Exact Diagonal Completion on Reachable Subspaces: Application to QAOA Placement

2026-09-29

AI Summary

This study addresses the redundant overhead of quantum circuits when applying the Quantum Approximate Optimization Algorithm (QAOA) to chip placement, proposing an operator optimization method based on reachable subspaces. Specifically, it constructs an exact Manhattan distance operator via a weighted L1 norm and simplifies the phase circuit by combining Walsh coefficient analysis with Gray code synthesis. Furthermore, a sparse recursive strategy is introduced to circumvent dense constraint storage, exploiting the degrees of freedom in unused encoding states to simultaneously compress circuit depth and gate count. Experimental results demonstrate that this approach reduces CNOT gate counts by over 50% under specific conditions. Although the method has been integrated into the OpenROAD toolchain, its end-to-end performance advantages remain to be further validated.

Abstract

Unused encoding states offer opportunities to simplify quantum circuits. For algorithms restricted to reachable subspaces, unspecified diagonal-operator entries can be optimized without changing ideal computation. We investigate exact diagonal completion for the quantum approximate optimization algorithm (QAOA) applied to placement, using permutation-preserving register swaps. We construct exact Manhattan-distance operators through weighted-$\ell_1$ optimization of Walsh coefficients and a sparse recurrence requiring $O(\sqrt{m})$ terms on balanced rectangles, avoiding $O(m^4)$ dense constraint storage. Across 160 geometries, weighted-$\ell_1$ completion reduces controlled-NOT (CX) counts relative to four alternative extensions in all 96 cases with unused binary codes under Gray-code synthesis. On an independently specified 60-case cohort, median reductions relative to virtual-coordinate extension are 28.0\%, 53.9\%, and 21.6\% at six, nine, and twelve sites. Under generic diagonal synthesis, reductions decrease to 10.9\%, 1.3\%, and 0.7\%, demonstrating compiler dependence. Additional ancillas reduce mixer serialization, but token circuits remain deeper than one-hot baselines. Ideal placement simulations show baseline-dependent solution quality, with classical search performing better. OpenROAD integration takes 72 QAOA and 216 classical placements across six RTL designs through clock-tree synthesis and global routing with zero overflow. Completion improves phase construction; no end-to-end advantage is established.

Problem

• QAOA
• exact diagonal completion
• reachable subspaces
• quantum circuit optimization
• placement

Innovation

• Exact Diagonal Completion
• QAOA
• Walsh Coefficients
• Weighted L1 Optimization
• Manhattan Distance Operator