To break, or not to break: Symmetries in adaptive quantum simulations, a case study on the Schwinger model

TL;DR

This work analyzes how symmetry properties in ADAPT-VQE operator pools impact quantum-resource efficiency for simulating the lattice Schwinger model. By contrasting top-down pools (varying translation, charge, locality, and time-reversal) with bottom-up tiling approaches that yield translation-invariant or charge-conserving pools, the study quantifies convergence behavior, CNOT-depth requirements, and sensitivity to symmetry breaking. The main finding is that, for near-term devices, pools that break translation invariance while preserving charge and using Z-preserving (or qubit-local) operators offer shallower circuits and faster convergence, whereas translation-invariant pools may be preferable on error-corrected hardware due to reduced shot counts. Time-reversal breaking is generally disfavored, as symmetry tends to be restored quickly, and boundary effects play a significant role in open-chain Schwinger lattices; the results inform strategies for constructing resource-efficient variational ansätze in gauge-theory simulations.

Abstract

We investigate the role of symmetries in constructing resource-efficient operator pools for adaptive variational quantum eigensolvers. In particular, we focus on the lattice Schwinger model, a discretized model of $1+1$ dimensional electrodynamics, which we use as a proxy for spin chains with a continuum limit. We present an extensive set of simulations comprising a total of $11$ different operator pools, which all systematically and independently break or preserve a combination of discrete translations, the conservation of charge (magnetization) and the fermionic locality of the excitations. Circuit depths are the primary bottleneck in current quantum hardware, and we find that the most efficient ansätze in the near-term are obtained by pools that $\textit{break}$ translation invariance, conserve charge, and lead to shallow circuits. On the other hand, we anticipate the shot counts to be the limiting factor in future, error-corrected quantum devices; our findings suggest that pools $\textit{preserving}$ translation invariance could be preferable for such platforms.

Figures (14)

• Figure 1: Examples of various types of symmetries in particle physics and their effects on a given particle. Preservation of any of these symmetries means that they correspond to a conserved quantity, i.e. if the Hamiltonian is invariant under charge conjugation, it means that charges are conserved.
• Figure 2: Schematic depicting the hierarchy of relaxing symmetries in the operator pools in this work for ADAPT-VQE. The pools shown here all preserve time-reversal $\mathcal{T}$. (a) The pools at the top level preserve every symmetry in the problem Hamiltonian, and we relax these symmetries to form new pools, using a top-down approach. (b) The ${\boxplus_\Lambda}$ and ${\boxplus_Q}$ pools are formed using a different, bottom-up approach, in which elementary tiled Pauli operators are used to construct more complex operators conserving translation invariance and charge.
• Figure 3: Evolution of the ADAPT energy density error for $L=9$ with respect to the number of ADAPT iterations for $\xi_{C}$. The markers indicate where the surface operators are selected for the $\Lambda$-pools. The top panel is an inset of the full optimization for $L=9$ which shows the trajectories in the initial stage of the algorithm, and where the first surface operators are selected.
• Figure 4: Energy density error evolution versus largest pool gradient during ADAPT-VQE for $L=9$, $\xi_{C}$. The gradient magnitudes are shown using the color scale on the right.
• Figure 5: Charge variation in the ansätze as energy density error decreases during ADAPT-VQE for $L=9$.