FlowMeas is introduced, which uses a generative flow network to directly sample finite ensembles of shallow Clifford measurement circuits subject to a prescribed shot budget and hardware constraints, and establishes generative learning as a flexible and unified framework for quantum measurement design under practical resource constraints.
Abstract
Extracting quantum information from a quantum state is a fundamental task of quantum computation, often requiring the estimation of many non-commuting observables under a finite measurement budget. For both near-term and early fault-tolerant settings, the measurement protocol must balance statistical efficiency against implementation resources such as circuit depth, connectivity, and entangling-gate count. Many existing strategies focus on two extremes: hardware-friendly product measurements with high sampling cost, and fully commuting measurements with deep circuits. Here we recast resource-constrained measurement design as a generative learning problem. We introduce FlowMeas, which uses a generative flow network to directly sample finite ensembles of shallow Clifford measurement circuits subject to a prescribed shot budget and hardware constraints. At zero entangling depth, FlowMeas learns qubit-wise commuting measurement schedules and already matches or improves leading product-measurement methods on nearly all molecular benchmarks. Allowing one or two entangling gate layers yields further reductions in energy estimation error of up to $27\%$ relative to the strongest state-independent product-measurement baseline. The learned policy can also be reused across related Hamiltonians, substantially accelerating retraining along a molecular potential-energy surface. We further obtain results for molecular Hamiltonians with up to 20 qubits and apply the framework to a compactly encoded 54-qubit interacting fermionic model, extending the demonstrated scale beyond prior molecular benchmarks. These results establish generative learning as a flexible and unified framework for quantum measurement design under practical resource constraints.
Noisy Intermediate-Scale Quantum (NISQ) devices impose structural constraints on the training of Variational Quantum Algorithms (VQAs). In the absence of full error correction, each quantum gate introduces a non-zero probability of error that accumulates with circuit depth, while gradient estimation through finite sampling adds additional statistical variability. As a result, convergence depends on a delicate balance between physical coherence and estimation variance. In this setting, the depth of the variational ansatz not only determines model expressivity, but also its operational feasibility under noise. Without an explicit characterization of the interaction between error accumulation, number of measurements (shots), and control strategies, increasing experimental resources may fail to improve performance and can even become counterproductive. In this work, we empirically analyze how circuit depth affects the practical signal of the gradient under depolarizing noise. Across multiple configurations ($q=4-7$ qubits), we observe a pattern consistent with an effective exponential attenuation of the coherent gradient contribution, characterized by a per-layer rate $\epsilon_{\text{eff}}$. The estimated magnitude of $\epsilon_{\text{eff}}$ depends on circuit complexity and remains largely independent of the measurement budget, indicating that it primarily reflects architectural and physical factors rather than sampling effects. The analysis of adaptive control strategies further suggests that accuracy gains are constrained by gradient attenuation, rather than scaling significantly with circuit depth. However, a crossmode comparison across three distinct training strategies reveals that, while accuracy improvements remain modest, adaptive perturbations substantially improve the statistical detectability of the attenuation signal, acting as diagnostic probes rather than mere optimization heuristics. Taken together, these results suggest that the usable depth of a variational ansatz can be interpreted as an emergent property of the effective signal-to-noise regime, governed by a measurable structural parameter whose ranking across circuit configurations is preserved independently of the training mode. This framework enables principled comparison of circuit architectures in terms of their effective trainability under noise, providing guidance for architectural and control design in Quantum Machine Learning under NISQ conditions.
C. Braga, Manuel A. Serrano, E. Fernández-Medina· 2026 IEEE International Conf...· 0 citations
As IQP circuits produce remarkably low intermediate magic relative to phase-randomised states with the same sampling distributions, this renders IQP-based quantum generative models as promising candidates for resource-efficient demonstrations of quantum advantage on early fault-tolerant architectures.
Quantum state and process tomography constitute essential diagnostic tools in quantum information science, yet their standard formulations suffer from prohibitive computational scaling as the number of qubits grows. In this work, we introduce a physics-constrained conditional generative adversarial network that bypasses iterative constrained inversion by directly learning a forward generative mapping conditioned on Pauli expectation values. The generator embeds a differentiable Cholesky layer at its output, which enforces Hermiticity, positive semidefiniteness, and unit trace by construction. Our experiments reveal that the strength of the $L^1$ penalty critically governs the emergence of GHZ coherence during training: an excessively large penalty postpones the coherence onset and yields a prolonged low-fidelity plateau, whereas an intermediate value enables the fastest stable convergence. Moreover, for high-temperature thermal states, an over-complete measurement basis proves necessary to prevent sustained late-stage fluctuations. By extending the same Cholesky constraint to the Choi-matrix representation, the framework naturally accommodates quantum process tomography. For systems with $n \ge 6$ qubits, the exponential growth of the underlying $2^n \times 2^n$ density matrix remains the fundamental bottleneck; we discuss how integrating tensor-network structures can contain the per-iteration cost while preserving reconstruction fidelity. Altogether, these results suggest that physically constrained generative learning offers a scalable and amortizable pathway toward data-driven tomography for noisy intermediate-scale quantum devices.
Random quantum objects are powerful resources for quantum information processing, yet exact Haar randomness is costly and typically unnecessary. We introduce an explicit sparse commuting circuit ensemble on $n$ qubits that reproduces low-order Haar moments in the stringent relative-error sense. The circuit consists of a sparse Clifford phase layer followed by independent single-qubit Clifford gates. Acting on a simple product state, the resulting ensemble forms $\epsilon$-approximate projective $2$- and $3$-designs in relative error, with the required logarithmic interaction degree being asymptotically optimal within this circuit family. It admits an ancilla-free implementation of quantum depth $O(\log(n/\epsilon))$ on an all-to-all architecture, as well as an adaptive constant-depth implementation---in fact, depth seven---using $O(n\log(n/\epsilon))$ ancilla qubits. Departing from existing shallow-design paradigms, our analysis exploits the intrinsic moment structure of commuting phase circuits; at third order, this requires a new block decomposition and combinatorial analysis that also suggests a route toward higher-order shallow designs. Our results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources, with applications to randomized characterization, quantum metrology, quantum algorithms, and many-body physics.
Qing-Yue Zhang, Jun-Jie Chen, Zhou You et al.· 0 citations
This work reveals and exploits this underexplored robustness property: how much non-Clifford and variational expressivity can be removed from the sampling circuit before SQD accuracy degrades, and answers through two complementary compression techniques: gradient-based operator pruning, which discards low-impact excitation operators, and Clifford rounding, which snaps remaining parameters to the nearest Clifford angle.
Kangyu Zheng, Yidong Zhou, Jinglei Cheng et al.· 0 citations
It is explained that the efficiently preparable states, device-generated distributions, variationally learned loading, and amortized preparation are required to get advantage from quantum machine learning and close with a checklist for evaluating input-dependent advantage claims.
M. Faryad· 0 citations
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