School of Mathematics and Statistics
Wuhan University
Fall 2025
August 20, 14:30--15:30pm, 雷军楼601
Speaker: Peixue Wu（University of Waterloo）
Title: From non-commutative optimal transport to limitations of quantum simulations
Abstract: We introduce a framework for quantifying the minimal resources required for quantum simulations based on the Lipschitz dual picture of non-commutative Wasserstein metric. This approach naturally leads to rigorous lower bounds on the circuit depth and volume necessary to implement quantum operations and prepare quantum states. In particular, we show that simulating a quantum channel whose Lipschitz constant scales linearly with the system size n requires a circuit depth lower bounded by Ω(logn). Moreover, applying this framework to Lindbladian-based algorithms for Gibbs or ground state preparation, we show that even for systems engineered to exhibit rapid mixing, the required circuit volume for implementing such algorithms scales at least linearly with n.
August 18, 14:00--15:00pm, 雷军楼601
Speaker: Sang-Jun Park （Université de Toulouse）
Title: Tensor free independence and central limit theorem
Abstract: Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In the joint work with Ion Nechita, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the “tensor distribution” limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of freeness, which we term “tensor freeness”. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables.
August 18, 11:00--12:00am, 雷军楼601
Speaker: Zhendong Xu（ Seoul National University ）
Title: Noncommutative L_p-Poincare inequalities
Abstract: We establish $L_p$-Poincar\'{e}-type inequalities for ergodic quantum Markov semigroups on finite von Neumann algebras, derived directly from the $L_2$-Poincar\'{e} inequality (PI) or from the mordified logarithmic Sobolev inequality (MLSI). Crucially, we obtain the asymptotic behaviors of best constants on $L_p$-Poincar\'{e}-type inequalities: $O(\sqrt{p})$ under the MLSI, while $O(p)$ under the PI, as $p\rightarrow\infty$. Applications encompass broad classes of ergodic quantum Markov semigroups, including but not limited to those on quantum tori, mixed $q$-Gaussian algebras, group von Neumann algebras, and compact quantum groups of Kac-type.
August 17, 16:00--17:00pm, 雷军楼601
Speaker: Jinge Bao （ School of Informatics, University of Edinburgh ）
Title: Learning junta distributions, quantum junta states, and QAC0 circuits
Abstract: In this work, we consider the problems of learning junta distributions, their quantum counterparts (quantum junta states) and circuits, which we show to be close to juntas.
(a). Junta distributions. A probability distribution is a -junta if it only depends on bits. We show that they can be learned with to error in total variation distance from samples, which quadratically improves the upper bound of Aliakbarpour et al. (COLT'16) and matches their lower bound in every parameter.
(b). Junta states. We initiate the study of -qubit states that are-juntas, those that are the tensor product of a -qubit state and an -qubit maximally mixed state. We show that these states can be learned with error ε in trace distance with single copies. We also prove a lower bound of copies. Additionally, we show that, for constant , copies are necessary and sufficient to test whether a state is ε-close or 7ε-far from being a k-junta.
(c) circuits. Nadimpalli et al. (STOC'24) recently showed that the Pauli spectrum of 𝖰 circuits (with a limited number of auxiliary qubits) is concentrated on low-degree. We remark that they implied something stronger, namely that the Choi states of those circuits are close to be juntas. As a consequence, we show that -qubit circuits with size , depth and auxiliary qubits can be learned from copies of the Choi state, improving the by Nadimpalli et al.
Spring 2025
May 28, 16:30--17:30pm, 雷军楼644
Speaker: Prof. Eric Bonnetier （Université Grenoble-Alpes）
Title: Small perturbations of the type of boundary condition of an elliptic PDE and application to shape optimization
Abstract: Asymptotic expansions of the solution to an elliptic PDE in the presence of inclusions of small size has been a topic of great interest in the past decades. Indeed, such expansions proved interesting for applications in inverse problems, as a means to build efficient and stable algorithms for detecting in homogeneities from boundary measurements. In this talk, we study the behavior of the solution to an elliptic equation when the boundary condition is perturbed on a small subset ω_ε of the boundary. We characterize the first term in the asymptotic expansion of the solution, in terms of the relevant measure of smallness of ω_ε, and we give explicit examples when ω_ε is a small surfacic ball in R^d,d=2,3.We use our asymptotic expansions to propose an algorithm for shape optimization problems, when the part of the boundary on which a specific boundary condition is prescribed is itself a design variable. This is joint work with Carlos Brito-Pacheco, Charles Dapogny, Rafael Estevez, and Michael Vogelius.
May 9, 10:00--11:00am, 雷军楼601
Speaker: Yangjing Dong (Nanjing University)
Title: The Computational Advantage of Mlp* Vanishes in the Presence of Noise
Abstract: Quantum multiprover interactive proof systems with entanglement MIP* are much more powerful than its classical counterpart MIP (Babai et al. '91, Ji et al. '20): while MIP = NEXP, the quantum class MIP* is equal to RE, a class including the halting problem. This is because the provers in MIP* can share unbounded quantum entanglement. However, recent works of Qin and Yao '21 and '23 have shown that this advantage is significantly reduced if the provers' shared state contains noise. This work attempts to exactly characterize the effect of noise on the computational power of quantum multiprover interactive proof systems. We investigate the quantum two-prover one-round interactive system MIP*[poly, O(1)], where the verifier sends polynomially many bits to the provers and the provers send back constantly many bits. We show noise completely destroys the computational advantage given by shared entanglement in this model. Specifically, we show that if the provers are allowed to share arbitrarily many noisy EPR states, where each EPR state is affected by an arbitrarily small constant amount of noise, the resulting complexity class is equivalent to NEXP = MIP. This improves significantly on the previous best-known bound of NEEEXP (nondeterministic triply exponential time) by Qin and Yao '21. We also show that this collapse in power is due to the noise, rather than the O(1) answer size, by showing that allowing for noiseless EPR states gives the class the full power of RE = MIP*[poly, poly]. Along the way, we develop two technical tools of independent interest. First, we give a new, deterministic tester for the positivity of an exponentially large matrix, provided it has a low-degree Fourier decomposition in terms of Pauli matrices. Secondly, we develop a new invariance principle for smooth matrix functions having bounded third-order Fréchet derivatives or which are Lipschitz continous.
April 25, 10:00~11:00am, 雷军楼601
Speaker: Zimu Li (Tsinghua University)
Title: Random Quantum Circuits and Unitary Designs
Abstract: Random quantum circuits, composed of random unitary gates, serve as canonical models of quantum dynamics. They provide critical insights into the nature of complex quantum many-body systems and have broad technological applications in quantum information. Situated at the intersection of representation theory, graph theory, stochastic processes, and functional analysis, the mathematical theory of random quantum circuits has attracted extensive study, significantly enriching the interplay between quantum information and mathematics.
In this talk, I will introduce the theoretical framework, known as the unitary design, to construct random quantum circuits. It can be seen as a special case of stochastic processes on groups, where we pay more attention to the unitary representation of the group on a quantum system and the convergence time to the desired (moments of) distribution. I will first review recent years' progresses in designing generic random circuits. Next, I will focus on unitary designs constrained by continuous and non-Abelian group symmetries, which are physically significant yet mathematically challenging. I will present our recent results on efficiently generating random circuits with and without symmetry constraints. Several illustrative examples and applications, including the dynamics of entanglement and correlation, the growth of quantum complexity, covariant random error correction codes, will be also discussed. Ref: arXiv:2411.04893, arXiv:2411.04898.
April 11, 14:00~15:00pm, 雷军楼601
Speaker: Ke Li (Harbin Institute of Technology)
Title: Reliability Function of Classical-Quantum Channels
Abstract: I will talk about the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Rényi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Rényi information. Based on joint work with Dong Yang.
April 5-6, Yangtze Analysis Seminar
April 2
Speaker: Qiang Zeng (Chinese Academy of Sciences)
Title: Replica symmetry breaking and landscape complexity for spin glasses
Abstract: In statistical physics, the study of spin glasses was initialized to describe the low temperature state of a class of magnetic alloys in the 1960s. The Sherrington-Kirkpatrick (SK) model is a mean field approximation of the physical short range spin glass model introduced in the 1970s. Starting in 1979, Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model. Since then, his method has been applied to study various complex systems, which eventually earned him the 2021 Nobel Prize in Physics.
In this talk, I will first introduce Parisi's conjectures and rigorous results by Talagrand and Panchenko. Then I will show that his prediction on infinite replica symmetry breaking holds at zero temperature for the more general mixed p-spin model. As an example for the application of Parisi's method, I will present Fyodorov and Le Doussal's prediciton on the Hessian spectrum at the global minimum of locally isotropic Gaussian random fields. A partial solution will be provided via landscape complexity.
March 21, 10:00am, 雷军楼A806
Speaker: Xiongfeng Ma (Tsinghua University)
Title: Embedded Quantum Circuit Complexity
Abstract: Quantum circuit complexity is a crucial metric across quantum information science, many-bodyphysics, and high-energy physics, traditionally focused on closed systems. In this talk, we delve intothe notion of embedded complexity in quantum circuits, characterizing interactions within largesystems. We introduce and analyze the complexity of projected states within subsystemsdemonstrating that it is significantly influenced by the circuit volume the aggregate of gatesaffecting both the subsystem and its complement. Our results indicate that ancillary qubits andmeasurements typically do not diminish the preparation cost of these projected states. We explorethe implications of circuit volume for embedded complexity and present a spacetime conversionmethod that leverages random gate teleportation to optimize circuit volume. Additionally, wepropose a streamlined shadow tomography protocol utilizing ancillary random states and Bell statemeasurements, providing valuable insights for practical implementations in quantum circuit designand analysis.Details are available in arXiv: 2408.16602.
March 14, 11-12am, 雷军楼 601
Speaker: Gupta Shubham (Tsinghua University)
Title: Discrete rearrrangement inequalities
Abstract: In the continuum, rearrangement inequalities is a classical theory that has been quite useful in solving problems coming from various parts of analysis: spectral geometry, calculus of variations, mathematical physics, spectral theory, to name a few. In my talk, I will discuss possible extensions of this theory to the discrete setting of graphs. It is a fairly new topic and most of the results in the area are proved in the last few years. I will talk about these developments, its connections with discrete isoperimetric inequalities, and its possible applications to problems concerning "analysis on graphs". The talk will be at the interface of discrete mathematics and analysis, and will be based on results obtained jointly with Stefan Steinerberger.
March 14, 10-11am, 雷军楼 601
Speaker: Kun Fang (Tsinghua University)
Title: Generalized quantum asymptotic equipartition
Abstract: We establish a generalized quantum asymptotic equipartition property (AEP) beyond the i.i.d. framework where the random samples are drawn from two sets of quantum states. In particular, under suitable assumptions on the sets, we prove that all operationally relevant divergences converge to the quantum relative entropy between the sets. More specifically, both the smoothed min- and max-relative entropy approach the regularized relative entropy between the sets. Notably, the asymptotic limit has explicit convergence guarantees and can be efficiently estimated through convex optimization programs, despite the regularization, provided that the sets have efficient descriptions.
Fall 2024
(2025) January 8，10:00-11:00 am, 雷军楼 601
Speaker: Jiabin Liu (天津工业大学)
Title: From Mazur-Ulam Theorem to Wigner's Theorem
Abstract: Whether metric structures can determine linear structures is a classic and important topic in the isometric theory of Banach spaces. The Wigner’s theorem states that a mapping between inner product spaces preserves the modulus of the inner product if and only if it is the product of a function of modulus one and a linear or anti-linear isometry. Recently, this theorem was generalized to the setting of real normed spaces. In this seminar, We will present two recent Wigner-type results obtained on complex smooth spaces.
December 23，10:00-11:00 am, 东北楼 404
Speaker: Hao Zhang (University of Illinois at Urbana-Champaign)
Title: Schatten class membership of commutators
Abstract: Commutators are well-known as generalizations of Hankel type operators. In this talk, we are concerned with the Schatten class membership of commutators involving nondegenerate singular integral operators. We describe their Schatten class membership in terms of Besov spaces. Our approach is based on the dyadic representation theorem of singular integral operators and the complex median method. If time permits, I will also introduce the endpoint estimate of the weak-type Schatten class of commutators.
November 11，10:00-11:00 am, 东北楼 404
Speaker: Bowen Li (City University of Hong Kong)
Title: Polynomial-Time Preparation of Low-Temperature Gibbs States for 2D Toric Code
Abstract: We propose a polynomial-time algorithm for preparing the Gibbs state of the two-dimensional toric code Hamiltonian at any temperature, starting from any initial condition, significantly improving upon prior estimates that suggested exponential scaling with inverse temperature. Our approach combines the Lindblad dynamics using a local Davies generator with simple global jump operators to enable efficient transitions between logical sectors. We also prove that the Lindblad dynamics with a digitally implemented low temperature local Davies generator is able to efficiently drive the quantum state towards the ground state manifold. Despite this progress, we explain why protecting quantum information in the 2D toric code with passive dynamics remains challenging.
September 9，2:00-3:00 pm, 东北楼 404
Speaker: Peixue Wu (Waterloo)
Title: Lower bound for simulation cost ofquantum markov semigroups
Abstract: In this work, we present a general framework to calculate the lower bound for simulating a broad class of quantum Markov semigroups. Given a fixed accessible unitary set, we introduce the concept of con- vexified circuit depth to quantify the quantum simulation cost and analyze the necessary circuit depth to construct a quantum simulation scheme that achieves a specific order. Our framework can be applied to both unital and non-unital quantum dynamics, and the tightness of our lower bound technique is illustrated by showing that the upper and lower bounds coincide in several examples. The talk is based on arXiv:2303.11304
August 19, 9:00-10:00 am, 东北楼 404
Speaker: Bang Xu (Houston)
Title: Noncommutative quantitative mean ergodic inequalities
Abstract: We establish the quantitative mean ergodic theorems for two subclasses of power bounded operators on a fixed noncommutative Lp-space with 1<p<∞, which mainly concerns power bounded invertible operators and Lamperti contractions. Our approach to the quantitative ergodic theorems relies on noncommutative square function inequalities. The establishment of the latter involves several new ingredients such as the almost orthogonality and Calder\'on-Zygmund arguments for non-smooth kernels from semi-commutative harmonic analysis, the extension properties of the operators under consideration from operator theory, and a noncommutative version of transference method due to Coifman and Weiss.
Spring 2024
July 4, 2:00-3:00 pm, 东北楼 209
Speaker: Yifan Jia（Technical University of Munich）
Title: A generic quantum Wielandt's inequality for matrix and Lie algebras
Abstract: Quantum Wielandt's inequality gives an optimal upper bound on the minimal length $k$ such that length-$k$ products of elements in a generating system span $M_n(\mathbb{C})$. It is conjectured that $k$ should be of order $\mathcal{O}(n^2)$ in general. In this talk, we first give an overview of how the question has been studied in the literature so far and its relation to a classical question in linear algebra, namely the length of the matrix algebra $M_n(\mathbb{C})$. We provide a generic version of Quantum Wielandt's inequality, which gives the optimal length with probability one. Specifically, we prove, based on [Klep and Spenko, 16] that $k$ generically is of order $\Theta(\log n)$, as opposed to the general case, where the best bound to date is $\mathcal O(n^2 \log n)$. Our result implies a new bound on the primitivity index of a random quantum channel. Furthermore, we shed new light on a long-standing open problem for Projected Entangled Pair State by concluding that almost any translation-invariant PEPS (in particular, Matrix Product State) with periodic boundary conditions on a grid with side length of order $\Omega( \log n )$ is the unique ground state of a local Hamiltonian.
Additionally, we observe similar characteristics for matrix Lie algebras, which we define as the Lie-length of a matrix Lie algebra. Numerical results for random Lie-generating systems indicate that Lie-lengths generically scale as $\Theta(\log n)$ for some typical $\Theta(n^2)$-dimensional Lie algebras e.g. $\mathfrak{su}(n)$, with a small number of Lie-generators. We prove this upper bound for almost any Lie-generating system $S\subseteq sl_n(\mathbb{C})$ with $\Omega(\log n)$ elements and indicate how this generic result for Lie algebras would apply to mathematical models of some physical problems, especially also to gate implementation of quantum computers.
June 25, 11:15-11:55 am, 东北楼 209
Speaker: Haonan Zhang（University of South Carolina）
Title: On the quantum KKL theorem and related inequalities
Abstract: The KKL theorem is a fundamental result in Boolean analysis, stating that any Boolean function has an influential variable. Montanaro and Osborne proposed a quantum extension of Boolean functions. In this context, some classical results have been extended to the quantum setting, such as Talagrand's- inequality. However, a quantum version of the KKL theorem seems to be missing, as conjectured by Montanaro and Osborne. In this talk, I will present an alternative answer to this question, saying that every balanced quantum Boolean function has a geometrically influential variable. This is based on joint work with Cambyse Rouzé (Inria) and Melchior Wirth (IST Austria).
June 25, 10:30-11:10 am, 东北楼 209
Speaker: Youming Qiao（University of Technology Sydney）
Title: Graphs and tensors
Abstract: Graphs are a prime object in combinatorics, and tensors appear naturally in linear algebra and geometry. Some recent works reveal intriguing connections between graphs and tensors, the first of which trace back to the works of Tutte and Lovász. In this talk we will present some such connections, and how these connections lead to results in quantum information, computational complexity, cryptography, invariant theory, and group theory.
June 25, 9:45-10:25 am, 东北楼 209
Speaker: Penghui Yao（Nanjing University）
Title: On Testing and Learning Quantum Junta Channels
Abstract: We consider the problems of testing and learning quantum k-junta channels, which are n-qubit to n-qubit quantum channels acting non-trivially on at most k out of n qubits and leaving the rest of qubits unchanged. We present the algorithms for testing and learning quantum junta channels. This answers an open problem raised by Chen et al. (2023). In order to settle these problems, we develop a Fourier analysis framework over the space of superoperators and prove several fundamental properties, which extends the Fourier analysis over the space of operators introduced in Montanaro and Osborne (2010).
June 25, 9:00-9:40 am, 东北楼 209
Speaker: Anurag Anshu（Harvard University）
Title: Learning theory in the “physical” corner of the Hilbert space
Abstract: We will discuss recent progress in the learnability of various classes of important quantum many-body states, such as Gibbs states, shallow quantum circuits and matrix product states. Our primary focus will be provably efficient algorithms, which often reveal new structures in the quantum states. Locality play a key role in these results, requiring measurements that only act on a few qubits in an entangling way. We will also highlight key challenges in further progress, in particular the search for optimal algorithms.