Variational Learning with Sparse Long-range Entangling Gates

Rigorous Time-dependent Hamiltonian Learning via Continuous Weak Measurements

Characterizing the Hamiltonian that a quantum processor actually implements is central to calibrating and validating current quantum hardware. Many devices, however, operate with generators that are time dependent by design. Here we develop a rigorous and experimentally friendly protocol for learning time-dependent many-body Hamiltonians from continuous weak measurement records. The key observation is that interaction sparsity reduces the global reconstruction to a set of local inverse problems, whose number is controlled by the interaction connectivity rather than by the system size. Pure separable probe states suffice to drive these inversions, and a graph-coloring construction embeds them into a small number of global product-state preparations. We derive explicit reconstruction-error bounds and a sample-complexity theorem that cleanly separates the finite-sampling statistical noise from the deterministic bias of the iterative state update, and we validate the protocol on time-dependent spin chains with up to $n=8$ qubits. Beyond these results, our analysis provides a rigorous foundation for time-dependent Hamiltonian learning from continuous monitoring in many-body systems, establishing a framework that extends naturally to many platforms and probe ensembles.

Expressibility and trainability of a two-dimensional pairwise quantum-circuit ansatz

Parameterized quantum circuits~(PQCs) constitute a central building block of variational quantum algorithms~(VQAs) and quantum machine learning~(QML) methods. Existing ansatz designs often adopt hardware-agnostic or simplified 1D chain/ring entanglement patterns. However, as quantum hardware continues to develop, native 2D connectivity patterns, such as planar superconducting-qubit architectures, are becoming increasingly important. Inspired by this hardware structure, we construct a native 2D pairwise ansatz and compare its expressibility and trainability with representative 1D ansatze at identical layer depths, despite their different circuit depths. For the fixed 16-qubit system, the 2D ansatz has the smallest KL divergence at $L=1$ and $2$, and its second-order frame potential approaches the theoretical lower bound more rapidly at shallow layer counts than the frame potentials of the three 1D ansatze. We also evaluate the gradient variance of the Pauli-$Z$-string expectation value $\langle Z_0\otimes\cdots\otimes Z_{15}\rangle$ with respect to the first $R_y$ angle. For this Pauli-$Z$ string and fixed parameter, the gradient variance is smaller for the 2D circuit at $L=1$--$4$. The differences narrow at $L=5$, and the four ansatze yield statistically compatible variances at $L=6$.

Learning Depth-Structured Parameterizations for Efficient Optimisation of Deep Quantum Circuits

Symmetry Constraints Regularize Neural Quantum State Learning

Neural quantum states (NQS) offer highly expressive variational wavefunctions, but their optimization is frequently bottlenecked by redundant parameters and poorly conditioned landscapes. We demonstrate that embedding Hamiltonian symmetries directly into the variational parameterization geometrically regularizes this learning problem. For Boltzmann-family NQS, we enforce symmetries by tying local Pauli-$Z$ generators along physical geometric orbits, analytically collapsing the trainable coefficient space prior to optimization. To quantify the resulting optimization geometry, we introduce a geometric metric built on the Jacobian and Hessian of the optimization landscape. This framework evaluates the fraction of the physically accessible state space that corresponds to high-quality, low-energy solutions. Evaluating our approach on transverse-field Ising (TFIM) and XXZ spin chains shows that symmetry compilation excises the vast majority of parameters while maintaining ground-state accuracy within the resolution of the reported benchmarks. In large TFIM systems, strong spatial constraints compress thousands of parameters down to tens, delivering substantial runtime accelerations. Our geometric diagnostics indicate that symmetry produces a more favorable target-aware geometry by concentrating the reachable state space around low-energy solutions while retaining broad target basins. Together, our results indicate that symmetry compilation concentrates the expressive power of NQS on states relevant to the target problem, thereby reducing model size and training cost without sacrificing accuracy.

Circuit Design Informed Adaptive Variational Quantum Algorithms

This work analyzes how circuit design constraints can systematically reduce the measurement overhead associated with repeated evaluations of the candidate gate pool in adaptive algorithms by focusing on the Hadamard test circuit architecture, hardware-aware qubit connectivity, and problem-specific adaptive framework.

Universal scaling framework for parameterized quantum evolutions at criticality

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.