Evidence for effectively constant shot complexity in the quantum approximate optimization algorithm without per-instance optimization

Abstract

We study a modified fixed-point version of the Quantum Approximate Optimization Algorithm (fpQAOA), where parameters are trained classically on small instances and then transferred to larger problems. Our scheme combines three ingredients: (i) targeting approximate solutions via a prescribed approximation ratio (AR), (ii) scaling the circuit depth linearly with the problem size using a two-parameter sin-cos angle encoding, and (iii) normalizing QUBO Hamiltonians by their Frobenius norm. Noiseless numerical simulations (for system sizes up to 30 qubits) across a variety of random QUBO ensembles show that with these modifications the median number of quantum circuit runs ("shots") required to achieve AR=0.95 counterintuitively decreases towards a nearly constant value as the problem size increases, while the per-shot time remains polynomial. Extrapolation of this finite-size behavior is consistent with an effectively constant sampling complexity. Moreover, removing any single component of the scheme restores rapid growth of the required number of shots, highlighting the synergistic nature of the three modifications. These empirical findings suggest that fpQAOA, equipped with the proposed protocol, may achieve scalable approximate performance with polynomial-depth circuits for the considered problem classes.

arXiv comment13 pages, 7 figures

References in corpus (23)

• Barren plateaus in quantum neural network training landscapes
• Noisy intermediate-scale quantum (NISQ) algorithms
• Quantum algorithms: an overview
• Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices
• Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor
• QAOA for Max-Cut requires hundreds of qubits for quantum speed-up
• Quantum Approximate Optimization of the Long-Range Ising Model with a Trapped-Ion Quantum Simulator
• Barren Plateaus in Variational Quantum Computing
• Evidence of Scaling Advantage for the Quantum Approximate Optimization Algorithm on a Classically Intractable Problem
• Parameter Concentration in Quantum Approximate Optimization
• Counterdiabaticity and the quantum approximate optimization algorithm
• Parameter Transfer for Quantum Approximate Optimization of Weighted MaxCut
• Parameter Setting in Quantum Approximate Optimization of Weighted Problems
• Evaluation of QAOA based on the approximation ratio of individual samples
• Towards a Linear-Ramp QAOA protocol: Evidence of a scaling advantage in solving some combinatorial optimization problems
• Comparing Three Generations of D-Wave Quantum Annealers for Minor Embedded Combinatorial Optimization Problems
• Quantum Approximate Multi-Objective Optimization
• Barren plateaus are swamped with traps
• Transforming optimization problems into a QUBO form: A tutorial
• Optimisation-Free Recursive QAOA for the Binary Paint Shop Problem
• A Practically Scalable Approach to the Closest Vector Problem for Sieving via QAOA with Fixed Angles
• Experimental factoring integers using fixed-point-QAOA with a trapped-ion quantum processor
• Evaluating the Limits of QAOA Parameter Transfer at High-Rounds on Sparse Ising Models With Geometrically Local Cubic Terms