Parameter Concentration in Quantum Approximate Optimization

TL;DR

This paper analytically and numerically demonstrates that optimal parameters in QAOA concentrate as an inverse polynomial with problem size, enabling more efficient training by using fewer qubits, which has practical significance.

Contribution

It provides the first analytical proof of parameter concentration in QAOA for depths p=1,2 and extends the understanding to higher depths with numerical verification.

Findings

Optimal QAOA parameters concentrate as an inverse polynomial in problem size.

Parameter concentration enables training on fewer qubits to approximate larger systems.

Numerical verification confirms the analytical results for higher depths.

Abstract

The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary quantum applications development. In QAOA, a quantum circuit is trained -- by repeatedly adjusting circuit parameters -- to solve a problem. Several recent findings have reported parameter concentration effects in QAOA and their presence has become one of folklore: while empirically observed, the concentrations have not been defined and analytical approaches remain scarce, focusing on limiting system and not considering parameter scaling as system size increases. We found that optimal QAOA circuit parameters concentrate as an inverse polynomial in the problem size, providing an optimistic result for improving circuit training. Our results are analytically demonstrated for variational state preparations at $p=1,2$ (corresponding to 2 and 4 tunable parameters respectively). The technique is also applicable for higher depths and the concentration effect is cross verified numerically. Parameter concentrations allow for training on a fraction $w < n$ of qubits to assert that these parameters are nearly optimal on $n$ qubits. Clearly this effect has significant practical importance.