We propose a technique for optimizing parameterized circuits in variational quantum algorithms based on the probabilistic tensor sampling optimization. This method allows one to relax random initialization issues or heuristics for generating initial guess of variational parameters, and can be used to avoid local minima. We illustrate our approach on the example of the quantum approximate optimization algorithm (QAOA) applied to the max-
k-cut problem based on the binary encoding efficient in the number of qubits. We discuss the advantages of our technique for searching optimal variational parameters of QAOA circuits in comparison to classical optimization methods.
Combinatorial optimization is one of the promising applications of quantum computation. Variational Quantum Algorithm (VQA) is a classical-quantum hybrid algorithm framework for noisy quantum devices. However, the quality of the solution is readily affected by statistical fluctuations and physical noise, making it challenging to maintain applicability for largescale problems. In contrast, sampling-based quantum algorithms have recently been successfully applied to large-scale quantum chemistry problems. In the sampling-based framework, the quantum device is used only for sampling, while the ground state and its energy are estimated on the classical device. In this study, we propose the Sampling-based Quantum Optimization Algorithm (SQOA) that leverages these advantages for combinatorial optimization. There are two challenges in constructing SQOA. The first challenge is that we need to encode the optimization problem in a non-diagonal Hamiltonian, even though many VQAs encode it into the Ising Hamiltonian, which is diagonal. The second challenge is that we need a method for efficient preparation of the input state to be sampled. To address the first challenge, we utilize the Quantum Relaxation (QR) method for encoding an optimization problem within a Hamiltonian. This method achieves a higher encoding density by mapping multiple variables to a single qubit, thus requiring fewer qubits for larger problems compared to the Ising Hamiltonian approach. Moreover, to address the second challenge, we investigate the parameter transferability within the Quantum Alternating Operator Ansatz for QR Hamiltonians. We show that restricting parameters to a linear form exhibits moderate transferability for 3-regular Max-Cut problems, similar to transferability observed in the Quantum Approximate Optimization Algorithm. Therefore, this property allows us to efficiently prepare the input state for a large problem instance using the parameters obtained from a small problem instance. We leveraged transferability to create input states and applied SQOA with QR to the Max-Cut instances. By transferring parameters from a 20-node problem, we demonstrate that SQOA with QR provides high-quality solutions for 40-node problems without variational parameter optimization.
Demonstrating quantum heuristics that outperform strong classical solvers on large-scale optimization remains an open challenge. Here we introduce Regularized Warm-Started QAOA (RWS-QAOA), which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling. We further propose a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm in which the quantum circuit parameters are fixed and only a classical warm starting step is instance-dependent. We evaluate RWS-QAOA on the Max-Cut problem for random regular graphs, where this protocol yields a constant-depth quantum circuit, across three complementary settings. First, on Quantinuum's trapped-ion processor, RWS-QAOA outperforms the classical algorithms with the best provable guarantees for Max-Cut on
3-regular graphs, namely Goemans-Williamson and Halperin-Livnat-Zwick, on
96-node instances. Second, tensor-network simulations on graphs with up to
N=10,000 nodes show that depth-
6 RWS-QAOA, achieving an average cut fraction of
0.9167, surpasses the best classical heuristics under matched restrictions (no local-search post-processing and no iterative refinement). Third, we remove these restrictions and benchmark against the strongest unrestricted classical heuristics, including an optimized parallel Burer-Monteiro solver that improves upon the MQLib implementation. Even against this stronger baseline, we project that surface-code RWS-QAOA reaches a quantum-classical runtime crossover below
0.2 seconds on
3,000-node graphs with fewer than
1.3 million physical qubits. Our results show that constant-depth quantum circuits combined with a classical warm start have a credible potential to surpass classical solvers on the Max-Cut problem when executed on future quantum computers.
In this work, we present a Bayesian Optimization (BO) approach for tuning the parameters of the Quantum Approximate Optimization Algorithm (QAOA) applied to max-cut problems. Within our black-box optimization framework, BO achieves competitive solutions while requiring significantly fewer quantum circuit evaluations compared to standard non-Bayesian global optimizers. These results highlight the potential of BO to enhance both the efficiency and the overall performance of variational quantum algorithms.
Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparameterization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA) -- a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems -- MAX-CUT and MAX-2-SAT. For MAX-CUT we observe that overparametriation is both sufficient and (statistically) necessary for attaining exact solutions, as confirmed numerically for up to
20 qubits. In fact, for MAX-CUT on 2-regular graphs we show the necessity to be exact, based on the analytically found optimal depth. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.
Variational Quantum Algorithms have emerged as a leading paradigm for near-term quantum computation. In such algorithms, a parameterized quantum circuit is controlled via a classical optimization method that seeks to minimize a problem-dependent cost function. Although such algorithms are powerful in principle, the non-convexity of the associated cost landscapes and the prevalence of local minima means that local optimization methods such as gradient descent typically fail to reach good solutions. In this work we suggest a method to improve gradient-based approaches to variational quantum circuit optimization, which involves coupling the output of the quantum circuit to a classical neural network. The effect of this neural network is to peturb the cost landscape as a function of its parameters, so that local minima can be escaped or avoided via a modification to the cost landscape itself. We present two algorithms within this framework and numerically benchmark them on small instances of the Max-Cut optimization problem. We show that the method is able to reach deeper minima and lower cost values than standard gradient descent based approaches. Moreover, our algorithms require essentially the same number of quantum circuit evaluations per optimization step as the standard approach since, unlike the gradient with respect to the circuit, the neural network updates can be estimated in parallel via the backpropagation method. More generally, our approach suggests that relaxing the cost landscape is a fruitful path to improving near-term quantum computing algorithms.