Exact Diagonal Completion on Reachable Subspaces: Application to QAOA Placement
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-ℓ1 optimization of Walsh coefficients and a sparse recurrence requiring O(√m) terms on balanced rectangles, avoiding O(m4) dense constraint storage. Across 160 geometries, weighted-ℓ1 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.
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BibTeX
@misc{friedewald2026exact,
title = {{Exact Diagonal Completion on Reachable Subspaces: Application to QAOA Placement}},
author = {Owen Friedewald and Ali Shiri Sichani and Chi-Ren Shyu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38734},
eprint = {2609.38734},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38734},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Diagonal Completion on Reachable Subspaces: Application to QAOA Placement
AU - Owen Friedewald
AU - Ali Shiri Sichani
AU - Chi-Ren Shyu
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38734
UR - https://arxiv.org/abs/2609.38734
AB - 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-ℓ_1 optimization of Walsh coefficients and a sparse recurrence requiring O(√m) terms on balanced rectangles, avoiding O(m^4) dense constraint storage. Across 160 geometries, weighted-ℓ_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.
ER -