Symmetry-Preserving Hierarchical Quantum Approximate Optimization Algorithm with Subproblems Beyond the Qubit Limit

Abstract

The quantum approximate optimization algorithm (QAOA) is a paradigmatic variational quantum algorithm for combinatorial optimization problems, yet on current quantum devices the available qubit count and achievable circuit depth remain the main bottlenecks to its practical application. Divide-and-merge reduces the number of variables encoded in a single quantum circuit, while contraction based on coarse graining extends the effective range of shallow circuits, but existing work has not addressed both challenges simultaneously. We propose symmetry-preserving hierarchical QAOA (SP-HQAOA), in which parity sector contraction preserves the zero-field Ising structure of the effective objective at every contraction step, thereby addressing both within a single algorithm. We further prove that this reduction is the unique one with this property. Successive contractions further decouple the maximum leaf block size $B$ from the qubit limit $Q$ per quantum call, so every quantum circuit uses at most $Q$ qubits even when $B>Q$. We also establish a deterministic repair-risk bound on the possible loss in optimal value caused by each sector restriction, and the resulting bounds can be summed across contraction steps and levels of the recursive hierarchy. With depth $p=1$, SP-HQAOA attains $99.4\%$ to $99.7\%$ of the exact optimum on $N=40$ random regular MaxCut leaf blocks and, with at most $10$ qubits per circuit, $96\%$ of the classical reference for $N=500$. SP-HQAOA demonstrates the capacity to use shallow quantum circuits to address combinatorial optimization problems whose variable count far exceeds the qubit limit of a single quantum call, and may enable practical quantum optimization advantages in the near term.