Tim Möbus

Discrete and continuous variables oftentimes require different treatments in many learning tasks. Identifying the Hamiltonian governing the evolution of a quantum system is a fundamental task in quantum learning theory. While previous works mostly focused on quantum spin systems, where quantum states can be seen as superpositions of discrete bit-strings, relatively little is known about Hamiltonian learning for continuous-variable quantum systems. In this work we focus on learning the Hamiltonian of a bosonic quantum system, a common type of continuous-variable quantum system. This learning task involves an infinite-dimensional Hilbert space and unbounded operators, making mathematically rigorous treatments challenging. We introduce an analytic framework to study the effects of strong dissipation in such systems, enabling a rigorous analysis of cat qubit stabilization via engineered dissipation. This framework also supports the development of Heisenberg-limited algorithms for learning general bosonic Hamiltonians with higher-order terms of the creation and annihilation operators. Notably, our scheme requires a total Hamiltonian evolution time that scales only logarithmically with the number of modes and inversely with the precision of the reconstructed coefficients. On a theoretical level, we derive a new quantitative adiabatic approximation estimate for general Lindbladian evolutions with unbounded generators. Finally, we discuss possible experimental implementations.

In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched Rényi divergences such as the sandwiched Rényi conditional entropy. We follow three different approaches: The first one is the axiomatic approach, which exploits the sub-/ superadditivity and joint concavity/ convexity of the exponential of the divergence. In our second approach, termed the "operator space approach", we express the entropic measures as norms and utilize their properties for establishing the bounds. These norms draw inspiration from interpolation space norms. We not only demonstrate the norm properties solely relying on matrix analysis tools but also extend their applicability to a context that holds relevance in resource theories. By this, we extend the strategies of Marwah and Dupuis as well as Beigi and Goodarzi employed in the sandwiched Rényi conditional entropy context. Finally, we merge the approaches into a mixed approach that has some advantageous properties and then discuss in which regimes each bound performs best. Our results improve over the previous best continuity bounds or sometimes even give the first continuity bounds available. In a separate contribution, we use the ALAAF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

We investigate quantum Markov semigroups on bosonic Fock space and identify a broad class of infinite-dimensional dissipative evolutions that exhibit instantaneous Sobolev-regularization. Motivated by stability problems in quantum computation, we show that for certain Lindblad operators that are polynomials of creation and annihilation operators, the resulting dynamics immediately transform any initial state into one with finite expectation in all powers of the number operator. A key application is in the bosonic cat code, where we obtain explicit estimates in the trace norm for the speed of convergence. These estimates sharpen existing perturbative bounds at both short and long times, offering new analytic tools for assessing stability and error suppression in bosonic quantum information processing. For example, we improve the strong exponential convergence of the (shifted) $2$-photon dissipation to its fixed point to the uniform topology.

We present a modular algorithm for learning external potentials in continuous-space free-fermion models including Coulomb potentials. Compared to the lattice-based approaches, the continuum presents new mathematical challenges: the state space is infinite-dimensional and the Hamiltonian contains the Laplacian, which is unbounded in the continuum and produces an unbounded speed of information propagation. Our framework addresses these difficulties through novel optimization methods and information-propagation bounds in combination with a priori regularity assumptions on the external potential. The resulting algorithm provides a unified and robust approach to learn both Coulomb interactions and other classes of physically relevant potentials, like trigonometric polynomials or general smooth functions. One possible application is to learn the charge and position of nuclei and ions distributed in continuous space as in quantum chemistry. Our results thus lay the foundations for a scalable and generalizable toolkit to explore fermionic systems governed by continuous-space interactions. Moreover, we provide numerical evidence supporting the classical post-processing algorithm of our Coulomb algorithm.