Models of quantum systems faithfully map system parameters to observations, but the inverse problem of parameter inference from measurement data presents a fundamental challenge: computationally intractable likelihoods due to an exponentially large Hilbert space. Here, we introduce simulation-based quantum system inference, a unified, likelihood-free framework that learns parameter posteriors directly from classical simulation data. The central idea is to pair polynomial-cost classical simulators, such as Pauli propagation and tensor networks, with normalizing flows or other neural density estimators for accurate, reusable inference. A single model, trained once, maps any new measurement record to its posterior in one forward pass---turning per-experiment inference into a fixed, up-front cost. We numerically demonstrate the framework's versatility across Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning, with examples involving 81-qubit shallow circuits and 735-parameter inference. In each case, the approach yields accurate estimates of identifiable parameters, while posterior uncertainty provides additional diagnostics of non-identifiability and indicates where further characterization is needed. Our framework reduces data-acquisition requirements in quantum experiments and accelerates parameter inference, providing a practical route to characterizing and improving large-scale quantum systems.
Models of quantum systems faithfully map system parameters to observations, but the inverse problem of parameter inference from measurement data presents a fundamental challenge: computationally intractable likelihoods due to an exponentially large Hilbert space. Here, we introduce simulation-based quantum system inference, a unified, likelihood-free framework that learns parameter posteriors directly from classical simulation data. The central idea is to pair polynomial-cost classical simulators, such as Pauli propagation and tensor networks, with normalizing flows or other neural density estimators for accurate, reusable inference. A single model, trained once, maps any new measurement record to its posterior in one forward pass---turning per-experiment inference into a fixed, up-front cost. We numerically demonstrate the framework's versatility across Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning, with examples involving 81-qubit shallow circuits and 735-parameter inference. In each case, the approach yields accurate estimates of identifiable parameters, while posterior uncertainty provides additional diagnostics of non-identifiability and indicates where further characterization is needed. Our framework reduces data-acquisition requirements in quantum experiments and accelerates parameter inference, providing a practical route to characterizing and improving large-scale quantum systems.
To extract complete information about experimental noisy quantum states, as well as correlations among different physical parameters, we develop an efficient machine learning-assisted framework with the normalizing flow based Bayesian parameter estimation (BPE). By embedding a physically interpretable parameter vector, s, into the quantum state ansatzes, ρ(s), the corresponding parameter distribution p(s∣D) is induced from BPE, based on the available experimental data D. The BPE allows us to perform parameter correlation analyses, providing powerful insights about how experimental parameters and related noise sources affect quantum features of the system. As an example, the flexibility of our framework is demonstrated using two different physical ansatzes on the optical cat state experiments, where a noisy single-photon state is added to impure squeezed state. With correlations among the Wigner function negativity, photon-addition fraction, noisy fraction, and pump powers, our framework gives interpretations for complicated experimental mechanisms in generating noisy non-Gaussian states. By leveraging normalizing flow-based neural networks and Bayesian uncertainty estimation, along with physical and experimental constraints incorporated naturally, the valuable information inferred in the posterior learning provides guidelines for experimentalists, as well as theorists, in identifying promising parameters to enhance desirable features.
The advancement and scaling of quantum technology has made the learning and identification of quantum systems and devices in highly multidimensional parameter spaces a pressing task for a variety of applications. In many cases, the integration of real-time feedback control and adaptive choice of measurement settings places strict demands on the speed of this task. Here we present a joint model selection and parameter estimation algorithm that is fast and operable on a large number of model parameters. The algorithm is based on variational Bayesian inference, which approximates the target posterior distribution by optimizing a tractable family of distributions, making it more scalable than exact inference methods relying on sampling and that generally suffer from high variance and computational cost in high-dimensional spaces. We show how a regularizing prior can be used to select between competing models, each comprising a different number of parameters, identifying the simplest model that explains the experimental data. The regularization can further separate the degrees of freedom, e.g., quantum systems in the environment or processes, which contribute to major features in the observed dynamics, with respect to others featuring small coupling, which only contribute to a background. As an application of the introduced framework, we consider the problem of the identification of multiple individual nuclear spins with a single electron spin quantum sensor, relevant for nanoscale nuclear magnetic resonance and for the implementation of multiqubit quantum networking nodes. With the number of environmental spins unknown a priori, our Bayesian approach is able to correctly identify the model, i.e., the number of spins and their couplings. We benchmark the algorithm on both simulated and experimental data, using standard figures of merit, and demonstrate that we can estimate dozens of parameters within minutes. Our methodology is compatible with the implementation of real-time adaptive choice of experimental settings in multiparameter estimation, which has the potential to greatly decrease the estimation time in a variety of metrology, tomography, and characterization tasks by prioritizing the measurements with the highest content of information.
High-dimensional regression problems, prevalent in diverse fields like genomics, finance, and medical diagnostics, pose a significant challenge to classical Bayesian inference. The computational cost of calculating posterior distributions often scales exponentially with the number of parameters, rendering many real-world applications intractable. This paper introduces a novel Bayesian inference algorithm inspired by the principles of quantum mechanics, specifically superposition and interference. We represent the prior distribution as a superposition of states, allowing for simultaneous exploration of the entire parameter space. Inference is then performed through a "quantum-inspired" sampling process that mimics the probabilistic nature of quantum measurements. This approach offers a potential pathway to circumvent the computational bottlenecks inherent in traditional Bayesian methods, particularly in the context of high-dimensional data. We outline the key components of the algorithm, provide a theoretical framework, and discuss its potential advantages and limitations. The core of the method lies in effectively harnessing the parallel exploration capability suggested by quantum mechanics to drastically reduce the computational burden associated with estimating the posterior distribution.
Readout errors are a major bottleneck to extracting reliable information from near-term quantum processors, especially when spatial correlations are non-negligible. We present a unified tensor-network framework that models the readout process as a matrix product operator (MPO), enabling efficient characterization and mitigation beyond uncorrelated approximations. The MPO model is trained via likelihood optimization on calibration data and applies to multiple tasks, including nonlocal observable estimation, random circuit sampling, and random-measurement protocols, such as classical shadows and learning-based tomography. Experiments on a superconducting processor and numerical simulations up to 20 qubits show that the MPO model captures correlated readout errors that uncorrelated models miss, with a sample cost that grows only near-linearly with system size. When extended to two-dimensional systems, the framework can also be integrated with tensor-network quantum error-correction decoders by performing joint inference over data and readout errors. These results establish tensor-network readout error mitigation as a scalable and versatile approach for noise-aware quantum data processing.
Quantum computers could solve problems beyond the reach of classical devices, but this potential depends on quantum error correction (QEC) to protect fragile quantum states from noise. A central challenge in QEC is decoding: inferring likely physical errors from syndrome patterns generated by repeated stabilizer measurements. Existing decoders, including graph-based and neural approaches, typically return a single correction hypothesis and therefore discard the richer posterior structure of the error distribution conditioned on the observed syndrome. Here we recast QEC decoding as posterior inference using discrete denoising diffusion, exploiting the analogy between stochastic error accumulation and the forward diffusion process. We introduce DiffQEC, a generative decoder that combines a syndrome processor for multi-round spatial-temporal syndrome histories with syndrome feature modulation to condition denoising on the observed syndrome throughout inference. On experimental data from Google's superconducting quantum processor, DiffQEC reduces logical error rates by up to 10.2% relative to minimum-weight perfect matching and by about 5% relative to tensor-network decoding. These improvements persist for larger code distances up to 17 under depolarizing noise and for logical circuits of increasing depth. Beyond accuracy, the learned posterior provides confidence estimates for post-selection and reveals physically meaningful error structure, establishing posterior generative decoding as a practical framework for QEC.
Bayesian learning is ubiquitous for implementing classification and regression tasks, however, it is accompanied by computationally intractable limitations when the feature spaces become extremely large. Aiming to solve this problem, we develop a quantum bayesian learning framework of the restricted Boltzmann machine in the quantum-enhanced feature spaces. Our framework provides the encoding phase to map the real data and Boltzmann weight onto the quantum feature spaces and the training phase to learn an optimal inference function. Specifically, the training phase provides a physical quantity to measure the posterior distribution in quantum feature spaces, and this measure is utilized to design the quantum maximum a posterior (QMAP) algorithm and the quantum predictive distribution estimator (QPDE). It is shown that both quantum algorithms achieve exponential speed-up over their classical counterparts. Furthermore, it is interesting to note that our framework can figure out the classical bayesian learning tasks, i.e. processing the classical data and outputting corresponding classical labels. And a simulation, which is performed on an open-source software framework for quantum computing, illustrates that our algorithms show almost the same classification performance compared to their classical counterparts. Noting that the proposed quantum algorithms utilize the shallow circuit, our work is expected to be implemented on the noisy intermediate-scale quantum (NISQ) devices, and is one of the promising candidates to achieve quantum supremacy.