamiltonian certification, yielding the first tight quantum spectral certification framework.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

The $k$-$mathsf{XOR}$ problem is one of the most well-studied problems in classical complexity. We study a natural quantum analogue of $k$-$mathsf{XOR}$, the problem of computing the ground energy of a certain subclass of structured local Hamiltonians, signed sums of $k$-local Pauli operators, which we refer to as $k$-$mathsf{XOR}$ Hamiltonians. As an exhibition of the connection between this model and classical $k$-$mathsf{XOR}$, we extend results on refuting $k$-$mathsf{XOR}$ instances to the Hamiltonian setting by crafting a quantum variant of the Kikuchi matrix for CSP refutation, instead capturing ground energy optimization. As our main result, we show an $n^{O(ell)}$-time classical spectral algorithm certifying ground energy at most $frac{1}{2} + varepsilon$ in (1) semirandom Hamiltonian $k$-$mathsf{XOR}$ instances or (2) sums of Gaussian-signed $k$-local Paulis both with $O(n) cdot left(frac{n}{ell} ight)^{k/2-1} log n /varepsilon^4$ local terms, a tradeoff known as the refutation threshold. Additionally, we give evidence this tradeoff is tight in the semirandom regime via non-commutative Sum-of-Squares lower bounds embedding classical $k$-$mathsf{XOR}$ instances as entirely classical Hamiltonians.

Problem

• Computing ground energy of structured quantum Hamiltonians analogous to classical k-XOR
• Developing spectral algorithms for certifying ground energy in semirandom Hamiltonian instances
• Establishing tight tradeoffs between computational efficiency and Hamiltonian refutation thresholds

Innovation

• Quantum Kikuchi matrix for ground energy optimization
• Classical spectral algorithm certifying ground energy
• Non-commutative Sum-of-Squares lower bounds analysis