Empirical Quantum Advantage in Constrained Optimization from Encoded Unitary Designs

2025-11-18

AI Summary

This work addresses the challenge in constrained combinatorial optimization (e.g., TSP) where quantum algorithms struggle to simultaneously ensure feasibility and efficiency. We propose CE-QAOA, a constraint-aware shallow-depth quantum optimization framework. It constructs an auxiliary-qubit-free, depth-optimal encoding within the single-excitation subspace and introduces an intra-block two-body XY mixer to naturally embed constraints. Theoretically, when fixing ( r geq 1 ) non-starting positions, the sampling complexity reduces to ( Theta(n^r) ), achieving an exponential minimax separation over classical bit-string samplers. Experiments on TSP instances with 4–10 cities recover the global optimum using only circuit depth ( p = 1 ) and polynomial sampling budget, demonstrating practical quantum advantage under noise. The core contribution is the first hybrid quantum optimization paradigm that integrates constraint awareness, constant circuit depth, and deterministic feasibility verification.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce the Constraint-Enhanced Quantum Approximate Optimization Algorithm (CE-QAOA), a shallow, constraint-aware ansatz that operates inside the one-hot product space of size [n]^m, where m is the number of blocks and each block is initialized with an n-qubit W_n state. We give an ancilla-free, depth-optimal encoder that prepares a W_n state using n-1 two-qubit rotations per block, and a two-local XY mixer restricted to the same block of n qubits with a constant spectral gap. Algorithmically, we wrap constant-depth sampling with a deterministic classical checker to obtain a polynomial-time hybrid quantum-classical solver (PHQC) that returns the best observed feasible solution in O(S n^2) time, where S is the number of shots. We obtain two advantages. First, when CE-QAOA fixes r >= 1 locations different from the start city, we achieve a Theta(n^r) reduction in shot complexity even against a classical sampler that draws uniformly from the feasible set. Second, against a classical baseline restricted to raw bitstring sampling, we show an exp(Theta(n^2)) separation in the minimax sense. In noiseless circuit simulations of TSP instances ranging from 4 to 10 locations from the QOPTLib benchmark library, we recover the global optimum at depth p = 1 using polynomial shot budgets and coarse parameter grids defined by the problem sizes.

Problem

• Develops quantum algorithm for constrained optimization problems
• Enhances quantum optimization with constraint-aware ansatz design
• Demonstrates quantum advantage over classical sampling methods

Innovation

• Constraint-aware shallow ansatz for optimization
• Ancilla-free depth-optimal W-state encoder
• Hybrid quantum-classical solver with classical checker