A Neural-Guided Variational Quantum Algorithm for Efficient Sign Structure Learning in Hybrid Architectures

Abstract

Variational quantum algorithms hold great promise for unlocking the power of near-term quantum processors, yet high measurement costs, barren plateaus, and challenging optimization landscapes frequently hinder them. Here, we introduce sVQNHE, a neural-guided variational quantum algorithm that decouples amplitude and sign learning across classical and quantum modules, respectively. Our approach employs shallow quantum circuits composed of commuting diagonal gates to efficiently model quantum phase information, while a classical neural network learns the amplitude distribution and guides circuit optimization in a bidirectional feedback loop. This hybrid quantum-classical synergy not only reduces measurement costs but also achieves high expressivity with limited quantum resources and improves the convergence rate of the variational optimization. We demonstrate the advancements brought by sVQNHE through extensive numerical experiments. For the 6-qubit J1-J2 model, a prototypical system with a severe sign problem for Monte Carlo-based methods, it reduces the mean absolute error by 98.9% and suppresses variance by 99.6% relative to a baseline neural network, while requiring nearly 19x fewer optimization steps than a standard hardware-efficient VQE. Furthermore, for MaxCut problems on 45-vertex Erdos-Renyi graphs, sVQNHE improves solution quality by 19% and quantum resource efficiency by 85%. Importantly, this framework is designed to be scalable and robust against hardware noise and finite-sampling uncertainty, making it well-suited for both current NISQ processors and future high-quality quantum computers. Our results highlight a promising path forward for efficiently tackling complex many-body and combinatorial optimization problems by fully exploiting the synergy between classical and quantum resources in the NISQ era and beyond.

Figures (7)

• Figure 1: (a) The algorithm seamlessly integrates quantum circuits with neural networks. The quantum circuits interleaves diagonal layers ($W_1 \cdots W_n$) with shallow, simultable layers ($G_1 \cdots G_n$). Parameters of the $G$ layers are optimized classically using a gradual transfer mechanism, while those of the $W$ layers are adjusted via the parameter shift rule on the actual hardware. Leveraging the commuting properties of the diagonal layers, the number of measurements needed to compute gradients for the $d$ parameters of $W$ drops from $\mathcal{O}(2d)$ to $\mathcal{O}(1)$. (b) Illustration of sVQNHE's advantages: The green dashed circle represents the coverage of the hybrid quantum-classical scheme. The star represents the optimal solution. (b1) describes the suboptimal situation, that is, the PQC cannot connect the solution region with the optimal solution; (b2) describes the potential quantum advantage. Similar to (b1), but the star is now inside the solution space, indicating successful coverage. (b3) Adjustable ansatz shows a transition from (b1) to (b2) via parameter adjustment, including gradient shading to indicate the evolution. (c) Illustration of the energy optimization process using the hierarchical optimization scheme. When entering the next layer, the local optimal solution will jump out due to the newly introduced parameters.
• Figure 2: Scaling test for the 1D J1-J2 model. The x-axis shows the number of qubits, while the y-axis displays the average relative error from the sVQNHE method (sign-VQNHE, blue line) and the NN method (NN, orange line) after 20 runs of 2000 steps each. The green line illustrates the difference in average relative error between the NN and sign-VQNHE methods as the qubit count increases. Shaded regions represent the relative error bar scaled by 0.5.
• Figure 3: Comparison of normalized energy and fidelity for $H_2O$ molecules at various bond angles (angle). The x-axis labels denote different geometric configurations (r, angle), where (r) is the bond length. The top panel (a) presents the average normalized energy $E_{\mathrm{pred}} / E_{\mathrm{exact}}$ for each group, while the bottom panel (b) shows the average fidelity between the predicted and exact wavefunctions. Blue bars represent results from the VQE method, and orange bars correspond to the sVQNHE method. Error bars indicate 0.5 standard deviation over 5 independent runs.
• Figure 4: Comparative performance of sVQNHE and VQE variants under a 3-qubit 1D Heisenberg Hamiltonian. (a) Finite sampling (1200 shots); (b) Noisy conditions (1200 shots, two-qubit: $0.005$; single-qubit: $0.0013$). Iteration count (x-axis) versus energy (y-axis). The solid black dashed line marks the ground state energy value. For layered optimization (sVQNHE, layered-VQE), blue and orange lines denote first- and second-layer parameter updates, respectively; standard-VQE (overall optimization) reflects sequential parameter refinement. For sVQNHE, coefficients of variation after stabilization are $cv_1 = 0.01618, cv_2 = 0.015912$ (a) and $cv_1 = 0.01534, cv_2 = 0.01487$ (b), indicating reduced variability in the second layer.
• Figure 5: Scaling test of sVQNHE using the MaxCut problem as an example. (a) The relative solution quality for the MaxCut problem, comparing parameter sets 1 and 2 across different problem sizes. The methods sign-VQNHE, brickwork-VQE, and sign-VQE correspond to the sVQNHE method with a 2-layer sign ansatz, the VQE method with brickwork ansatz, and the VQE method with 2-layer sign ansatz, respectively. The depth of brickwork ansatz is determined according to the reference sciorilli2025towards, that is, the circuit depth is sublinearly related to the number of vertices in the graph. For sign-VQE, the ansatz is the same as the circuit finally generated by the corresponding sVQNHE scheme finally generated, utilizing a two-layer sign ansatz with two diagonal and two non-diagonal layers. The value range of relative solution quality is $[0,1]$. The closer to 1, the better the performance is compared with other algorithms (see Equation \ref{['eq:R_e']} in the Method section for details). For example, in the case of 45 vertices, the quality of the solution of sVQNHE is relatively optimal regardless of whether the parameter set is 1 or 2. (b) The single-step measurement consumption for the same problem sizes. Parameter set 2 features an increased number of parameters in the NN component and doubles the number of iterations compared to parameter set 1. The circuit depth in the brickwork architecture scales sub-linearly with the number of vertices. (c) Exact state-vector simulation for MaxCut on Erdős–Rényi graphs with $m=1485$ vertices. Gray error bars represent one standard deviation. (d) Exact state-vector simulation for MaxClique on Erdős–Rényi graphs with $m=135$ vertices. Gray error bars represent one standard deviation.

Theorems & Definitions (3)

• Theorem 1
• Proposition 1
• Theorem 2