Quantum-Classical Neural Networks for Solving Lindblad Master Equations
Authors: Rafael Florencio Díaz, Julio Guerrero García, Elvira Romera Gutiérrez
Organizations: Departamento de Matem´aticas, Universidad de Ja´en, Edificio de Servicios Generales - Campus Cient´ıfico Tecnol´ogico de Linares. Avda. de la Universidad (Cintur´on Sur), s/n, Linares, 23700, Ja´en, Espa˜na. · C´atedra de Energ´ıa y Tecnolog´ıas Cu´anticas, Universidad de Granada, Spain. · Departamento de Matem´aticas, Universidad de Ja´en, Campus Las Lagunillas, Ja´en, 23071, Ja´en, Espa˜na. · Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Granada, Avenida de Fuentenueva s/n, Granada, 18071, Granada, Espa˜na.
Variational quantum algorithms have emerged as a promising approach for simulating open quantum systems, yet their expressive power and trainability remain poorly understood. We propose a variational framework for solving Lindblad master equations by minimizing a residual cost function using a compositional ansatz obtained by combining a classical neural network with a parameterized quantum circuit. The density matrix is represented through a purification on an enlarged Hilbert space, ensuring positivity, Hermiticity, and unit trace by construction, while all gradients are evaluated exactly via the parameter-shift rule. Beyond the algorithmic formulation, we derive an upper bound for the covering number of the proposed compositional model and compare it with that of purely quantum neural networks. Our analysis shows that the classical parametrization enlarges the effective hypothesis class while providing finer control over its complexity, thereby establishing a theoretical connection between expressivity, covering numbers, effective dimension, and the barren plateau phenomenon. Numerical experiments validate both the accuracy of the proposed method for representative Lindblad master equations and the theoretical predictions regarding gradient concentration. These results provide a unified variational and theoretical framework for the simulation of dissipative quantum dynamics using compositional quantum machine learning models.
Figures & tables
Figure 1: Schematic representation of the quantum neural network (the case nrgq=1 is shown for simplicity)
Figure 2: Classical neural network schematic.
Figure 3: Quantum neural network scheme of the two-qubit architecture used in the simulation.
Figure 4: Comparison between the expected values ⟨σx⟩ and ⟨σy⟩ as functions of time t , obtained from the analytic solution and from the proposed model in Figure 3 at the final iteration for nw=34 . Numerical results are those produced by the proposed model.
Figure 5: Comparison between the values of the cost function given in ( 38 ) as a function of the number of iterations of the gradient descent method, obtained using nw=34 , nw=64 , and nw=124 .
Figure 6: Comparison of the absolute error between the analytic and numerical expected values of ⟨σx⟩ and ⟨σy⟩ as functions of time t at the final iteration, for different values of nw .
Figure 7: Quantum neural network scheme used in the simulation.
Figure 8: Logarithm of the variance of the gradient with respect to the rotation angles for the expected values of a global observable and for the Lindblad residual cost function defined in ( 46 ), as a function of the number of qubits. Linear least-squares fits are shown for the global observable (solid line) and for the Lindblad residual cost function (dashed line).
Figure 9: Variance of the gradient with respect to the CNN parameters w as a function of the number of qubits, for the expected values of a global observable and for the Lindblad residual cost function defined in ( 46 ). Results are shown for different intervals of nw . Linear least-squares fits are shown for the global observable (solid lines) and for the Lindblad residual cost function (dashed lines).
Figure 10: Variance of the gradient with respect to the CNN parameters w , in logarithmic scale, as a function of nw for different numbers of qubits. Results are shown for both cases: the global observable and the Lindblad residual cost function. Linear least-squares fits are shown for the global observable (solid lines) and for the Lindblad residual cost function (dashed lines).
Variational quantum algorithms use parameterized quantum circuits and classical optimization to approximate solutions to quantum and optimization problems.
Inferring the dynamical generator of a many-body quantum system from measurement data is essential for the verification, calibration, and control of quantum processors. When the system is open, this task becomes considerably harder than in the purely unitary case, because coherent and dissipative mechanisms can produce similar measurement statistics and long-time data can be insensitive to coherent couplings. Here we tackle this so-called Lindbladian learning problem of open-system characterisation with maximum-likelihood on Pauli measurements at multiple experimentally friendly transient times, exploiting the richer information content of transient dynamics. To navigate the resulting non-convex likelihood loss-landscape, we augment the physical model neural differential-equation term, which is progressively removed during training to distil an interpretable Lindbladian solution. Our method reliably learns open-system dynamics across neutral-atom (with 2D connectivity) and superconducting Hamiltonians, as well as the Heisenberg XYZ, and PXP models on a spin-1/2 chain. For the dissipative part, we show robustness over phase noise, thermal noise, and their combination. Our algorithm can robustly infer these dissipative systems over noise-to-signal ratios spanning four orders of magnitude, and system sizes up to N=6 qubits with fewer than 5×105 shots.
Timothy Heightman, Roman Aseguinolaza Gallo, Edward Jiang +3
ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain · Quside Technologies SL, Carrer d’Esteve Terradas 1, 08860 Castelldefels, Barcelona, Spain · CIC nanoGUNE Consolider, Tolosa Hiribidea 76, E-20018 Donostia-San Sebastian, Spain +3
Quantum operations on pure states can be fully represented by unitary matrices. Variational quantum circuits, also known as quantum neural networks, embed data and trainable parameters into gate-based operations and optimize the parameters via gradient descent. The high cost of training and low fidelity of current quantum devices, however, restricts much of quantum machine learning to classical simulation. For few-qubit problems with large datasets, training the matrix elements directly, as is done with weight matrices in classical neural networks, can be faster than decomposing data and parameters into gates. We propose a method that trains matrices directly while maintaining unitarity through a single regularization term added to the loss function. A second training step, circuit alignment, then recovers a gate-based architecture from the resulting soft-unitary. On a five-qubit supervised classification task with 1000 datapoints, this two-step process produces a trained variational circuit in under four minutes, compared to over two hours for direct circuit training, while achieving lower binary cross-entropy loss. In a second experiment, soft-unitaries are embedded in a hybrid quantum-classical network for a reinforcement learning cartpole task, where the hybrid agent outperforms a purely classical baseline of comparable size.
Basil Kyriacou, Mo Kordzanganeh, Maniraman Periyasamy +1
Variational quantum circuits have been central to many proposed near-term applications of quantum computing, but a growing body of evidence suggests that trainability and quantum advantage are fundamentally at odds: ansätze expressive enough to resist efficient classical simulation tend to exhibit barren plateaus, while structures that provably rule out barren plateaus typically render them classically simulable. We propose a stacked linear combination of unitaries (S-LCU) as a variational ansatz which provides a tunable trade-off between barren plateaus and classical simulability. Using a diagrammatic analysis, we bound the loss-landscape variance of the Free Fermion S-LCU, whose elements are fermionic Gaussian unitaries. We prove a variance lower bound of Ω(1/(nk3l)), with a simulation cost of O(k2ln3) using the best known classical algorithm, compared to a quantum gate complexity of only O(lkn2). The number of layers l serves as a single dial that trades computational complexity against the rate of cost concentration. This offers practitioners a systematic method for constructing ansätze with a complexity-trainability trade-off that best suits their application and hardware.
Nikhil Khatri, Stefan Zohren, Gabriel Matos
Machine Learning Research Group, University of Oxford, United Kingdom · 2Quantinuum, London, United Kingdom