Variational ans\"atze are a cornerstone of quantum many-body physics, providing compact approximations to complex ground states using finite resources. Recent quantum-technology advances have introduced a new class based on layered parameterized evolutions. Assessing whether they can represent critical ground states is challenging: correlations span all length scales, while finite circuit depth limits how far they extend. Building on finite-resource scaling from tensor networks, we assign each ansatz an emergent correlation length $\xi_D$, the longest range over which it faithfully captures critical correlations. Its growth with refinement parameter $D$, $\xi_D \propto D^\kappa$, defines an exponent $\kappa$ measuring how efficiently an architecture converts resources into long-distance correlations. Applying this framework to the critical transverse-field Ising model, with $D$ the circuit depth of parameterized evolutions, we compare ans\"atze with nearest-neighbor and long-range generators, and layers where generators act separately or combined. All ans\"atze are compatible with algebraic growth, but fitted exponents range from $\kappa\simeq1$ to $\kappa\simeq3$. Exponential interactions give the largest exponents, while power-law interactions stay close to nearest-neighbor behavior, showing long-range support alone gives no scaling advantage. Combined-generator layers generally outperform separable ones, so layer organization matters alongside interaction range. Finally, a quasiparticle analysis shows finite depth leaves an unresolved window of width $\xi_D^{-1}$ around the low-energy modes responsible for long-distance correlations. The emergent correlation length thus acts as an infrared resolution scale, providing a benchmark for critical-state preparation and a guide for designing resource-efficient variational architectures.
Simulating strongly correlated quantum materials often involves not a single Hamiltonian, but a family of Hamiltonians whose ground states evolve across experimentally tunable couplings. Foundation neural quantum states (FNQS) offer a promising route to amortizing many-body calculations across such families, but can lose accuracy near phase transitions and still incur non-negligible sampling costs that grow with the number of target couplings. We introduce the Foundation Neural Effective Hamiltonian (FNEH), which projects a Hamiltonian family onto a compact subspace spanned by FNQS sampled at selected couplings. By variationally combining FNQS across parameter space, FNEH systematically improves their ground-state approximation and can recover phase boundaries that the foundation model misidentifies. Once the required operator matrix elements are sampled, FNEH enables sweeps over couplings, observables, and phase boundaries at a cost governed by the small effective-Hamiltonian dimension, without repeated neural-network sampling at every target coupling. We demonstrate FNEH in strongly correlated moir\'e materials, where it accurately resolves competing phases, enables high-resolution multidimensional phase scans, and substantially reduces the computational cost of exploring many target Hamiltonians. The results open a new avenue for studying strongly correlated quantum materials with foundation models.
We benchmark eight classical optimizers for exact-statevector VQE calculations on a controlled hierarchy of frustrated spin models, ranging from a diagonal Ising glass to transverse-field Ising and anisotropic Heisenberg models. The benchmark includes local, stochastic-gradient, evolutionary, covariance-adaptation, and swarm-based optimization methods under matched function-evaluation budgets. To understand their performance beyond final energies, we characterize the underlying Hamiltonian--ansatz landscapes in terms of local minima, gradients, curvature, and ground-state reachability. We use simple variational circuits, from an $R_y$ product-state ansatz for the diagonal model to shallow $R_y$--CNOT hardware-efficient circuits for the noncommuting models, and study how increasing circuit depth changes their expressivity, reachability, and optimization geometry. We find that optimizer performance changes substantially across the model hierarchy and is closely connected to landscape structure, while the variational gap represents a separate source of error. These results show how classical optimization, variational expressivity, and landscape geometry jointly determine VQE performance for frustrated spin models.
V. Novák, Ivan Zelinka, Swagatam Das et al.· 0 citations
This work constructs a native 2D pairwise ansatz and compares its expressibility and trainability with representative 1D ansatze at identical layer depths, despite their different circuit depths.
Projected entangled pair states (PEPS) provide an efficient variational ansatz for two-dimensional quantum phases, but computing observables remains challenging because PEPS contraction is generally costly. Here, we parameterize two-dimensional quantum states using variational PEPS subject to isometric constraints and map the resulting ansatz onto monitored quantum circuits, replacing tensor-network contraction with circuit sampling. For infinite cylinders, the transfer matrix defines a quantum channel on the virtual boundary. We use a fixed-point treatment and a monitored-circuit unraveling of this channel to evaluate observables efficiently. Using a constant number of variational parameters and a number of qubits that scales only with the cylinder width, our method yields a phase diagram for the $J_1$-$J_2$ model in qualitative agreement with DMRG results. Because the monitored circuits are compatible with near-term quantum hardware, this approach provides a hybrid quantum-classical framework for simulating two-dimensional quantum many-body systems.
Yu-Qing Rong, Huanhai Zhou, Guo-Yi Zhu et al.· 0 citations
Mapping quantum phase diagrams through classical simulation is notoriously resource-intensive, as even small systems far from the thermodynamic limit demand prohibitive computational effort. The variational quantum eigensolver (VQE) offers a compelling alternative, exploiting approximate ground states to distinguish phases. An appealing proposal, dubbed as Delta-VQE, determines critical points by contrasting variational energies optimized from reference states of distinct phases. Intriguingly, the diagnostic sharpens as circuit depth decreases, highlighting its promise as a resource-conscious probe of quantum criticality. To probe the broader applicability and underlying mechanisms of this approach, we investigate the one-dimensional transverse-field Ising model with a three-spin cluster interaction, a setting in which the Ising transitions are generally situated beyond the self-dual line. We demonstrate that, whenever dual ans\"atze are employed, Delta-VQE invariably detects the self-dual points rather than the true criticality. In contrast, when ans\"atze are carefully tailored to embody the competing phases across the boundary, the genuine Ising critical point can be successfully identified with only minor finite-size effects. Our results establish that, while Delta-VQE provides a resource-efficient probe of quantum criticality without requiring precise ground-state preparation, its diagnostic power is fundamentally contingent upon the judicious selection of physically representative ans\"atze.
Multipartite Bell tests provide a correlation-only route to benchmarking quantum processors, but their application at large scales is hindered by the rapid decay of many-body correlators under noise and exponentially many terms in conventional Bell expressions. Here we address these scalability obstacles by introducing a finite-setting generalized Mermin family of state-tailored Bell inequalities with analytic certification bounds, in which the measurement-setting number $m$ provides an additional certification dimension complementary to the system size $n$. We show that, for the powers-of-two setting choices considered here, increasing $m$ leaves the ideal normalized multipartite quantum value unchanged while lowering the relevant classical bounds, thereby strengthening the Bell-violation ratios and yielding an improved noise-robustness scaling compared to the standard Mermin inequality. We test this construction experimentally on a programmable superconducting processor by preparing Greenberger-Horne-Zeilinger (GHZ) states of up to 80 qubits. Using randomized sampling for direct Bell-operator estimation, we observe Bell ratios that grow exponentially with system size, certify a nonlocality depth of 14, and show that increasing $m$ strengthens both the Bell ratio and depth certification. All results are obtained solely from measured correlators and analytical bounds, without readout correction, tomography, or model-based mitigation. Generalized Mermin inequalities therefore provide a sharper Bell benchmark for noisy large-scale GHZ states.
Jianbin Cai, Jun-Xiang Huang, F. Otto et al.· 0 citations
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