[Paper Review] Local algorithms and the failure of log-depth quantum advantage on sparse random CSPs

Antares Chen, Neng Huang+1 more|arXiv (Cornell University)|Oct 2, 2023

Markov Chains and Monte Carlo MethodsMathematics3 citations

TL;DR

This paper constructs a classical local message-passing algorithm for sparse random constraint satisfaction problems (CSPs) at high clause density, showing it achieves the optimal satisfaction fraction among algorithms obstructed by the overlap gap property (OGP). The algorithm, derived from a dual stochastic control problem linked to an extended Parisi variational principle, proves that no quantum advantage exists for the Quantum Approximate Optimization Algorithm (QAOA) or bounded-depth quantum circuits up to log n depth, even in the average case.

ABSTRACT

We construct and analyze a message-passing algorithm for random constraint satisfaction problems (CSPs) at large clause density, generalizing work of El Alaoui, Montanari, and Sellke for Maximum Cut [arXiv:2111.06813] through a connection between random CSPs and mean-field Ising spin glasses. For CSPs with even predicates, the algorithm asymptotically solves a stochastic optimal control problem dual to an extended Parisi variational principle. This gives an optimal fraction of satisfied constraints among algorithms obstructed by the branching overlap gap property of Huang and Sellke [arXiv:2110.07847], notably including the Quantum Approximate Optimization Algorithm and all quantum circuits on a bounded-degree architecture of up to $ε\cdot \log n$ depth.

Motivation & Objective

• To close the gap in understanding quantum advantage for optimization on sparse random CSPs.
• To establish that classical local algorithms can achieve the theoretical limit under the overlap gap property (OGP).
• To demonstrate that quantum advantage fails even for log-depth quantum circuits on bounded-degree architectures.
• To extend prior work on message-passing and spin glass duality to general even-predicate CSPs.
• To provide a rigorous framework for optimal algorithm design in the presence of OGP obstructions.

Proposed method

• Develops a message-passing algorithm on sparse hypergraphs using state evolution to track message distributions.
• Solves a dual stochastic control problem to determine optimal message nonlinearities.
• Uses a sparse-to-dense connection to link random CSPs to mean-field Ising spin glasses.
• Applies a finite difference approximation to the continuous-time control problem, enabling discrete algorithm design.
• Employs pseudo-Lipschitz functions and concentration inequalities to prove algorithmic value concentration.
• Relies on Assumption 1 (technical condition on measure continuity) to ensure convergence and optimality.

Experimental results

Research questions

• RQ1 Can classical local algorithms achieve the theoretical performance limit in sparse random CSPs under the OGP?
• RQ2 Does the QAOA or log-depth quantum circuits offer any advantage over classical algorithms in this regime?
• RQ3 Can the extended Parisi variational principle be used to derive optimal message-passing rules for general even-predicate CSPs?
• RQ4 What is the role of the overlap gap property in obstructing both classical and quantum algorithms?
• RQ5 How do message-passing algorithms relate to spin glass models and stochastic control in the mean-field limit?

Key findings

• The proposed message-passing algorithm asymptotically achieves the optimal fraction of satisfied constraints among algorithms obstructed by the OGP.
• The algorithm’s performance matches the extended Parisi variational principle value, proving optimality within the OGP-obstructed class.
• Quantum advantage via QAOA or log-depth quantum circuits is ruled out for sparse random CSPs with even predicates.
• The algorithm’s value concentrates tightly around its mean, ensuring high-probability success as n → ∞.
• The method extends prior results on Max-Cut to general even-predicate CSPs using a unified framework based on stochastic control duality.
• The result holds under Assumption 1, which remains an open technical challenge to fully prove, though expected to hold.