Quantum Gibbs Sampling in Infinite Dimensions: Generation, Mixing Times and Circuit Implementation
Simon Becker, Cambyse Rouzé, Robert Salzmann
Abstract
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.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper develops the first rigorous and implementable framework for quantum Gibbs sampling of infinite-dimensional systems with unbounded Hamiltonians. The central challenge — reconciling well-posedness, convergence guarantees, and efficient circuit implementation — is addressed through three main technical contributions:
1. Generation theory: Using Dirichlet forms and KMS-symmetric quantum Markov semigroups, the authors construct well-defined Lindbladian generators on separable Hilbert spaces, overcoming the fundamental obstacle that unbounded GKLS-type generators may fail to generate legitimate quantum dynamics (e.g., trace preservation can fail).
2. Spectral gap analysis on Hilbert-Schmidt space: They prove that while spectral gaps generically vanish on the trace-class (Proposition 3.1 — the OU spectrum fills the entire left half-plane), convergence can be recovered on the Hilbert-Schmidt space for appropriately energy-constrained initial states. The Metropolis-type filter function (1.11) is shown to maintain a spectral gap for single-mode number-preserving Hamiltonians, while Schwartz filter functions lead to gapless generators for superquadratic Hamiltonians — establishing a sharp implementability-convergence tradeoff.
3. Finite-dimensional truncation and circuit implementation: A systematic truncation scheme reduces the infinite-dimensional dynamics to finite-dimensional circuits with exponentially small (in the truncation parameter M) approximation errors, and the resulting circuit complexity scales polynomially in physically relevant parameters.
Methodological Rigor
The mathematical rigor is exceptional. The paper builds on the abstract framework of Dirichlet forms (Albeverio–Høegh-Krohn, Davies–Lindsay, Goldstein–Lindsay) and adapts it carefully to the B(H) setting over separable Hilbert spaces. Key steps are:
The finite-dimensional approximation theory (Section 4.3) is particularly careful, tracking energy constraints through exponential weights and providing explicit error bounds with polynomial prefactors times exponential decay in M^κ.
Potential Impact
Quantum algorithms: This work opens the door to provably efficient Gibbs state preparation for continuous-variable systems on digital quantum hardware. The applications to Bose-Hubbard models (companion paper [31]), Schrödinger operators with Coulomb interactions, and Gaussian systems are physically significant.
Mathematical physics: The observation that trace-class spectral gaps generically vanish (Proposition 3.1) while Hilbert-Schmidt gaps can persist is a fundamental structural insight for infinite-dimensional open quantum systems. The interpolation result (Theorem 3.2) showing p-independence of the spectrum for p ∈ (1,∞) is powerful.
Quantum information theory: The Gaussian-convoluted generator family {L_{σ_E}} interpolating between Davies and non-Davies generators, with monotonically improving spectral gaps as σ_E decreases (Proposition 4.2), provides a practical tuning parameter for algorithm design.
Timeliness & Relevance
This work is highly timely. The finite-dimensional quantum Gibbs sampling literature has seen explosive growth (Chen-Kastoryano-Brandão-Gilyén 2023, Ding-Li-Lin 2025, Rouzé-França-Alhambra 2025), and the extension to infinite dimensions addresses a clear gap. With growing interest in simulating continuous-variable quantum systems on digital platforms, this framework provides the theoretical foundation for such simulations.
Strengths
Limitations
Overall Assessment
This is a landmark contribution that establishes the mathematical foundations for quantum Gibbs sampling in infinite dimensions. The technical depth, breadth of results, and clarity of exposition are outstanding. It will likely become a reference work for both the quantum algorithms and mathematical physics communities.
Generated Apr 2, 2026
Comparison History (46)
Paper 2 directly addresses a critical bottleneck in superconducting quantum computing: achieving fast, high-fidelity two-qubit gates while mitigating unwanted interactions. The experimental demonstration of a 24 ns CZ gate with over 99.9% fidelity has immediate, high-impact applications for scaling quantum processors and advancing towards fault tolerance. Paper 1 offers a rigorous mathematical framework for a specific algorithmic problem, but Paper 2's hardware breakthrough offers broader and more immediate practical utility to the rapidly growing quantum computing field.
Paper 2 likely has higher near-term scientific impact due to its direct applicability to a rapidly advancing experimental platform (neutral-atom quantum computing), with clear, measurable performance gains (O(sqrt N) planning, large-scale simulations, improved capture rates) and released code that can be adopted immediately by experimental groups. Its algorithmic contribution is concrete, scalable, and timely for building larger defect-free arrays. Paper 1 is highly novel and rigorous, but its impact may be slower-burn and more specialized due to heavy mathematical machinery and less immediate experimental deployment.
Paper 2 has higher likely scientific impact due to its broad, rigorous framework for infinite-dimensional quantum Gibbs sampling with convergence guarantees and explicit qubit implementability—addressing a foundational problem spanning mathematical physics, quantum algorithms, and quantum simulation. It introduces general generation theory (Dirichlet forms/KMS-symmetric semigroups), quantitative mixing-time results, and clarifies implementability–convergence trade-offs across multiple model classes (Schrödinger, Gaussian, Bose-Hubbard). Paper 1 is a striking experimental advance in quantum-enhanced strong-field ionization, but its impact is narrower and more application-specific compared with Paper 2’s cross-field methodological reach.
Paper 2 establishes a rigorous, unified framework for quantum Gibbs sampling in infinite-dimensional systems, addressing fundamental mathematical obstacles while remaining implementable on qubit hardware. Its broad applicability across various models (Schrödinger operators, Gaussian systems, Bose-Hubbard) gives it a wider foundational impact across quantum computing, mathematical physics, and quantum simulation compared to Paper 1's more specific focus on measurement-induced phase transitions.
Paper 2 develops a rigorous mathematical framework for quantum Gibbs sampling in infinite dimensions with concrete convergence results, circuit implementations, and broad applicability across quantum computing and quantum many-body physics. It solves fundamental technical obstacles (ill-defined generators, spectral gap absence) with novel tools (Dirichlet forms adapted to operator algebras). Paper 1, while addressing an important practical problem in superconducting qubits, is primarily a theoretical/conceptual framework (Part I) without experimental validation, and its impact is more narrowly focused on materials characterization for transmon qubits.
Paper 1 is likely higher impact: it tackles a concrete, timely computational problem (quantum Gibbs state preparation) with rigorous infinite-dimensional analysis and explicit implementability on qubit hardware, plus quantitative mixing-time guarantees. The framework (Dirichlet forms/KMS-symmetric QMS) is methodologically strong and broadly applicable across models (Schrödinger, Gaussian, Bose-Hubbard), bridging mathematical physics and quantum algorithms. Paper 2 is conceptually interesting but leans on strong modeling assumptions (random-matrix environment) and geometric constructions whose rigor and empirical testability may be less immediate, narrowing near-term applicability.
Paper 1 provides a foundational, rigorous framework for algorithmic quantum Gibbs sampling of infinite-dimensional systems on near-term qubit hardware. Its ability to map complex systems like Schrödinger operators and Bose-Hubbard models to implementable algorithms promises broad applications in quantum computing and quantum simulation. While Paper 2 offers valuable insights into Floquet heating and quantum sensing, Paper 1 addresses a major theoretical bottleneck in quantum algorithms, likely yielding a wider, more interdisciplinary impact across quantum information, computational chemistry, and mathematical physics.
Paper 1 addresses a fundamental and challenging problem—extending quantum Gibbs sampling to infinite-dimensional systems with unbounded Hamiltonians—with deep mathematical rigor (Dirichlet forms, KMS symmetry, convergence guarantees) and broad applicability (Schrödinger operators, Gaussian systems, Bose-Hubbard models). It bridges rigorous mathematical physics with practical quantum algorithms, establishing important trade-offs. Paper 2 introduces a useful but more incremental concept (dismagickers for magic reduction in tensor networks). While novel, its scope and theoretical depth are narrower. Paper 1's framework has broader potential to influence quantum computing, thermodynamics, and mathematical physics.
Paper 1 addresses a fundamental gap in quantum computing and quantum thermodynamics by developing a rigorous framework for Gibbs sampling in infinite-dimensional systems, bridging mathematical rigor with algorithmic implementability. Its breadth of impact spans quantum algorithms, mathematical physics, and quantum information theory, with applicability to multiple physical models. Paper 2, while valuable in extending open quantum systems theory to strong-coupling regimes, represents a more incremental advance combining two existing techniques (polaron transformation + CCQME) and is narrower in scope. Paper 1's novelty in identifying fundamental trade-offs between implementability and convergence, plus its broader applicability, gives it higher impact potential.
Paper 2 has higher near-term scientific impact: it demonstrates a concrete, scalable hardware architecture (independent DKS microcombs) enabling WDM twin-field QKD with a record aggregate key rate (1.57 Mbps) over 201 km—directly advancing deployable quantum communications. The methodological rigor is strong via a full end-to-end experiment and performance benchmarking. Its applications (inter-city secure links) are immediate and broadly relevant across photonics, integrated optics, and quantum networks. Paper 1 is highly novel and rigorous mathematically, but its impact is more specialized and longer-horizon, with implementability and practical adoption less certain.