Albert H. Werner

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.

Hamiltonian learning protocols are quickly establishing themselves as valuable tools to benchmark and verify quantum computers and simulators. However, virtually no rigorous protocols exist to learn time-dependent Hamiltonians and Lindbladians, despite their widespread applications. In this work, we address this gap and show how to learn the time-dependent evolution of a locally interacting $n$-qubit system arranged on a graph $\mathsf{G}$ of effective dimension $D$ by resorting only to the preparation of product Pauli eigenstates, evolution by the time-dependent generator for given times and measurements in product Pauli bases. We assume that the time-dependent parameters are well-approximated by functions in a known space of dimension $m$ and for which we can efficiently perform stable interpolation, say by polynomial functions. Our protocol outputs an expansion in that basis that approximates the parameters up to $\epsilon$ in an interval. The protocol only requires $\widetilde{\cO}\big(\epsilon^{-2}\,\poly{m}\,\log(n\delta^{-1})\big)$ samples and $\poly{n,m}$ preprocessing and postprocessing to learn the parameters with probability of success $1-\delta$, making it highly scalable. Importantly, the scaling in the dimension $m$ is polynomial, whereas naive extensions of previous methods yield a dependency that is exponential in $m$. Like previous protocols for the time-independent case, ours is mostly based on estimating time derivatives of expectation values of various observables through interpolation techniques. We then obtain well-conditioned linear equations that allow us to evaluate the value of the time-dependent function for a local generator. However, whereas in the time-independent case it sufficed to only consider derivatives at time $t=0$, here we need to evaluate them at finite times while still being able to relate the derivatives to parameters of the evolution. Thus, besides dealing with technical intricacies related to the time-dependent case, our main innovation is to show how to combine Lieb-Robinson bounds, process shadows and semidefinite programs to estimate the parameters of the evolution efficiently at constant times. Along the way, we extend state-of-the-art Lieb-Robinson bounds on general graphs to the time-dependent, dissipative setting, a result of independent interest. In addition, we show how our technique can be used to verify the outputs of time-dependent dynamics for polynomial times from access to short-time dynamics for cases of interest like linear adiabatic schedules. As such, our protocol is a valuable tool to verify various state preparation procedures on quantum computers and simulators, such as adiabatic preparation, or to characterize time-dependent Markovian noise.

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.

Abstract We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an asymptotic variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum. These strong constraints allow us to classify and explicitly describe all such monotones, leading to a rate formula expressed as an optimization involving sandwiched Renyi divergences. As an application we give a new derivation of the strong converse error exponent in quantum hypothesis testing.

We introduce a general model of stochastically generated matrix product states (MPS) in which the local tensors share a common distribution and form a strictly stationary sequence, without requiring spatial independence. Under natural conditions on the associated transfer operators, we prove the existence of thermodynamic limits of expectation values of local observables and establish almost-sure exponential decay of two-point correlations. In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially in the distance between the two sites, with a deterministic rate. In the i.i.d.\ case, the exponential decay still holds with a deterministic rate, with the probability approaching one exponentially fast in the distance. For strictly stationary ensembles with decaying spatial dependence, the correlation decay quantitatively reflects the mixing profile: $\rho$–mixing yields polynomial bounds with high probability, while stretched-exponential (resp. exponential) decay in $\rho$ (resp. $\beta$) yields stretched-exponential (resp. exponential) decay of the two-point function, again with correspondingly strong high-probability guarantees. Altogether, the framework unifies and extends recent progress on stationary ergodic ensembles and Gaussian translation-invariant ensembles, providing a transfer-operator route to typical correlation decay in random MPS.

Open many-body quantum systems play an important role in quantum optics and condensed-matter physics, and capture phenomena like transport, interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behaviour, for various open quantum systems ranging from few to many bodies.

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and black hole scrambling, as well as in process design. In this work, we present a theory for continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analyzing local random quantum processes. Much of the progress reported is of technical nature: in particular, by relying on representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. We also introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes.