北京雁栖湖应用数学研究院

The talk will be focused on examples and some general features of Hybrid quantum-classical integrable systems. In this systems classical inferable dynamics, commuting flows is lifted to a quantum “fibers” resulting in a geometric structure similar to a flat connection on a vector bundle.

In this talk, I will introduce a partition function defined on bi-colored three-manifolds decorated by tensor diagrams from a spherical fusion category C. This partition function yields three-manifold invariants and three-dimensional topological quantum field theories (TQFTs). I will discuss how well-known invariants and TQFTs, such as Turaev-Viro theory and Reshetikhin-Turaev theory, can be naturally embedded within our framework. Furthermore, our bi-colored theory provides topological interpretations for fundamental concepts in tensor categories, including the Drinfeld center and Frobenius-Schur indicators.

In this talk, we will give a sufficient condition for the existence of tensor category structure on the representation categories of vertex operator algebras, and consequently provide a few examples of tensor categories. We will also discuss how to prove rigidity of such tensor categories.

In this talk, we focus on the weighted Hardy spaces of polynomial growth, which cover the classical Hardy space, weighted Bergman spaces, weighted Dirichlet spaces and much broader. We discuss the boundedness of the composition operators with symbols of analytic automorphisms of unit open disk (finite Blaschke product and analytic functions on the unit closed disk) acting on weighted Hardy spaces of polynomial growth. Moreover, we study the norms, spectra and (semi-)Fredholmness of composition operators induced by disc automorphisms. Furthermore, the Jordan decomposition theorem and similar classification for the representation of analytic functions on the unit closed disk as multiplication operators are obtained.

Entanglement distillation is crucial in quantum information processing. But it remains challenging to estimate the distillable entanglement and its closely related essential quantity, the quantum capacity of a noisy quantum channel. This work proposes methods for evaluating these two central quantities in quantum Shannon theory. We also apply our methods to investigate purifying the maximally entangled states under practical noises and notably establish improvements. Our bounds also offer useful benchmarks for evaluating the quantum capacities of basic quantum channels of interest, including the Pauli channels and the random mixed unitary channels.

C*-algebras as noncommutative spaces have deep application in differential geometry and mathematical physics. I this talk, I will brief discuss the basic ideas in non-commutative geometry and present my joint work with Huaxin Lin (and part of them are also with George Elliott and Zhuang Niu) on the classification of simple separable C*-algebras with finite nuclear dimensions.

The construction and classication of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this talk, I will discuss how to construct and classify crystalline topological superconductors (TSC) and topological insulators (TI) in interacting fermion systems. I will also discuss the relationship between internal symmetry protected SPT phases and crystalline symmetry protected SPT Phases.

Conformal blocks associated to a vertex operator algebras (VOA) V and a Riemann surface are chiral halves of 2d conformal field theory in the sense of Segal. It is the key ingredient of the tensor category of VOA modules Mod(V). Assume that the VOA V and its modules are unitary. Then on the space of conformal blocks one can define (algebraically) a non-degenerate Hermitian product. If this Hermitian product is positive definite (which is expected to be always true) then Mod(V) is a unitary modular tensor category. In this talk, I will explain how to understand this Hermitian product geometrically in terms of the complex conjugate structures of Riemann surfaces. This geometric understanding will help us prove the positive definiteness for certain examples that cannot be proved using other methods.

The standard notion of morphisms between operator algebras is (*-)homomorphisms. A more general notion called c.p.c. order zero maps was introduced by W. Winter and J. Zacharias. Extensive theories have been built on top of this notion, including various dimensions for C*-algebras and C*-dynamical systems (even with quantum group actions). It has played a pivotal role in the classification program of C*-algebras and is seeing more applications to C*-dynamical systems and quantum principal bundles. I will survey some of the ideas and results around this notion.

Unravelling the source of quantum computing power has been a major goal in the field of quantum information science. In recent years, the quantum resource theory (QRT) has been established to characterize various quantum resources, yet their roles in quantum computing tasks still require investigation. The so-called universal quantum computing model (UQCM), e.g., the circuit model, has been the main framework to guide the design of quantum algorithms, creation of real quantum computers etc. In this work, we combine the study of UQCM together with QRT. We find, on one hand, using QRT can provide a resource-theoretic characterization of a UQCM, the relation among models and inspire new ones, and on the other hand, using UQCM offers a framework to apply resources, study relation among resources and classify them. We develop the theory of universal resources in the setting of UQCM, and find a rich spectrum of UQCMs and the corresponding universal resources. Depending on a hierarchical structure of resource theories, we find models can be classified into families. In this talk, I will present four natural families of UQCMs: the amplitude family, the quasi-probability family, the Hamiltonian family, and the evolution family. They include some well known models, like the measurement-based model and adiabatic model, and also inspire new models such as the contextual model and von Neumann architecture we introduce. Each family contains at least a triplet of models, and such a succinct structure of families of UQCMs offers a unifying picture to investigate resources and design models. It also provides a rigorous framework to resolve puzzles, such as the role of entanglement vs. interference, and unravel resource-theoretic features of quantum algorithms. See arXiv:2303.03715, arXiv:2304.03460.

Higher Kazhdan projections were introduced by Li-Nowak-Pooya, associated to higher degree group cohomology, in comparision to the classical Kazhdan projection appeared in the degree zero cohomology. In the joint work with Sanaz Pooya, we explicitly describe the K-theory class of the higher Kazhdan projections of certain free product groups and their Cartesian products. The explicit description enables us to obtain new calculations of Lott’s delocalized l^2-Betti numbers, which are generalizations of the classical Betti numbers and l^2-Betti numbers. In particular, we establish the first non-vanishing results for infinite groups.

I will discuss some recent results about relative entropies in QFT, with particular emphasis on the singular limits of such entropies.

Each of the two classical series of Lie groups, SL(n) and OSp(n), gives a two-parameter quantum invariant of knots (and suitable graphs). There is also a conjectural third classical series, the exceptional series, containing all the exceptional Lie groups (and thus having only finitely many points). We look at the quantum version of this third series, and show that it satisfies a simple quantum Jacobi relation, giving a (conjectural) skein-theoretic description for a third two-parameter quantum exceptional polynomial invariant. We can unconditionally use these to compute previously out-of-reach knot polynomials (for particular exceptional groups) for all knots with 12 or fewer crossings.

Distillation, or purification, is central to the practical use of quantum resources in noisy settings often encountered in quantum communication and computation. Conventionally, distillation requires using some restricted “free” operations to convert a noisy state into one that approximates a desired pure state. Here, we propose to relax this setting by only requiring the approximation of the measurement statistics of a target pure state, which allows for additional classical postprocessing of the measurement outcomes. We show that this extended scenario, which we call virtual resource distillation, provides considerable advantages over standard notions of distillation, allowing for the purification of noisy states from which no resources can be distilled conventionally. We show that general states can be virtually distilled with a cost (measurement overhead) that is inversely proportional to the amount of existing resource, and we develop methods to efficiently estimate such cost via convex and semidefinite programming, giving several computable bounds. We consider applications to coherence, entanglement, and magic distillation, and an explicit example in quantum teleportation (distributed quantum computing). This work opens a new avenue for investigating generalized ways to manipulate quantum resources.

Randomized benchmarking (RB) is the gold standard for experimentally evaluating the quality of quantum operations. The current framework for RB is centered on groups and their representations but this can be problematic. For example, Clifford circuits need up to O(n^2) gates and thus Clifford RB cannot scale to larger devices. Attempts to remedy this include new schemes such as linear cross-entropy benchmarking (XEB), cycle benchmarking, and nonuniform RB but they do not fall within the group-based RB framework. In this work, we formulate the universal randomized benchmarking (URB) framework, which does away with the group structure and also replaces the recovery-gate-plus-measurement component with a general “post-processing” positive operator-valued measurement (POVM). Not only does this framework cover most of the existing benchmarking schemes but it also gives the language for and helps inspire the formulation of new schemes. We specifically consider a class of URB schemes called twirling schemes. For twirling schemes, the post-processing POVM approximately factorizes into an intermediate channel, inverting maps, and a final measurement. This leads us to study the twirling map corresponding to the gate ensemble specified by the scheme. We prove that if this twirling map is strictly within unit distance of the Haar twirling map in induced diamond norm, the probability of measurement as a function of gate length is a single exponential decay up to small error terms. The core technical tool we use is the matrix perturbation theory of linear operators on quantum channels. As an application, we investigate the theoretical foundation of the linear XEB, and propose a variant using Clifford circuits, that allows efficient classical post-processing and supports holistic benchmarking of over 1,000 qubits.

Duan, Severini, and Winter proposed the study of a specific matrix space as a quantum generalization of graphs, which allows for the formulation and study of a quantum version of Shannon’s zero-error capacity problem. In this talk, we further develop the combinatorial theory of matrix spaces through the lens of graph theory. Initially, we introduce basic correspondences between matrix space properties and graph-theoretical properties, such as nilpotency versus acyclicity, irreducibility versus connectivity, and dimension expansion versus vertex expansion. Subsequently, we demonstrate how these correspondences can be enhanced to the so-called inherited correspondences, which lead to extremal problems for matrix spaces and have applications in invariant theory and noncommutative algebra. Finally, we discuss applications in quantum information processing and provide examples of graph-theoretic properties that are no longer valid in the matrix space setting.

We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C*-algebras. We show that for inductive limits of finite dimensional actions of fusion categories on unital AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on AF-algebras.

In this talk, Pan Zhang will introduce a general framework for decoding quantum error correction codes using generative modeling. The model learns the joint probability of logical operators of multiple logical qubits and all syndromes using autoregressive neural networks, specifically with casual transformers. The training is in an unsupervised way without requiring any labeled training data, so is termed as pre-training, after which, the model offers fast computation of the likelihood of the logical operators, and directly generates the most-likely logical operators for all logical qubits in the manner of maximum likelihood decoding, for all syndromes. Based on the pre-trained model, Pan Zhang will futher introduce refinement to compute a more accurate likelihood of logical operators for a given syndrome by using reinforcement learning and by directly sampling the stabilizer operators. The framework is general, it applies to arbitrary error models and applies in the same way to quantum codes with different topologies such as surface code and general quantum LDPC code. It also utilizes the parallelization power of GPUs to decode a large number of syndromes simultaneously. The approach sheds like on the efficient decoding of quantum error correction codes using generative artificial intelligence and modern computational power.

I will explain why topological defects in an n+1D topological order form a fusion n-category with a trivial E_1-center. I will present some basic results on separable n-categories, E_m-monoidal fusion n-categories, E_n-centers and their physical applications obtained in arXiv:2011.02859 and arXiv:2107.03858.

Moore’s law states that the number of transistors on a microprocessor chip will double every two years or so. It was worldwide acknowledged that Moore’s law was nearing its end. In other words, the miniaturization of transistors has been an essential progress in computers mainly to speed up their computation. Such miniaturization has approached its fundamental limits. Fortunately, the development of quantum computing brings light to solve this problem. The method for high accuracy surface modelling (HASM) is an approach to reinforced machine learning. It can be transformed into a large sparse linear system and combined with the Harrow-Hassidim-Lloyd (HHL) quantum algorithm, by which a HASM-HHL algorithm was developed for quantum machine learning. HASM has been successfully operated on classical computers to conduct spatial interpolation, upscaling, downscaling, data fusion and model-data assimilation of eco-environmental surfaces, such as digital terrain models, climate change, carbon stocks, CO2 concentrations, soil properties, COVID-19, species diversity, and ecosystems. In all of these applications, HASM has produced more accurate results than other methods, thereby foreshadowing the advantages that would likely follow the adoption and use of HASM-HHL for such applications. Ideally, HASM-HHL can maintain the high accuracy of classical algorithms, and meanwhile it can achieve exponential speedup compared to the classical algorithms, which has been demonstrated by several case-studies in Poyang Lake Basin.

Trapped atomic ions is among the leading platforms in building a scalable quantum computer. By encoding naturally identical qubits in the atoms, we entangle them by a controlled laser drive. We show how to tailor the inter-qubit couplings to perform quantum simulations of non-equilibrium physics. Combining global interactions and individual single qubit addressing, a universal quantum computer can be realized. Building upon these tools, we show how numerical optimizations can aid the design of qubit couplings in higher dimensions, even when the physical system is one-dimensional. We discuss further potentials of building a graph-model based quantum computer.

It has been known that all bipartite pure quantum state can be self-tested, i.e., any such state can be certified completely by first measuring both subsystems of this state by proper local quantum measurements and then observing that the correlation between the choices of measurements and their outcomes satisfies certain condition, where the conclusion can be reliable even if the involved quantum measurements are untrusted. In such protocols, quantum nonlocality is crucial and plays a central role, which means that each party has to conduct at least two different quantum measurements to produce a desirable correlation. Here, we prove that when the underlying Hilbert space dimension is known beforehand, any d x d bipartite pure state can be pinned down completely (up to local transformations) by a certain correlation generated by only one measurement setting on each subsystem, where each measurement produces only 3d outcomes. In our protocol, there is no any quantum nonlocality involved.