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
O~(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/logd) 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/logd) queries, a bound we nearly match with an explicit
Θ(d2logd)-query classical algorithm.Numerical simulations on deflected corrugated spring and Easom functions illustrate the promising performance of CV-QAOA on more general problems.
Continuous-variable (CV) quantum systems offer a natural framework for continuous optimization through their infinite-dimensional Hilbert spaces. In this paper, we propose the Complex Continuous-Variable Quantum Approximate Optimization Algorithm (CCV-QAOA), a variational framework operating in the complex domain that optimizes over complex decision variables. The method efficiently solves real and complex multivariate optimization problems. To demonstrate its versatility, we apply CCV-QAOA across a broad suite of optimization use cases, including convex quadratic minimization, scaling studies with circuit depth and cutoff dimension, constrained quadratic programs using penalty constructions, and non-convex benchmarks such as the Styblinski-Tang function and complex quartic landscapes.
Variational quantum algorithms (VQAs) provide a promising approach to achieving quantum advantage for practical problems on near-term noisy intermediate-scale quantum (NISQ) devices. Thus far, most studies on VQAs have focused on qubit-based systems, but the power of VQAs can be potentially boosted by exploiting infinite-dimensional continuous-variable (CV) systems. Here, we implement the CV version of one VQA, a quantum approximate optimization algorithm, by developing an automated collaborative computing system between a programmable photonic quantum computer and a classical computer. We experimentally demonstrate that this algorithm solves the minimization problem of simple continuous functions by implementing the quantum version of gradient descent to localize an initially broadly distributed wave function to the minimum. This method allows the execution of a practical CV quantum algorithm on a physical platform. Our work can be extended to the minimization of more general functions, providing an alternative to achieve the quantum advantage in practical problems.
Optimization of continuous-variable objectives on standard gate-based quantum computers via variational algorithms such as QAOA is typically approached by first discretizing each decision variable into a finite binary representation. This increases qubit requirements and restricts solution precision through fixed-resolution encodings. We propose a qubit-efficient variational framework for continuous optimization that instead encodes each decision variable into the squared amplitude or equivalently, the measurement probability of a single qubit state. This removes explicit discretization from the variable representation while remaining entirely within the standard qubit circuit model unlike methods like CV-QAOA employing qumode based hardware to achieve the same. To read out encoded variables, we propose using amplitude estimation rather than naive sampling or tomographic reconstruction, with the goal of improving precision scaling for continuous-value recovery. We outline how amplitude-estimation error propagates to decision-variable error and then to objective-value error under standard regularity assumptions, suggesting a distinct width-versus-precision tradeoff relative to discretized approaches. In particular, the framework replaces the logarithmic increase in qubits needed for finer binary precision with a constant cost of one qubit per decision variable, while shifting accuracy requirements into the estimation procedure. We position this approach relative to traditional discretized variational formulations, and argue that it provides a promising new direction for continuous optimization on standard qubit architectures.
Due to its unique scalability potential, continuous-variable quantum optics is a promising platform for large-scale quantum computing. In particular, very large cluster states with a two-dimensional topology that are suitable for universal quantum computing and quantum simulation can be readily generated in a deterministic manner, and routes towards fault tolerance via bosonic quantum error correction are known. In this article we propose a complete measurement-based quantum computing architecture for the implementation of a universal set of gates on the recently generated two-dimensional cluster states [M. V. Larsen et al., Science 366, 369 (2019); W. Asavanant et al., Science 366, 373 (2019)]. We analyze the performance of the various quantum gates that are executed in these cluster states as well as in other two-dimensional cluster states (the bilayer-square lattice and quad-rail lattice cluster states [R. N. Alexander et al., Phys. Rev. A 94, 032327 (2016); N. C. Menicucci, Phys. Rev. A 83, 062314 (2011)]) by estimating and minimizing the associated stochastic noise addition as well as the resulting gate error probability. We compare the four different states and find that, although they all allow for universal computation, the quad-rail lattice cluster state performs better than the other three states, which all exhibit similar performance.
This paper introduces a novel variational quantum algorithm built upon the established Quantum Approximate Optimization Algorithm also known as QAOA. Since the known parameter fixing strategy imposes constraints on QAOA to enhance tractability at the cost of some expressive power, the proposed layerwise approach integrates it with the existing Multi-Angle QAOA technique, which is characterized in turn by height-ened expressiveness through an increased number of parameters, albeit with increased optimization challenges. Consequently, the proposed layerwise-Multi-Angle QAOA combines the strengths of one variant with the limitations of the other, striking a balance in algorithmic design. The effectiveness of the proposed algorithm is assessed through experimental evaluations on the Maximum Cut problem. This study reveals promising results in heuristic predictions, with robustness both in terms of approximation ratio and optimization capabilities.