We develop a rigorous and implementable framework for Gibbs sampling of infinite-dimensional quantum systems governed by unbounded Hamiltonians. Extending dissipative Gibbs samplers beyond finite dimensions raises fundamental obstacles, including ill-defined generators, the absence of spectral gaps on natural Banach spaces, and tensions between implementability and convergence guarantees. We overcome these issues by constructing KMS-symmetric quantum Markov semigroups on separable Hilbert spaces that are both well-posed and efficiently implementable on qubit hardware. Our generation theory is based on the abstract framework of Dirichlet forms, adapted here to the case of algebras of bounded operators over separable Hilbert spaces. Leveraging the spectral properties of our self-adjoint generators, we establish quantitative convergence results in trace distance, including regimes of fast thermalization. In contrast, we also identify Hamiltonians for which a naive choice of generators guaranteeing implementability generally comes at the cost of losing convergence of the associated evolutions, thereby establishing a strong trade-off between implementability and convergence. Our framework applies to a wide class of models, including Schrödinger operators, Gaussian systems, and Bose-Hubbard Hamiltonians, and provides a unified approach linking rigorous infinite-dimensional analysis with algorithmic Gibbs state preparation. less

Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation. less

A strong converse bound for the classical identification capacity of a quantum channel is an upper bound on the asymptotic identification rate of classical messages sent through the channel, such that, above this rate, the probability of an identification error necessarily converges to one. Converse bounds for identification are notoriously difficult to obtain for fully quantum channels. The only previously known converse bound, due to Atif, Pradhan and Winter [Int.~J.~Quantum Inf.~22(5):2440013, 2024], has the unsatisfactory feature of remaining strictly positive even for a completely noisy channel, for which identification is clearly impossible. We derive strong (and hence also weak) converse bounds, for the qubit depolarizing channel with noise parameter $p$, that vanish as $p\to 1$, thereby yielding the correct behavior in the completely noisy limit. Moreover, in the setting of simultaneous classical identification under the constraint of complete product measurements, our converse bound matches the corresponding achievability bound, and establishes that in this case the identification capacity equals the classical capacity of the channel. less

Quantum sensing technologies offer transformative potential for ultra-sensitive biomedical sensing, yet their clinical translation remains constrained by classical noise limits and a reliance on macroscopic ensembles. We propose a unifying generational framework to organize the evolving landscape of quantum biosensors based on their utilization of quantum resources. First-generation devices utilize discrete energy levels for signal transduction but follow classical scaling laws. Second-generation sensors exploit quantum coherence to reach the standard quantum limit, while third-generation architectures leverage entanglement and spin squeezing to approach Heisenberg-limited precision. We further define an emerging fourth generation characterized by the end-to-end integration of quantum sensing with quantum learning and variational circuits, enabling adaptive inference directly within the quantum domain. By analyzing critical parameters such as bandwidth matching and sensor-tissue proximity, we identify key technological bottlenecks and propose a roadmap for transitioning from measuring physical observables to extracting structured biological information with quantum-enhanced intelligence. less

In this work, we study the problems of certifying and learning quantum $k$-local Hamiltonians, for a constant $k$. Our main contributions are as follows: - Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only $O(1/\varepsilon)$ evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of $Ω(1/\varepsilon)$. To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis. - Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature. - Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022). less

Recent studies have shown that Trotter errors are highly initial-state dependent and that standard upper bounds often substantially overestimate them. However, the mechanism underlying anomalously small Trotter errors and a systematic route to identifying error-resilient states remain unclear. Using interaction-picture perturbation theory, we derive an analytical expression for the leading-order Trotter error in the eigenbasis of the Hamiltonian. Our analysis shows that initial states supported on spectrally commensurate energy ladders exhibit strongly suppressed error growth together with persistent Loschmidt revivals. We refer to such states as Trotter scars. To identify such states in practice, we further introduce a general variational framework for finding error-minimizing initial states for a given Hamiltonian. Applying this framework to several spin models, we find optimized states whose spectral support and dynamical behavior agree with the perturbative prediction. Our results reveal the spectral origin of Trotter-error resilience and provide a practical strategy for discovering error-resilient states in digital quantum simulation. less