Quantum Computers Efficiently Simulate Complex Materials’ Thermal Behaviour

A new approach to simulating the thermal properties of complex bosonic systems on quantum computers has been achieved. Simon Becker and colleagues at Bocconi University in collaboration with Institut Polytechnique de Paris, RWTH Aachen University and Claude Bernard University Lyon 1 present the first rigorous framework for Gibbs sampling in infinite-dimensional systems, addressing a key gap in current quantum algorithms. The research shows that physically relevant bosonic models, including the Bose-Hubbard Hamiltonian, can efficiently generate thermal states with exponential convergence. This mathematically controlled method opens avenues for quantum simulation, offering potential advantages in understanding thermalisation and many-body complexity.

Controlling Dissipative Dynamics for Efficient Gibbs State Preparation and Thermal Property

A finite-rank reduction of the dissipative dynamics controls the generator via compact perturbations, deducing the discreteness of the spectrum and the stability of the gap. This allows for efficient preparation of the corresponding Gibbs state on qubit hardware, and enables a quantum algorithm to compute thermal properties of the associated model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.

The simulation of ground and thermal states of many-body quantum systems is widely regarded as one of the most promising routes toward near-term quantum advantage. Advances in the complexity-theoretic understanding of quantum many-body systems, the development of quantum algorithms for estimating physically relevant observables, and increasingly sophisticated experimental demonstrations of quantum simulations have all contributed to recent progress. Thermal state preparation, in particular, has recently seen major progress.

Quantum Gibbs samplers based on Lindbladian dynamics have been introduced in a series of works, with subsequent results establishing efficient convergence in a variety of physically relevant regimes. However, classical algorithms can efficiently approximate thermal expectation values and partition functions in most regimes where quantum algorithms are known to be efficient. This raises whether genuine quantum advantages can emerge in alternative physical settings.

While most existing results focus on locally finite-dimensional systems, such as spin models or fermionic lattices, bosonic systems have received far less attention in the computational complexity field. Historically, discrete systems have dominated quantum algorithmic studies because they parallel the classical digital-computation model. Bosonic systems were mainly explored in quantum communication and information theory, but a rigorous complexity-theoretic framework for these systems has begun to emerge, revealing phenomena that differ substantially from those of finite-dimensional systems.

Bosonic systems arise naturally in many areas of physics, including condensed matter, quantum optics, atomic physics and quantum chemistry, and their dynamics have therefore been extensively studied from the perspective of Hamiltonian simulation and mathematical physics. Existing quantum runtime guarantees for Gibbs sampling in the bosonic setting are largely restricted to Gaussian systems, for which quantum advantage is not generally expected. Bosonic systems present unique challenges from a computational standpoint.

Classical approximation techniques that are highly effective for spin systems often fail in infinite-dimensional settings. For example, semidefinite-programming relaxations, which provide powerful approximation methods for ground-state problems, do not readily extend to bosonic systems. Similarly, cluster-expansion techniques that yield efficient classical algorithms for partition functions of finite-dimensional systems encounter major obstacles in bosonic models due to the unboundedness of the Hamiltonians and observables.

Moreover, Gibbs states of bosonic systems can remain entangled at arbitrarily high temperatures, in sharp contrast with discrete variables. These challenges suggest that bosonic systems may offer natural regimes where quantum algorithms outperform classical approaches, motivating a systematic study of quantum Gibbs sampling in continuous-variable many-body systems. This letter initiates such a program, studying a family of efficiently implementable quantum Gibbs samplers for infinite-dimensional systems and establishing positive spectral-gap results, and hence exponential convergence to equilibrium.

The focus is on the Bose-Hubbard model, which is central to many-body bosonic physics because of its fundamental theoretical significance and its direct experimental relevance. The results provide a rigorous framework for thermal-state preparation and for quantum algorithms estimating thermodynamic observables in bosonic many-body systems. The primary task considered is the preparation of the Gibbs state σβ(H) := e−βH/ Try e−βH of the Hamiltonian H of an infinite dimensional quantum system at inverse temperature β >. To achieve this, a family of dissipative quantum Gibbs samplers extended into infinite-dimensional systems is considered.

The jumps associated with the generator LσE, b f,H of the dissipation formally consist of the following dressing of a family of bare jump operators {Aα}α∈A = {(Aα)†}α∈A: Lα(H):= Z R eitHAαe−itHf(t) dt, where the filter function f is chosen so that the Gibbs state is stationary: LσE, b f,H(σβ(H)) =. In fact, the so-called KMS condition bf(ν)= bf(−ν)e−βν/2 on the Fourier transform bf(ν) = R f(t)eitνdt ensures that the generator is self-adjoint with respect to a well-chosen inner product ⟨., .⟩σβ(H) (see Appendix A). The parameter σE > 0 controls both the Hamiltonian simulation time required to implement the dissipation on a quantum computer and the spectral gap, gap(LσE, b f,H), of the generator acting on the Hilbert space associated with ⟨., .⟩σβ(H). The latter controls the mixing time of the dynamics when initialised in a well-chosen state ρini of the system with ρini ≤c σβ(H): tmix(ε) := inf t ≥0: etLσE, b f,H( ρini ) −σβ(H) 1 ≤ε ≤ 2 log c/ε gap(LσE, b f,H), given ε ∈(0, 1). Intuitively, as σE →0, the amount of resources required to implement the evolution up to time tmix(ε) blows up, while the spectral gap decreases as σE increases. In a companion paper, single-mode settings where a bad choice of filter f may lead to a vanishing gap, resulting in infinite mixing times, are also identified. Building on these insights, the main goal is to prove that, in the limit σE →∞, the resulting generator L b f,H remains gapped for a broad class of physically relevant bosonic systems, and to develop end-to-end schemes for computing their key physical properties.

The findings concerning the positivity of the spectral gap for the Bose-Hubbard model are detailed in Section III. The common strategy underlying the results is as follows: first, solvable or approximately solvable reference models, typically Gaussian or number-diagonal ones, whose associated Gibbs samplers admit explicit spectral control are identified. Second, it is shown that physically relevant perturbations preserve positivity of the dissipative gap and their fixed points stay close to that of the unperturbed dynamics. This allows for efficient preparation of the corresponding Gibbs state on qubit hardware, providing a quantum algorithm to compute thermal properties of the associated model.

This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise. Simulating thermal properties of Bose-Hubbard models on a quantum computer is considered. Recent progress has occurred in understanding the complexity of quantum many-body systems, developing quantum algorithms for estimating physically relevant observables, and demonstrating quantum simulations.

The preparation of thermal states has seen major progress, with a series of works introducing quantum Gibbs samplers based on Lindbladian dynamics and establishing efficient convergence in various regimes. However, classical algorithms can efficiently approximate thermal expectation values and partition functions in regimes where quantum algorithms are known to be efficient. This raises whether genuine quantum advantages can emerge in alternative physical settings.

While most existing results focus on locally finite-dimensional systems, such as spin models or fermionic lattices, bosonic systems have received less attention in computational complexity literature. Historically, discrete systems have dominated quantum algorithmic studies due to their parallel with classical digital computation, while bosonic systems were mainly explored in quantum communication and information theory. A rigorous complexity-theoretic framework for bosonic quantum systems has begun to emerge recently, revealing phenomena differing substantially from those of finite-dimensional systems.

Bosonic systems arise naturally in many areas of physics, including condensed matter, quantum optics, atomic physics and quantum chemistry. Their dynamics have been extensively studied from the perspective of Hamiltonian simulation and mathematical physics. Existing quantum runtime guarantees for Gibbs sampling in the bosonic setting are largely restricted to Gaussian systems, for which quantum advantage is not generally expected. Bosonic systems present unique challenges from a computational standpoint, as classical approximation techniques effective for spin systems often fail in infinite-dimensional settings.

The researchers demonstrated the first general framework for rigorously sampling Gibbs states in bosonic many-body systems. This is important because it extends efficient thermal state preparation, previously limited to finite-dimensional systems, to the more complex realm of infinite-dimensional bosonic models like the Bose-Hubbard Hamiltonian. The work establishes exponential convergence to the thermal state via gapped dissipative generators, and provides a quantum algorithm for computing thermal properties on qubit hardware. The authors suggest this controlled route to Gibbs sampling has implications for quantum simulation, thermalization, and understanding many-body complexity.