[Paper Review] Classical symmetries and QAOA

Ruslan Shaydulin, Stuart Hadfield+2 more|arXiv (Cornell University)|Dec 8, 2020

Quantum Computing Algorithms and Architecture60 references15 citations

TL;DR

This paper establishes a formal link between classical symmetries of an objective function and the quantum dynamics of the QAOA, showing that symmetries in the classical problem lead to invariant measurement probabilities in QAOA regardless of parameters or circuit depth. It demonstrates that graph symmetry properties alone can predict the minimum QAOA depth needed for target MaxCut approximation ratios under linear parameter schedules.

ABSTRACT

We study the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the underlying symmetries of the objective function to be optimized. Our approach formalizes the connection between quantum symmetry properties of the QAOA dynamics and the group of classical symmetries of the objective function. The connection is general and includes but is not limited to problems defined on graphs. We show a series of results exploring the connection and highlight examples of hard problem classes where a nontrivial symmetry subgroup can be obtained efficiently. In particular we show how classical objective function symmetries lead to invariant measurement outcome probabilities across states connected by such symmetries, independent of the choice of algorithm parameters or number of layers. To illustrate the power of the developed connection, we apply machine learning techniques towards predicting QAOA performance based on symmetry considerations. We provide numerical evidence that a small set of graph symmetry properties suffices to predict the minimum QAOA depth required to achieve a target approximation ratio on the MaxCut problem, in a practically important setting where QAOA parameter schedules are constrained to be linear and hence easier to optimize.

Motivation & Objective

• To formalize the relationship between classical symmetries of an objective function and quantum symmetry properties in QAOA dynamics.
• To identify how classical symmetries constrain QAOA measurement outcome probabilities independently of algorithm parameters or circuit depth.
• To investigate whether symmetry properties of graphs can serve as predictive features for QAOA performance, particularly in constrained optimization settings.
• To evaluate the effectiveness of machine learning models trained on symmetry features for predicting the minimum QAOA depth required to achieve a target approximation ratio on MaxCut.
• To demonstrate practical utility of symmetry-based prediction in real-world QAOA applications with linear parameter schedules.

Proposed method

• Formalizing the connection between the group of classical symmetries of the objective function and the quantum symmetry properties of the QAOA unitary evolution.
• Deriving that measurement outcome probabilities remain invariant under transformations induced by classical symmetries, regardless of QAOA parameters or number of layers.
• Extracting graph symmetry features (e.g., automorphism group properties) as input to machine learning models for QAOA performance prediction.
• Training supervised machine learning models on symmetry features to predict the minimum number of QAOA layers required to reach a specified MaxCut approximation ratio.
• Applying the models in a constrained setting where QAOA parameter schedules are restricted to linear functions, simplifying optimization and enhancing practicality.

Experimental results

Research questions

• RQ1 How do classical symmetries of the objective function influence the quantum dynamics and measurement outcomes in QAOA?
• RQ2 Can symmetry properties of a graph’s automorphism group predict the minimum QAOA depth required to achieve a target approximation ratio for MaxCut?
• RQ3 To what extent do symmetry-based features outperform other features in predicting QAOA performance under linear parameter schedules?
• RQ4 Does the invariance of measurement probabilities under classical symmetries hold independently of QAOA parameters and circuit depth?
• RQ5 Can machine learning models trained on symmetry features generalize across different MaxCut instances with varying symmetry structures?

Key findings

• Classical symmetries of the objective function induce invariance in QAOA measurement outcome probabilities, regardless of the number of layers or choice of parameters.
• A small set of graph symmetry properties—specifically those related to the automorphism group—suffices to predict the minimum QAOA depth required for a target MaxCut approximation ratio.
• Numerical evidence shows that symmetry-based machine learning models achieve strong predictive performance even under the constraint of linear QAOA parameter schedules.
• The invariance of measurement probabilities under symmetry transformations holds universally across all QAOA parameter values and circuit depths.
• The proposed symmetry-based approach enables efficient prediction of QAOA performance without requiring full circuit optimization, reducing computational cost in practical settings.
• The results demonstrate that symmetry considerations can serve as a powerful, low-dimensional proxy for QAOA performance prediction in structured optimization problems.