Beijing Institute of Mathematical Sciences and Applications

Quantum computers have the potential to gain algebraic and even up to exponential speed up compared with its classical counterparts, and can lead to technology revolution in the 21st century. Since quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators. The most efficient quantum PDE solver is quantum simulation based on solving the Schrodinger equation. It will be interesting to explore what other problems in scientific computing, such as ODEs, PDEs, and linear algebra that arise in both classical and quantum systems, can be handled by quantum simulation. We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs. For non-autonomous PDEs and ODEs, or Hamiltonian systems with time-dependent Hamiltonians, we also add an extra dimension to transform them into autonomous PDEs that have only time-independent coefficients, thus quantum simulations can be done without using the cumbersome Dyson’s series and time-ordering operators. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing.

It is anticipated that quantum computing will be able to tackle hard real-world problems. Fluid dynamics, a highly challenging problem in classical physics and various applications, emerges as a good candidate for showing quantum utility. We report our recent progress on quantum computing of fluid dynamics. In theory, we propose a quantum spin representation of fluid dynamics, which transforms the Navier-Stokes equation into the Schrödinger-Pauli equation through the generalized Madelung transformation. In this way, the fluid flow can be regarded as a special quantum system, which is feasible for flow simulation on a quantum computer. In terms of algorithm, we propose a quantum Hamiltonian simulation algorithm, which is able to simulate compressible or incompressible flows and scalar convection-reaction-diffusion problems with quantum acceleration. In terms of hardware implementation, we have realized the quantum simulation of two-dimensional unsteady flow on a quantum processor. These results demonstrate the potential of quantum computing to simulate complex flows, including turbulence, in future endeavors.

The rapid development of quantum technology brings the quantum measurement and noise as new ingredients to the quantum many-body physics, asking for deeper understanding of the mixed state phase of matter in open quantum systems. In this talk I will first discuss how the paradigmatic Ising phase transition is transformed to the strongly disordered Nishimori transition in an open quantum system, about its theory and experimental realization in the GHZ state preparation or decoherence of toric code. Then I will focus on the Kramers Wannier self-duality, a non-invertible algebraic symmetry, being generalized to the mixed state as an average symmetry that governs a new type of criticality.

Reversing an unknown unitary evolution remains a formidable challenge, as conventional methods necessitate an infinite number of queries to fully characterize the quantum process. Here we introduce the Quantum Unitary Reversal Algorithm (QURA), a deterministic and exact approach to universally reverse arbitrary unknown unitary transformations using $O(d^2)$ calls of the unitary, where d is the system dimension. Our construction resolves a fundamental problem of time-reversal simulations for closed quantum systems by affirming the feasibility of reversing any unitary evolution without knowing the exact process. The algorithm also provides the construction of a key oracle for unitary inversion in quantum algorithm frameworks such as quantum singular value transformation. We also propose the framework of virtual combs that exploit the unknown process iteratively with additional classical post-processing to simulate the process inverse.

The branching algorithm is a fundamental technique for designing fast exponential-time algorithms to solve combinatorial optimization problems exactly. It divides the entire solution space into independent search branches using predetermined branching rules and ignores the search on suboptimal branches to reduce the time complexity. The complexity of a branching algorithm is primarily determined by the branching rules it employs, which are often designed by human experts. In this paper, we show how to automate this process with a focus on the maximum independent set problem. The main contribution is an algorithm that efficiently generates optimal branching rules for a given sub-graph with tens of vertices. Its efficiency enables us to generate the branching rules on-the-fly, which is provably optimal and significantly reduces the number of branches compared to existing methods that rely on expert-designed branching rules. Numerical experiment on 3-regular graphs shows an average complexity of $O(1.0441^n)$ can be achieved, better than any previous methods.

One of the most promising types of quantum algorithms are those that solve combinatorial optimization problems. There are a number of difficulties that stand in the way: small gaps to excited states, barren plateaus in the energy, and incomplete expressibility of the states. These obstacles can be dealt with by making the quantum algorithms adaptable. I will describe several ways to do this, and show that considerable improvements over non-adaptive algorithms are possible.

Quantum entanglement is a distinctive feature of quantum physics and is key resource in many quantum information tasks. Entanglement in continuous-variable non-Gaussian states are of particular interest in quantum technology due to their potential applications in quantum computing and quantum metrology. However, how to create such states and detect its non-Gaussian entanglement remain a challenge since the sheer amount of information in such states grows exponentially and makes a full characterization impossible. Here, I would introduce our recent progress for creating non-Gaussian states and experimentally feasible approach to detect non-Gaussian entanglement.

Variational quantum optimization is a pivotal approach for leveraging near-term quantum devices to achieve practical quantum advantages. However, it faces two major challenges that impede its scalability and effectiveness. The first challenge arises when quantum circuits become deep, leading to the barren plateau phenomenon, where the optimization landscape becomes exponentially flat. The second challenge emerges with shallow circuits, which are susceptible to exponential local traps, hindering the convergence to desirable solutions. In this talk, I will present two strategies addressing these distinct challenges. To mitigate the barren plateaus associated with deep circuits, we propose leveraging many-body localization (MBL) within Floquet-initialized quantum circuits. By initializing the circuit in the MBL phase, we inhibit the formation of a unitary 2-design, thereby maintaining area-law entanglement instead of volume-law entanglement. This approach effectively circumvents barren plateaus during the optimization process. In the presence of the second challenge—exponential local traps in shallow circuits—we explore new applications in quantum phase transition characterization. We introduce a hybrid algorithm that integrates quantum optimization with classical machine learning techniques—utilizing LASSO for identifying conventional phase transitions and Transformer models for detecting topological transitions. This method employs a sliding window of Hamiltonian parameters to learn appropriate order parameters and accurately estimate critical points.

Parameterized Quantum Circuits (PQCs) are crucial in quantum machine learning and circuit synthesis, enabling the practical realization of complex quantum tasks. However, the learning of PQCs is primarily limited to classical optimization methods, facing issues such as gradient vanishing. In this work, we introduce a nested optimization model that utilizes quantum gradients to enhance the learning of PQCs for polynomial-type cost functions. Our approach leverages quantum algorithms to effectively navigate the optimization landscape, identifying and overcoming a typical type of gradient vanishing in learning a PQC. Through numerical experiments, we demonstrate the feasibility of our method on two tasks: the Max-Cut problem and polynomial optimization. Our method excels in generating circuits without gradient vanishing and effectively optimizes the cost functions. In addition, from the perspective of quantum algorithms, our model improves quantum optimization for polynomial-type cost functions, addressing the challenge of exponential growth in sample complexity.

I will discuss a construction of a family of 1D quantum lattice models that respect unitary fusion category symmetry. This family can be thought of as edge models of 2D symmetry-enriched topological states. An interesting feature of these models is that they often (but may not always) exhibit a gapless critical phase (i.e., a gapless region of codimension zero in the parameter space) due to the presence of category symmetry. I will discuss numerical results of some examples.

In the first part, we review the classic non-commutative polynomial optimization, the Navascues-Pironio-Acin hierarchy of semidefinite relaxations, and its application in quantum information science – computing the maximum quantum violation of linear Bell inequalities. State polynomials (polynomials in operators and expectations) are a further generalization of non-commutative polynomials. In the second part, we introduce the theory of state polynomial optimization, and present the generalization of Navascues-Pironio-Acin hierarchy for solving state polynomial optimization problems. Finally, we give a method for computing the maximum quantum violation of nonlinear Bell inequalities based on state polynomial optimization.

A rational two-dimensional conformal field theory often gives a modular invariant partition function through the action of $SL(2,\mathbb{Z})$ on the upper half plane. We present a new physical interpretation of such modular invariance and relate modular invariance, violation of Haag duality, and the Renyi entropy.

Masahito Hayashi received the B.S. degree from the Faculty of Sciences, Kyoto University, Japan, in 1994, and the M.S. and Ph.D. degrees in mathematics from Kyoto University, Japan, in 1996 and 1999, respectively. He worked in Kyoto University as a Research Fellow of the Japan Society of the Promotion of Science (JSPS) from 1998 to 2000, and worked in the Laboratory for Mathematical Neuroscience, Brain Science Institute, RIKEN from 2000 to 2003, and worked in ERATO Quantum Computation and Information Project, Japan Science and Technology Agency (JST) as the Research Head from 2000 to 2006. He worked in the Graduate School of Information Sciences, Tohoku University as Associate Professor from 2007 to 2012. In 2012, he joined the Graduate School of Mathematics, Nagoya University as Professor. He worked in the Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology as Chief Research Scientist from 2020 to 2023. In 2023, he joined School of Data Science, The Chinese University of Hong Kong, Shenzhen as Professor and was granted as Presidential Chair Professor. He also joined Shenzhen International Quantum Academy as Chief Research Scientist. Also, he worked in Centre for Quantum Technologies, National University of Singapore as Visiting Research Associate Professor from 2009 to 2012 and as Visiting Research Professor from 2012 to 2024. He also worked in Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, China as a Visiting Professor from 2018 to 2020. He joined Department of Information Engineering, The Chinese University of Hong Kong as Adjunct Professor in 2024.

Quantum measurements are the key for extracting information from quantum systems and for connecting the quantum world with the classical world. For many applications in quantum information processing, randomized measurements have proved to be much more efficient in information extraction. In this talk I will discuss the applications of randomized measurements in various tasks, including quantum state verification, entanglement certification, and shadow estimation etc. As important mathematical tools underlying randomized measurements, the roles of complex projective designs and unitary designs will be highlighted.

In the noisy intermediate-scale quantum (NISQ) era, quantum computing devices are characterized by limited sizes, imperfect quantum gates, and the absence of quantum error correction (QEC). Despite these limitations, the quality and reliability of NISQ devices have been steadily improving, with the goal of achieving a significant increase in reliable circuit depths through qubit fabrication advancements error mitigation techniques. Therefore, it is crucial to explore quantum algorithms and applications that use NISQ devices and can demonstrate advantages over traditional approaches with improved performance before fully error-corrected fault-tolerant quantum computers become available. In this talk we will discuss quantum computational approaches refined to overcome the challenge of limited quantum resources in the NISQ era in the fields of quantum computational chemistry and quantum machine learning.

Boson-sampling, and its scattershot counterpart, is a non-universal quantum computer that is believed achieved significant computational advantage over classical computing. It is significantly more straightforward to build than any universal quantum computer proposed so far. In this talk, we review aspects of boson sampling and Gaussian boson sampling and its realization on integrated photonic chips.

Quantum resource distillation is a fundamental task in quantum information science. Minimizing the distillation overhead, i.e., the amount of noisy source states required to produce some desired output state within target error $\epsilon$, which typically scales as $O(\log^\gamma(1 / \epsilon))$, is crucial for the scalability of quantum computation and communication. A prior work [Phys. Rev. Lett. 125, 060405 (2020)] established a universal no-go theorem for resource distillation, indicating that no distillation protocol can achieve $\gamma < 1$ in the one-shot (finite-output) setting. Here, we show that this fundamental limit can be surpassed with the aid of quantum catalysts—an additional resource that facilitates the transformation but remains unchanged before and after the process. Specifically, we show that multi-shot distillation protocols can be converted into one-shot catalytic protocols, which hold significant practical benefits, while maintaining the distillation overhead. In particular, in the context of magic state distillation, our result indicates that the code-based low-overhead distillation protocols that rely on divergingly large batches can be promoted to the one-shot setting where the batch volume can be arbitrarily small for any accuracy. Combining with very recent results on asymptotically good quantum codes with transversal non-Clifford gates, we demonstrate that magic state distillation with constant overhead can be achieved with controllable output size using catalytic protocols. Furthermore, we demonstrate that catalysis enables a spacetime trade-off between overhead and success probability. Notably, we show that the optimal constant for constant-overhead catalytic magic state distillation can be reduced to 111 at the price of compromising the success probability by a constant factor. Finally, we present an illustrative example that extends the catalysis techniques to the study of dynamic quantum resources. This provides the channel mutual information with a one-shot operational interpretation, thereby addressing the open question posed by Wilming in [Phys. Rev. Lett. 127, 260402 (2021)].

Quantum computing is one of the most exciting future computing technologies based on fundamental law of matrix mechanics. However, the room of quantum algorithms, and when they could offer potential advantages against classical counterparts in practical applications, is highly obscure. In this talk, we will try to help illustrate the boundary of quantum advantages by novel methods provided from advanced tools developed in computer science. First, we will introduce GroverGPT, a large language model simulating quantum searching with 8 billion training parameters. On certain metrics, we could also show that they could outperform OpenAI’s GPT-4o. Moreover, we show that the model has significant generalization capabilities and could simulate the result of Grover’s quantum circuit with experiments up to 20 parameters, with evidence that the model could “learn” partially the nature of quantum algorithms, which helps illustrate the boundary of classical simulability of quantum Turing machines especially when they are noisy. Second, we show an end-to-end pipeline on the practical noiseless and fault-tolerant resource estimations of the HHL algorithm, a famous quantum algorithm for matrix inversion with provable quantum speedups and is BQP-complete. For a given quantum error correction code setup, we explicitly perform the resource count and identify the space, time, and energy costs for performing HHL algorithms fault-tolerantly, and illustrate when it will outperform classical counterparts for a general setup of condition numbers, row sparsity, precision and the size of matrix. Our works indicate that it is possible to illustrate a practical boundary between quantum and classical computing using most advanced tools in high-performance computing and large language models.

Floquet codes and dynamical codes offer a new avenue for quantum error correcting codes. In this talk, I will discuss a general formalism for analyzing the error correcting properties of these codes. Specifically, we extend the notion of distance of static stabilizer codes and subsystem codes to the unmasked distance of dynamical codes, and we develop an algorithm that determines what syndrome information can be learnt given an arbitrary dynamical code and use this to obtain the code’s unmasked distance. Further, we use the tools developed for the algorithm to reveal the structure of a generic Floquet code. Based on joint work with Daniel Gottesman.

In this talk, I present an overview of two ongoing projects. First, I discuss quantum circuits with gates that respect a global symmetry or, equivalently, conserve a global charge, such as the total energy of the system. Recent studies have shown that, in the presence of symmetry, the locality of gates imposes severe restrictions on the set of realizable unitaries. I explain how the nature of these restrictions strongly depends on the properties of the symmetry. For instance, some restrictions arise exclusively in the case of non-Abelian symmetries. Additionally, I briefly discuss recent work on the statistical properties of random circuits with symmetry-respecting gates (arXiv:2408.14463). In the second part of my talk, I present recent results on quantum Fisher information metrics and their applications in the resource theories of quantum thermodynamics and asymmetry. Specifically, I highlight a recent result that provides an operational interpretation of the RLD quantum Fisher information metric in the context of coherence distillation (arXiv:2409.05974).

I will go through our recent efforts in my group using photons and atoms to build increasingly large-scale quantum computers and, in turn, how these early quantum computers can already be used for studies of fundamental problems in mathematics, quantum physics, and condensed matter physics. We use the protocol of Gaussian boson sampling to demonstrate quantum computational advantage, with up to 255 detected photons [Zhong et al. Science 2020, PRL 2021, Deng et al. PRL 2023]. We develop an AI-enabled constant-time-overhead rearrangement protocol to prepare a 2024 defect-free atomic array [Lin et al. 2024]. Using a single atom trapped in an optical tweezer and cooled to the motional ground state in three dimensions, we faithfully realize the Einstein-Bohr recoiling-slit gedankenexperiment tunable at the quantum limit [Zhang et al. 2024]. Based on a bottom-up quantum engineering approach, we experimentally created the fractional quantum Hall state using strongly interacting photons [Wang et al. Science 2024]. We further use the quantum computing platform to rule out a real-value description of standard formalism of quantum theory [Chen et al. PRL 2022].

In this talk, we explore a class of three-dimensional (3D) fracton topological orders that exhibit exotic boundary phenomena called “Toeplitz braiding” in the thermodynamic limit. These systems are constructed by stacking 2D twisted $\mathbb{Z}_N$ topologically ordered layers along the z-direction, coupled while maintaining translation symmetry. The effective field theory is described by an infinite-component Chern-Simons theory with a block-tridiagonal Toeplitz K-matrix. A key finding is the connection between boundary zero modes in the K-matrix spectrum and the emergence of Toeplitz braiding, where mutual braiding phase angles between boundary anyons oscillate and remain nonzero in the thermodynamic limit. Interestingly, the integer-valued Hamiltonian matrix of the 1D Su-Schrieffer-Heeger model can serve as a nontrivial K-matrix, demonstrating the presence of robust boundary zero modes without relying on symmetry protection. We will also discuss numerical results and potential future directions, including the construction of 3D lattice models to realize this phenomenon.

The U.S. Decennial Census plays a critical role in policy-making, including federal funding allocation and redistricting. In 2020, the Census Bureau implemented differential privacy to protect individual data through a noise injection method. This raised the question of whether stronger privacy guarantees could be achieved or if the full privacy budget was utilized. In this paper, we address this by applying harmonic analysis and discrete mathematics to track privacy losses using fff-differential privacy. Our results show that between 8.50% and 13.76% of the privacy budget remained unused across various geographical levels. We demonstrate that the Census Bureau could reduce unnecessary noise injection by up to 24.82%, improving the accuracy of privatized census data without compromising privacy. This reduction in noise also enhances the utility of private census data in downstream applications, such as analyzing the relationship between earnings and education.

Recent studies have unveiled new possibilities for discovering intrinsic quantum phases that are unique to open systems, including phases with average symmetry-protected topological (ASPT) order and strong-to-weak spontaneous symmetry breaking (SWSSB) order in systems with global symmetry. In this work, we propose a new class of phases, termed the double ASPT phase, which emerges from a nontrivial extension of these two orders. This new phase is absent from prior studies and cannot exist in conventional closed systems. Using the recently developed imaginary-Lindbladian formalism, we explore the phase diagram of a one-dimensional open system with $\mathbb{Z}_{\sigma}\times \mathbb{Z}_{2\tau}$ symmetry. We identify universal critical behaviors along each critical line and observe the emergence of an intermediate phase that completely breaks the $\mathbb{Z}_{\sigma^2}$ symmetry, leading to the formation of two triple points in the phase diagram. These two triple points are topologically distinct and connected by a domain-wall decoration duality map. Our results promote the establishment of a complete classification for quantum phases in open systems with various symmetry conditions.

Large-scale variational quantum algorithms (VQA) are widely recognized as a potential pathway to achieve practical quantum advantages. However, the presence of quantum noise might suppress and undermine these advantages, which blurs the boundaries of classical simulability. To examine noise effects on VQAs, we introduce a new classical simulation approach based on the Pauli path integral. This approach enables approximate calculation of operator expectation values under single-qubit Pauli noise, with controllable truncation error and polynomial computational cost. Furthermore, we apply this method to simulate IBM’s 127-qubit quantum computations, yielding consistent results. This work is a collaboration with Fuchuan Wei, Song Chen, and Zhengwei Liu.