Variational Quantum Algorithms are promising candidates for near-term quantum advantage, but face significant scalability challenges due to barren plateaus. In this work, we address the trainability of the Variational Quantum Eigensolver for combinatorial portfolio optimization by evaluating three warm-start initialization strategies: Clifford, near-Clifford and a mixed approach. We compare these against a cold-start baseline initialized with the initial parameters close to zero across system sizes from [Formula: see text] to [Formula: see text]. Our results show that while the uninformed cold start fails to converge for larger systems, all warm-start strategies achieve a [Formula: see text] success rate and high initial-state fidelities. Crucially, we observe no clear difference in convergence behavior among the warm-start methods. Consequently, for this class of QUBO-based portfolio optimization problems, we identify the purely Clifford-based approach as the most efficient strategy; it consistently places the optimizer in a region from which the variational optimization converges successfully while remaining polynomially simulable on classical hardware during the quantum phase of the algorithm, indicating that the additional computational cost of non-Clifford resources offers negligible benefits for initialization in this specific context.
Variational Quantum Algorithms are promising candidates for near-term quantum advantage, but face significant scalability challenges due to barren plateaus. In this work, we address the trainability of the Variational Quantum Eigensolver for combinatorial portfolio optimization by evaluating three warm-start initialization strategies: Clifford, near-Clifford and a mixed approach. We compare these against a cold-start baseline initialized with the initial parameters close to zero across system sizes from [Formula: see text] to [Formula: see text]. Our results show that while the uninformed cold start fails to converge for larger systems, all warm-start strategies achieve a [Formula: see text] success rate and high initial-state fidelities. Crucially, we observe no clear difference in convergence behavior among the warm-start methods. Consequently, for this class of QUBO-based portfolio optimization problems, we identify the purely Clifford-based approach as the most efficient strategy; it consistently places the optimizer in a region from which the variational optimization converges successfully while remaining polynomially simulable on classical hardware during the quantum phase of the algorithm, indicating that the additional computational cost of non-Clifford resources offers negligible benefits for initialization in this specific context.
Variational Quantum Eigensolver (VQE) is widely used in near-term hardware. However, their performances remain limited by the poor trainability and are dependent on random parameter initialization. In this work, we propose a warm start method inspired by imaginary time evolution, allowing for determining initial parameters that prioritize lower energy states in a resource-efficient way. Using classical simulations, we demonstrate that this warm start method significantly improves the success rate and reduces the number of iterations required for the convergence of VQE. The numerical results also indicate that the warm start approach effectively mitigates statistical errors arising from a finite number of measurements, and to a certain extent alleviates the effect of barren plateaus.
Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA), offer a promising route to tackling combinatorial optimization problems on near and intermediate-term quantum devices. However, their performance critically depends on the choice of initial parameters, and the limited expressiveness of the QAOA ansatz makes identifying effective initializations both difficult and unscalable. To address this, we propose a framework, Scalable Parameter Initialization for QAOA (SPIQ), that employs a relaxed QAOA ansatz to enable classical search over a set of Clifford-preparable quantum states that yield high-quality solutions. These states serve as superior QAOA initializations, driving rapid convergence while significantly reducing the quantum circuit evaluations needed to reach high-quality solutions and consequently lowering quantum-device cost. We present a scalable, application-agnostic initialization framework that achieves an absolute accuracy improvement of up to 80% over state-of-the-art initialization and reduces initial-state diversity by up to 10,000x across QUBO, PUBO, and PCBO problems spanning tens to hundreds of qubits. We further benchmark its performance on a wide range of problem formulations and instances derived from real-world datasets, demonstrating consistent and scalable improvements. Furthermore, we introduce two complementary strategies for selecting high-quality Clifford points identified by our search procedure and using them to seed multi-start optimization, thereby enhancing exploration and improving solution quality.
Abstract Variational Quantum Eigensolver (VQE) is widely used in near-term hardware. However, their performances remain limited by the poor trainability and are dependent on random parameter initialization. In this work, we propose a warm start method inspired by imaginary time evolution, allowing for determining initial parameters that prioritize lower energy states in a resource-efficient way. Using classical simulations, we demonstrate that this warm start method significantly improves the success rate and reduces the number of iterations required for the convergence of VQE. The numerical results also indicate that the warm start approach effectively mitigates statistical errors arising from a finite number of measurements, and to a certain extent alleviates the effect of barren plateaus.
A portfolio optimization problem is highly important for the financial sector and critical in portfolio management. To solve the mean-variance portfolio optimization classically, the time resources needed grow exponentially with the number of stocks used. In practice, the mean-variance portfolio optimization can be solved for a few hundred assets with classical computers. Due to the exponential nature of the portfolio optimization problem, quantum computing can be utilized to improve the computation process. For solving the portfolio optimization with quantum computers we use hybrid algorithms, build of quantum and classical computing parts. More specifically, we consider the most promising hybrid algorithm, called Variational Quantum Eigensolver (VQE), which is particularly suitable for near-term quantum hardware. To maximize the potential of the near-term quantum computers, we aim to discover the suitable form of the most crucial components of the VQE algorithm. In addition to this, to further improve the performance of the VQE algorithm, we investigate the utilization of various warm-start techniques for maximizing the efficient use of the limited quantum resources. The crucial components of the VQE algorithm are the quantum circuit, i.e. ansatz, and classical optimizer. We approach the problem of finding the suitable ansatz for portfolio optimization from two different directions, with theoretical and hardware-efficient approaches. We start with the theoretically best possible, but complex structured chemistry-inspired ansatz and continue to study the capabilities of hardware efficient, but ad hoc, Dicke state ansatzes. For optimizing the classical component of the VQE algorithm, we will briefly consider the effect of various classical optimizers on the performance of the most suitable ansatzes. When the most promising components of the VQE algorithm are known, the application of various warm-start techniques of parameter initialization can begin. In initialization strategies, we investigate how the special features of Dicke state ansatzes can be used to develop tailored initialization strategies. We also investigate how optimal parameters of related problems can be utilized in parameter initialization. Our results draw clear differences in warm-startability between various ansatzes and optimizers. The results on the warm-startability of different ansatzes are important for the quantum algorithm design, because often when solving especially real-world optimization problems, the original problem can change just a little, and therefore warm-start techniques can be highly beneficial. Based on the results the Dicke state ansatzes can be surprisingly useful and their special features allow us to develop a successful initialization strategy.
Solving combinatorial optimization problems using variational quantum algorithms (VQAs) might be a promise application in the NISQ era. However, the limited trainability of VQAs could hinder their scalability to large problem sizes. In this paper, we improve the trainability of variational quantum eigensolver (VQE) by utilizing convex interpolation to solve portfolio optimization. Based on convex interpolation, the location of the ground state can be evaluated by learning the property of a small subset of basis states in the Hilbert space. This enlightens naturally the proposals of the strategies of close-to-solution initialization, regular cost function landscape, and recursive ansatz equilibrium partition. The successfully implementation of a 40-qubit experiment using only 10 superconducting qubits demonstrates the effectiveness of our proposals. Furthermore, the quantum inspiration has also spurred the development of a prototype greedy algorithm. Extensive numerical simulations indicate that the hybridization of VQE and greedy algorithms achieves a mutual complementarity, combining the advantages of both global and local optimization methods. Our proposals can be extended to improve the trainability for solving other large-scale combinatorial optimization problems that are widely used in real applications, paving the way to unleash quantum advantages of NISQ computers in the near future.
The barren plateau phenomenon, characterized by loss gradients that vanish exponentially with system size, poses a challenge to scaling variational quantum algorithms. Here we explore the potential of warm starts, whereby one initializes closer to a solution in the hope of enjoying larger loss variances. Focusing on an iterative variational method for learning shorter-depth circuits for quantum we conduct a case study to elucidate the potential and limitations of warm starts. We start by proving that the iterative variational algorithm will exhibit substantial (at worst vanishing polynomially in system size) gradients in a small region around the initializations at each time step. Convexity guarantees for these regions are then established, suggesting trainability for polynomial-size time steps. However, our study highlights scenarios where a good minimum shifts outside the region with trainability guarantees. Our analysis leaves open the question whether such minima jumps necessitate optimization across barren plateau landscapes or whether there exist gradient flows, i.e., fertile valleys away from the plateau with substantial gradients, that allow for training. While our main focus is on this case study of variational quantum simulation, we end by discussing how our results work in other iterative settings.