CV-QAOA: Efficient Low-Depth Quantum Optimization of Continuous Variables
Abstract
We study a Continuous-Variable Quantum Approximate Optimization Algorithm (CV-QAOA) for high-dimensional continuous optimization. Our formulation extends an earlier CV-QAOA proposal with a variationally optimized initial state and recovers the convergence guarantees of Quantum Hamiltonian Descent (QHD) in the high-depth limit. We prove rigorous performance guarantees of CV-QAOA on several families of cost functions. First, we show d-step CV-QAOA minimizes any d-dimensional strictly convex quadratic function with 2d quantum queries to the cost function. We then analyze a family of nonconvex "Rotated Double Well" (RDW) functions with 2d local minima introduced by arXiv:2311.00811. While prior work showed QHD reaches its global minimum with Õ(d3) queries, we prove that 1-step CV-QAOA solves RDW with just two quantum queries. Although general-purpose classical solvers need superpolynomial time for RDW and structure-awareness can reduce the cost to polynomial time, we show that the 1-step CV-QAOA protocol can be efficiently dequantized, and that a gradient-aligned line search succeeds with O(d) queries, nearly matching the information-theoretic Ω(d/log d) query lower bound. To move beyond the dequantizable regime, we introduce a ``Rotated Square Well'' (RSW) problem, whose globally flat landscape suppresses useful local gradient information. For this family, we show that an adiabatic evolution simulated by CV-QAOA can reach the global minimum using do(1) queries. On the other hand, any classical algorithm that learn the hidden rotation in RSW provably requires Ω(d2/log d) queries, a bound we nearly match with an explicit Θ(d2log d)-query classical algorithm.Numerical simulations on deflected corrugated spring and Easom functions illustrate the promising performance of CV-QAOA on more general problems.
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BibTeX
@misc{bharadwaj2026cvqaoa,
title = {{CV-QAOA: Efficient Low-Depth Quantum Optimization of Continuous Variables}},
author = {Sriram Bharadwaj and Di Luo and Leo Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06815},
eprint = {2610.06815},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06815},
}
RIS
TY - UNPB
TI - CV-QAOA: Efficient Low-Depth Quantum Optimization of Continuous Variables
AU - Sriram Bharadwaj
AU - Di Luo
AU - Leo Zhou
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06815
UR - https://arxiv.org/abs/2610.06815
AB - We study a Continuous-Variable Quantum Approximate Optimization Algorithm (CV-QAOA) for high-dimensional continuous optimization. Our formulation extends an earlier CV-QAOA proposal with a variationally optimized initial state and recovers the convergence guarantees of Quantum Hamiltonian Descent (QHD) in the high-depth limit. We prove rigorous performance guarantees of CV-QAOA on several families of cost functions. First, we show d-step CV-QAOA minimizes any d-dimensional strictly convex quadratic function with 2d quantum queries to the cost function. We then analyze a family of nonconvex "Rotated Double Well" (RDW) functions with 2^d local minima introduced by arXiv:2311.00811. While prior work showed QHD reaches its global minimum with Õ(d^3) queries, we prove that 1-step CV-QAOA solves RDW with just two quantum queries. Although general-purpose classical solvers need superpolynomial time for RDW and structure-awareness can reduce the cost to polynomial time, we show that the 1-step CV-QAOA protocol can be efficiently dequantized, and that a gradient-aligned line search succeeds with O(d) queries, nearly matching the information-theoretic Ω(d/log d) query lower bound. To move beyond the dequantizable regime, we introduce a ``Rotated Square Well'' (RSW) problem, whose globally flat landscape suppresses useful local gradient information. For this family, we show that an adiabatic evolution simulated by CV-QAOA can reach the global minimum using d^o(1) queries. On the other hand, any classical algorithm that learn the hidden rotation in RSW provably requires Ω(d^2/log d) queries, a bound we nearly match with an explicit Θ(d^2log d)-query classical algorithm.Numerical simulations on deflected corrugated spring and Easom functions illustrate the promising performance of CV-QAOA on more general problems.
ER -