Symbolic Pauli Propagation for Gradient-Enabled Pre-Training of Quantum Circuits

TL;DR

The paper introduces symbolic Pauli propagation to represent quantum observable expectations as analytic functions of circuit parameters, enabling off-line pre-training via gradient-based optimization and reducing on-chip training costs. It couples exact Heisenberg-propagation with practical truncations on Pauli weight and frequency to maintain tractable representations while preserving accuracy, and provides a rigorous error bound for joint truncations. Applied to a VQE task on the ANNNI model, the method demonstrates scalable, accurate energy estimates for reasonably large systems, with explicit guidance on how truncation parameters impact performance. The work offers a promising pathway to pre-train quantum circuits on classical resources before deployment on hardware, contingent on favorable propagation properties of the chosen ansatz.

Abstract

Quantum Machine Learning models typically require expensive on-chip training procedures and often lack efficient gradient estimation methods. By employing Pauli propagation, it is possible to derive a symbolic representation of observables as analytic functions of a circuit's parameters. Although the number of terms in such functional representations grows rapidly with circuit depth, suitable choices of ansatz and controlled truncations on Pauli weights and frequency components yield accurate yet tractable estimators of the target observables. With the right ansatz design, this approach can be extended to system sizes beyond the reach of classical simulation, enabling scalable training for larger quantum systems. This also enables a form of classical pre-training through gradient-based optimization prior to deployment on quantum hardware. The proposed approach is demonstrated on the Variational Quantum Eigensolver for obtaining the ground state of a spin model, showing that accurate results can be achieved with a scalable and computationally efficient procedure.

Figures (7)

• Figure 1: Distribution of the Pauli weights (left) and frequencies (right) for the propagated observable $Z_0$ of the local entangler ansatz at different depths for $12$ qubits.
• Figure 2: Local entangler Ansatz
• Figure 3: Mean absolute errors at truncation values $w_{\mathrm{cut}}$ and $\nu_{\mathrm{cut}}$ for uniformly sampled parameters of the Local Entangler Ansatz with 12 qubits and 3 repetitions of the unitary. The transparent red overlay represents the theoretical error bound obtained from Assumption \ref{['ass:weight-frequency-decay']}.
• Figure 4: Energies of the 18-spin ANNNI model at various values of $\kappa$ and $h$, obtained using exact diagonalization (left) and by minimizing the function of the propagated Hamiltonian via gradient descent (right).
• Figure 5: Relative Errors of the energies between the diagonalized energies and the VQE energies from Fig. \ref{['fig:vqephasediagE']}