SQAI-NCTS Workshop on Tensor Network and Quantum Embedding

25 Mar 2024, 09:00 → 29 Mar 2024, 18:00 Asia/Tokyo

Hongo campus, The University of Tokyo, Tokyo, Japan

Wei-Lin Tu (Keio University), Yi-Ping Huang (National Tsing Hua University), Chia-Min Chung (NSYSU), Satoshi Morita (Keio University)

About

This workshop focuses on the related topics of tensor network and its potential application in the wide research field such as the quantum algorithm. Topics covered by it include the quantum embedding, quantum machine learning, condensed matter problems, etc.

Dates

March 25 - 29, 2024  

Venue

Hongo campus, the University of Tokyo  

Registration fee

Free

Banquet 

Faculty: 4,000 Japanese yen
Student: 2,000 Japanese yen

Speakers (in alphabetical order)

• Didier Poilblanc (Univ. of Toulouse)
• Garnet Kin-Lic Chan (California Institute of Technology)
• Hiroshi Shinaoka (Saitama Univ.)
• Ian McCulloch (Natl. Tsing Hua Univ.)
• Ji-Yao Chen (Sun Yat-sen Univ.)
• Keisuke Fujii (Osaka University)
• Lei Wang (IOP, Chinese Academy of Science)
• Miles Stoudenmire (Flatiron Institute)
• Mingpu Qin (Shanghai Jiao Tong Univ.)
• Naoki Kawashima (ISSP, Univ. of Tokyo)
• Pochung Chen (Natl. Tsing Hua Univ.)
• Tao Xiang (IOP, Chinese Academy of Science)
• Thomas Ayral (Eviden Quantum Lab)
• Ying-Jer Kao (Natl. Taiwan Univ.)

Organizers

Organizing Committee Chair

• Synge Todo (Univ. of Tokyo) 
• Ian McCulloch (Natl. Tsing Hua Univ.) 

Vice Chair

• Yusuke Nomura (Keio Univ.)
• Chia-Min Chung (Natl. Sun Yat-sen Univ.) 
• Yi-Ping Huang (Natl. Tsing Hua Univ.) 

Program Chair

• Satoshi Morita (Keio Univ.) 
• Wei-Lin Tu (Keio Univ.) 

Local Arrangement Chair

• Tsuyoshi Okubo (Univ. of Tokyo) 

Support

Supported by the Center of Innovations for Sustainable Quantum AI (JST Grant Number JPMJPF2221) & NCTS-National Center for Theoretical Sciences Physics Division

 

Note: If you receive any emails offering you hotels or accommodation for the event, e.g. from "Global Travel Experts" please ignore it - this is a well-known scam targeting registrants to Indico events.

Monday, 25 March

Welcome: Opening

Convener: Wei-Lin Tu (Keio University)

1 Opening speech

Invited talks: Ji-Yao Chen

Convener: Wei-Lin Tu (Keio University)

2 Tensor networks: from ground states to excitations and back

Tensor networks provide a way to systematically study not only ground states but also excitation spectrum of quantum many-body systems. When dealing with the latter, diagrammatic summation would typically arise. In this talk, I will describe the idea of using generating functions to solve this problem in the context of both matrix product states and projected entangled-pair states. As an application of excitation ansatz, I will discuss the results for the Kitaev model. If time permits, a variant of this model and related questions will also be mentioned.

Speaker: Prof. Ji-Yao Chen (Sun Yat-sen University)

Invited talks: Garnet Kin-Lic Chan

Convener: Wei-Lin Tu (Keio University)

3 From tensor networks to quantum embedding to correlated materials

Speaker: Garnet Kin-Lic Chan (California Institute of Technology)

Invited talks: Pochung Chen

Convener: Yi-Ping Huang (National Tsing Hua University)

4 Tensor network based finite-size scaling for 2D classical models.

In this talk, we discuss how to perform tensor network based finite-size scaling analysis for 2D classical models. We first use HOTRG to renormalize the weight tensor of the partition function, then we use renormalized tensor to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain physical quantities such as the correlation length, magnetization, and energy density. These quantities are used in finite-size analysis to estimate the critical temperature and critical exponents. We show that they can be determined accurately and the results can be systematically improved by increasing the bond dimension. Furthermore, we show that the conformal data can also be estimated accurately and the low energy part of the conformal tower can be accurately obtained.
Finally, we define an entanglement for the 2D classical models and the entanglement show expected scaling behavior, from which the central charge can be estimated.

Speaker: Prof. Pochung Chen (NHTU)

Invited talks: Naoki Kawashima

Convener: Tsuyoshi Okubo (Univ. of Tokyo)

5 Compression of Tensor Networks

I review several attempts at effective compression of tensor networks includeing relatively new finidings of our own. Tensor network compression is a key technology in a few entirely different contexts. First, many of massive data sets, collection of images, customer preferences, etc., are naturally viewed as big tensors. In recognizing, correcting and compressing such data sets, decomposition into a network of smaller tensors is a promising candidate. We then encounter the need for optimizing/compressing the tensor network. Second, in the statistical mechanics, the real-space renormalization group method a la Kadanoff is recently re-formulated in terms of tensor networks, and produces highly accurate predictions on the critical information not only a first few relevant scaling dimensions but also higher ones together with operator-product-expansion coefficients. There, the renormalization mapping can be viewed as a compression of the tensor networks. We recently found that the same technique is useful in both contexts.

Speaker: Naoki Kawashima (Institute for Solid State Physics, University of Tokyo)

Invited talks: Ian McCulloch

Convener: Tsuyoshi Okubo (Univ. of Tokyo)

6 Environment expansion for matrix product states

I will present a general framework for incorporating degrees of freedom into a tensor network (i.e. bond expansion), with applications for DMRG, TDVP, and other algorithms. Our approach makes use of reduced rank singular value decompositions, such that all operations required for the bond expansion have computational complexity that is at most quadratic in the bond dimension $D$ and linear in the local Hilbert space dimension $d$, so much cheaper than other components of DMRG that scale as $D^3$. The 'pre-expansion' approach interpolates between single-site and 2-site DMRG, giving convergence similar to 2-site DMRG but otherwise identical performance to single-site DMRG. 'Post-expansion' is a successor to the single-site subspace-expansion (3S) algorithm for models with long-range interactions, with better convergence properties and easier to control. These algorithms perform better than conventional DMRG in all cases, but are especially useful for models where the local Hilbert space dimension is large, such as bosonic degrees of freedom.

Speaker: Ian McCulloch

Tuesday, 26 March

Invited talks: Tao Xiang

Convener: Chia-Min Chung (NSYSU)

7 Tensor Network Renormalization and its Applications

The tensor-network renormalization group is known for its profound implications for understanding and solving correlated quantum systems. I will explore sophisticated tensor-network techniques for assessing dynamical excitations in low-dimensional quantum lattice models:
1. Introduce a matrix-product representation for low-energy excited states and
present methods for their precise determination. [1]
2. Employ the triangular antiferromagnetic model to investigate the dynamical spectra of two-dimensional quantum lattice modes within the single-mode approximation. [2]
3. Discuss the kicked Ising problem, previously utilized to showcase the benefits of noisy quantum computing through error mitigation on the NISQ quantum computing platform, as a case study for efficiently and accurately calculating the time evolutions of two-dimensional tensor-network states. [3]

[1] X. Li, Z. Zhou, G. Xu, R. Chi, Y. Guo, T. Liu, H. Liao, T. Xiang, Accurate
determination of low-energy eigenspectra with multi-target matrix product states,
Phys. Rev. B 109, 045115 (2024).
[2] R. Chi, Y. Liu, Y. Wan, H. J. Liao, T. Xiang, Spin excitation spectra of anisotropic
spin-1/2 triangular lattice Heisenberg antiferromagnets, Phys. Rev. Lett. 129,
227201 (2022)
[3] H.-J. Liao, K. Wang, Z.-S. Zhou, P. Zhang, T. Xiang, Simulation of IBM's kicked
Ising experiment with Projected Entangled Pair Operator, arXiv:2308.03082

Speaker: Tao Xiang (Institute of Physics, Chinese Academy of Sciences)

Invited talks: Hiroshi Shinaoka

Convener: Chia-Min Chung (NSYSU)

8 Exploiting hidden low-rank structures in physics

Tensor networks are a powerful tool for compressing wave functions and density matrices of quantum systems in physics. Recent developments have shown that tensor network techniques can efficiently compress many functions beyond these traditional objects. Notable examples include the solutions to turbulence in Navier–Stokes equations [1] and the computation of Feynman diagrams [2,3]. These advancements have heralded a new era in the use of tensor networks for expediting the resolution of various complex equations in physics.

In this presentation, we will overview our recent research in this domain. Initially, we will discuss the compression of the space-time dependence of the correlation function in quantum systems [3] through the use of the “Quantics Tensor Train.” [4,5] This method leverages the inherent length-scale separation in the system to represent the function efficiently. Our approach demonstrates solving diagrammatic equations in a compressed format.

Subsequently, we will introduce a novel and robust tool named “Quantics Tensor Cross Interpolation.” [6] This method is designed to learn a quantics low-rank representation of a given function, showcasing the versatility and potential of tensor network techniques in handling complex functions in physics.

If time is allowed, we will briefly review our open-source libraries implementing these technologies.

[1] N. Gourianov et al., Nat. Comput. Sci. 2, 30 (2022).
[2] Y. N. Fernandez et al., PRX 12, 041018 (2022).
[3] H. Shinaoka et al., PRX 13, 021015 (2023).
[4] I. V. Oseledets, Dokl. Math. 80, 653 (2009).
[5] B. N. Khoromskij, Constr. Approx. 34, 257 (2011).
[6] M. K. Ritter, …, H. Shinaoka and X. Waintal, PRL 132, 056501 (2024).

Speaker: Hiroshi Shinaoka (Department of Physics, Saitama University)

Invited talks: Ying-Jer Kao

Convener: Yi-Ping Huang (National Tsing Hua University)

9 Hybrid Classical-Quantum Architectures for Quantum Machine Learning

In this talk, I will introduce a hybrid model combining a quantum-inspired tensor network (TN) and a variational quantum circuit (VQC) to perform supervised and reinforcement learning tasks. This architecture allows for the classical and quantum parts of the model to be trained simultaneously, providing an end-to-end training framework. We show that compared to the principal component analysis, a tensor network based on the matrix product state with low bond dimensions performs better as a feature extractor for the input data of the variational quantum circuit in the binary and ternary classification of MNIST and Fashion-MNIST datasets. We also use this architecture to perform quantum reinforcement learning on the MiniGrid environment with 147-dimensional inputs.The hybrid TN-VQC architecture provides a natural way to perform efficient compression of the input dimension, enabling further quantum machine learning applications on noisy intermediate-scale quantum devices. Finally, I will discuss some regularization methods to address the issue of barren plateaus during training for multi-layer VQC.

Speaker: Ying-Jer Kao (Center for Theoretical Physics and Department of Physics, National Taiwan University, Taipei 10607, Taiwan)

Invited talks: Lei Wang

Convener: Yusuke Nomura (Keio University)

10 CrystalFormer

We introduce CrystalFormer, a transformer-based autoregressive model specifically designed for space group-controlled generation of crystalline materials. The space group symmetry significantly simplifies the crystal space, which is crucial for data and compute efficient generative modeling of crystalline materials. Leveraging the prominent discrete and sequential nature of the Wyckoff positions, CrystalFormer learns to generate crystals by directly predicting the species and locations of symmetry-inequivalent atoms in the unit cell. Our results demonstrate that CrystalFormer matches state-of-the-art performance on standard benchmarks for both validity, novelty, and stability of the generated crystalline materials. Our analysis also shows that CrystalFormer ingests sensible solid-state chemistry information from data for generative modeling. The CrystalFormer unifies symmetry-based structure search and generative pre-training in the realm of crystalline materials. The simplicity, generality, and flexibility of CrystalFormer position it as a promising architecture to be the foundational model of the entire crystalline materials space, heralding a new era in materials modeling and discovery.

Speaker: Lei Wang (Institute of Physics, Chinese Academy of Sciences)

Invited talks: Thomas Ayral

Convener: Yusuke Nomura

11 Can tensor networks help quantum computers solve quantum many-body problems?

Speaker: Thomas Ayral (Eviden Quantum Lab)

Wednesday, 27 March

Symposia talks

Convener: Matthias Gohlke (Okinawa Institute of Science and Technology Graduate University)

12 Optimizing the structure of tree tensor network for quantum generative modeling using mutual information-based approach

Generative modeling is a crucial task in the field of machine learning. Recently, there have been several proposals for generative models on quantum devices. We can efficiently optimize generative models defined by tensor network states, but their performance largely depends on the geometrical structure of the tensor network. To tackle this issue, we have proposed an optimization method for the network structure in the tree tensor network class, based on the least mutual information principle. Generative modeling with an optimized network structure has better performance than a fixed network structure. Moreover, by embedding data dependencies into the tree structure based on the least mutual information principle, we can geometrically represent the correlations in the data.

Speaker: Prof. Kenji Harada (Graduate School of Informatics, Kyoto University)

13 Proposal of Channel Attention for Quantum Convolutional Neural Networks

Quantum Convolutional Neural Networks (QCNNs) have emerged in recent years, demonstrating success, especially in the domain of quantum phase recognition problems. However, the practical application of QCNNs to real-world problem-solving demands further cost reduction and improvement in performance. To address these challenges, this study introduces a novel solution through the introduction of a channel attention mechanism designed specifically for QCNNs. Drawing inspiration from its classical counterpart, our proposed attention mechanism generates multiple output channels based on the measurement of quantum bits. This approach not only enhances the performance of QCNNs beyond conventional methods but also addresses the need for cost reduction in model implementation.

Speaker: Dr Gekko Budiutama (Tokyo Institute of Technology, Quemix Inc.)

14 [CANCELED] Analysis of Data-encoding Induced Barren Plateau in Quantum Machine Learning

In recent years, Variational Quantum Algorithms (VQA) have been actively studied as a promising approach on Noisy Intermediate Scale Quantum computers (NISQ). VQA solve problems by minimizing a cost function through the updating of parameters in a variational quantum circuit on a classical computer. Quantum Machine Learning (QML), which utilizes variational quantum circuits for machine learning, is one such approach.

However, it has been noted that vanishing gradients of the cost function, referred to as Barren Plateau (BP), can occur in VQA, posing challenges for optimization. Various causes of Barren Plateaus have been investigated, including the structure of the ansatz, the cost function, noise, and data encoding. Nevertheless, the effect of data encoding remains not fully understood.

This study analytically investigates the effect of data encoding on the variance of the cost function gradient. Specifically, we derive upper and lower bounds on the variance of the cost function gradient for a given data encoding circuit and cost function. Additionally, we numerically confirm that the scaling of the variance of the cost function gradient is independent of the form of the cost function, such as mean absolute error, mean squared error, and cross-entropy.

Speaker: Kensuke Kamisoyama (Department of Physics, Graduate School of Science, The University of Tokyo)

Symposia talks

Convener: Prof. Kenji Harada (Graduate School of Informatics, Kyoto University)

15 Toward tensor renormalization group study of quantum chromodynamics

We discuss the recent progress in the application of the tensor renormalization group method toward lattice quantum chromodynamics. In the first half, we discuss the application of the Grassmann tensor network for gauge theories with multiple fermion flavors. With the multilayer tensor network construction, it is possible to compute the two-dimensional abelian gauge theories up to 4 flavors. In the second half, we discuss the novel formulation for non-abelian lattice gauge theory with a significantly reduced tensor size which can be generalized to higher dimensions. Preliminary results for three-dimensional SU(2) and SU(3) gauge theories are presented.

Speaker: Atis Yosprakob (Niigata University)

16 Parton Distributions in the Schwinger Model with Matrix Product States

Parton Distribution Functions (PDFs) describe universal properties of bound states in high energy physics and allow to predict scattering amplitudes in processes with large momentum transfer. The numerical calculation of PDFs involves the evaluation of a Wilson line along a lightcone. In contrast to Monte Carlo simulations in euclidean spacetime, the evolution on a lightcone can be directly computed in the Hamiltonian formalism. The necessary spatial- and time-evolution can be efficiently applied using established tensor network methods. We study PDFs in the Schwinger model using matrix product states.

Speaker: Manuel Schneider (NYCU)

17 Correlation functions of the Ising model on the stacked pentagon lattice

Thermodynamic properties of the classical Ising model on a hierarchical lattice is studied by tensor network methods. The lattice consists of pentagons, where 2, 3 or 4 of them meet at each vertex, which is the lattice site. Taking the spin configuration sum other than the lowest spin row, we obtain the boundary state at the bottom of the system. This summation can be performed numerically by means of the TEBD method, since the entanglement of the state is not strong. Power low decay of the correlation function is numerically confirmed in the relatively high temperature region. Besides, from the hierarchical structure of the lattice, it is also possible to take spin configuration sum partially from the bottom of the system, in the manner as the CTMRG or TRG method. Calculated results suggests the presence of the bulk phase transition at the top of the system. Generalization of the lattice can be considered such as the stacked square lattice drawn inside a triangle.

Speaker: Prof. Tomotoshi Nishino (Professor)

Symposia talks

Convener: Chisa Hotta (University of Tokyo)

18 Matrix product state methods for excitations

In recent years, matrix product state (MPS) numerics have emerged as the method of choice for examining the low-energy physics of many-body quantum systems in one spatial dimension, as well as small-width 2D systems. While the density matrix renormalisation group (DMRG) algorithm is used to calculate ground states, analysis of the low-lying excitations is typically doing using time-evolution simulations. In this talk, we will look at the MPS excitation ansatz, a complementary approach which efficiently represents low-energy particle-like excitations directly in the thermodynamic limit. We will highlight recent work in finding partlce-like excitations inside of scattering continua, as well as constructing real-space wavepackets to examine particle scattering.

Speaker: Jesse Osborne (University of Queensland)

slides.pdf

19 Stability of topological solitons in SSH Model under non-Hermitian deformations

The SSH model, describing a one-dimensional chain of atoms or sites with alternating coupling
strengths, exhibits topological solitons in the form of domain walls or edge states. These topological
solitons are the results of the topological nature of the SSH model, and different symmetries can protect
the existence and stability of these solitons. In this work, we explore the effects of non-Hermitian
perturbations on the stability and behaviour of these solitons by investigating the symmetries of the
underlying system. Furthermore, we explore the interplay between non-Hermitian perturbations
and other external parameters, such as disorder or lattice modifications. We investigate how these
additional factors affect the robustness and stability of the topological solitons and their associated
edge states in different configurations of soliton defects in the SSH model.

Speaker: Vikash Mittal (National Tsing Hua University, Hsinchu, Taiwan)

20 Sample complexity in matrix product states at finite temperature

Computational complexity of physical states is a critical topic because it relates to the preparation on classical or quantum computers, as well as in real experiments. From the perspective of computational complexity theory, thermal and ground states of one-dimensional systems are well-understood. There exists an efficient classical algorithm for calculating thermal states at temperatures with $\beta = \mathcal{o}(\log N)$, where $N$ is the system size [1]. On the other hand, estimating the ground state energy is generally challenging even for quantum computers [2]. This difference between thermal and ground states has not been thoroughly explored.

In this work, we derive an exact analytical form for the required number of samples in random samplings based on the matrix product state. Previous research has shown that the required number of samples is characterized by the normalized fluctuation of the partition function, denoted as $\delta z^2$ [3]. We find a qualitative change in $\delta z^2$ with temperature; at high temperatures $\beta \lesssim \beta_c$, $\delta z^2$ scales linearly with $N$, and at low temperatures $\beta \gtrsim \beta_c$, $\delta z^2$ scales as $N^2$. Here, the crossover temperature $\beta_c$ depends on the system size as
\begin{equation}
\beta_c \simeq \frac{1}{\Delta E} \log N
\end{equation}
where $\Delta E$ is the spectral gap. This result bridges the difference in computational complexity between thermal and ground states.
[1] T. Kuwahara, A. M. Alhambra, and A. Anshu, Phys. Rev. X 11, 011047
(2021).
[2] S. Hallgren, D. Nagaj, and S. Narayanaswami, Quantum Info. Comput.
13, 721–750 (2013).
[3] A. Iwaki and C. Hotta, Phys. Rev. B 106, 094409 (2022).

Speaker: Atsushi Iwaki (University of Tokyo)

240323_iwaki_SQAI_v2.pdf

Symposia talks

Convener: Tomotoshi Nishino (Kobe University)

21 The State Preparation of Multivariate Normal Distributions using Tree Tensor Network

The quantum state preparation of probability distributions is an important subroutine for many quantum algorithms. When embedding $D$-dimensional multivariate probability distributions by discretizing each dimension into $2^n$-point, we need a state preparation circuit comprising a total of $nD$ qubits, which is often difficult to compile. In this study, we propose a method to generate state preparation circuits for $D$-dimensional multivariate normal distributions, utilizing tensor networks. We represent the probability distribution with a tree tensor network and perform the task of quantum circuit compilation through the optimization of tensor networks. Especially, by employing structural optimization, we can search for a network structure that efficiently represents the correlations between variables. The numerical results suggest that our method can dramatically reduce the circuit depth while maintaining fidelity compared to existing approaches. Moreover, for normal distributions with one-dimensional correlations, we can construct state preparation circuits in a scalable manner, regardless of the number of variables, by using tensor cross interpolation.

Speaker: Hidetaka Manabe

22 Designing exact MPS ground state solutions in 1D and 2D

We propose a protocol to design an exact MPS as a ground state of the bulk Hamiltonian based on cluster units that share their sites with the neighboring clusters.
We first decide what kind of clusters we use, and define a state that we want to have on each cluster as constituents of the local density matrix.
By entangling these clusters by partially projecting out the components that we want to discard, we can obtain a highly entangled ground state.
Of course this treatment is not always successful for all given models or conditions, and the protocol tells us how to judge or search for such set of exact ground state and the corresponding bulk Hamiltonian.
We demonstrate many cases that we are able to successfully obtain an exact multicritical ground state in the form of MPS, and show that we could further construct a series of solutions in the two dimensional lattices.
This method is useful to design a reference system in numerical tensor network-based or other calculations and helps us to understand the similarities between different solutions belonging to apparently different models.

Speaker: Chisa Hotta (University of Tokyo)

23 Quantum phase transition between spin liquid and spin nematics in spin-1 Kitaev honeycomb model

Besides the exactly solvable spin-1/2 Kitaev model, higher spin-$S$ ones, not exactly solvable, are promising playgrounds for researches on the quantum spin liquid as well. As one of main interests in higher spin-$S$ cases, the interplay between the Kitaev spin liquid (KSL) and spin nematics has attracted attentions, which may lead to novel quantum properties of matters. However, this is far from understood since it is hard to investigate their hidden magnetism. In our work, we probe this interplay by utilizing the infinite Projected Entangled Pair State (iPEPS) to introduce quantum entanglements between spins. We here consider a spin-1 model on the honeycomb lattice with competing bilinear-biquadratic and Kitaev interactions. As a result, we map out the phase diagram with emphasis on parameter regions in the vicinity of two pure Kitaev limits, in which we discover the direct KSL--spin-nematics transitions. In particular, around the ferro-Kitaev limit, the KSL is stabilized under the influence of the almost pure spin-quadrupolar interaction. Also, the spin-nematic phase is extended to the parameter region near the antiferro-Kitaev limit. It is perhaps the first time that the direct phase transition between a quantum spin liquid phase and a spin nematic phase is discovered in higher spin-S systems. We expect that this phase transition may emerge when one control the extent of spin-phonon couplings in materials with strong spin-orbit couplings.

Speaker: Tohru Mashiko (Univ. of Tokyo)

24 Thermal pure matrix product state in two dimensions: tracking thermal equilibrium from paramagnet down to the Kitaev spin liquid state

We show that the matrix product state provides a thermal quantum pure state representation in equilibrium in two spatial dimensions over the entire temperature range. We use the Kitaev honeycomb model as a prominent, non-trivial example hosting a quantum spin liquid ground state. Our method is able to qualitatively capture the double-peak in the specific heat, which was previously obtained nearly exactly using a method tailored to the Kitaev honeycomb model. In contrast, our method can be applied to general systems including those with competing interactions. We also demonstrate, that the truncation process efficiently discards the high-energy states, eventually reaching the long-range entangled topological state with very low statistical errors.

Speaker: Matthias Gohlke (Okinawa Institute of Science and Technology Graduate University)

Poster: Short Presentation

Convener: Wei-Lin Tu (Keio University)

Poster: Poster Session & Banquet

25 Compressed NeRF Architecture with Tensor Networks

Neural Radiance Field (NeRF) is a well-known 3D reconstruction method capable of generating novel views of a target scene. NeRF model often employs a neural network trained by captured images to represent a 3D scene as a continuous function that maps a 3D coordinate and a view direction to color and density. In this work, we examine the potential of NeRF acceleration by replacing the MLP layers of a standard NeRF architecture with Matrix Product Operators (MPO). We show that our preliminary experiments with NeRF-MPO, our NeRF variant, can efficiently reduce model size with comparable performance, indicating the prospect of applying tensor networks to NeRF.

Speakers: Haruo Fujiwara (Hakuhodo DY Holdings Inc.), Ryutaro Nagai (blueqat inc.), Hiroshi Kato (Hakuhodo DY Holdings Inc.)

SQAI_flashtalk.pdf

26 Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction

We study the ground-state phase diagram of a spin-$\frac12$ XXZ model with a chirality-chirality interaction (CCI) on a two-leg ladder. This model offers a minimal setup to study an interplay between spin and chirality degrees of freedom. The spin-chirality duality transformation allows us to relate the regimes of weak and strong CCIs. By applying the Abelian bosonization and the duality, we obtain a rich phase diagram that contains distinct gapped featureless and ordered phases. In particular, Néel and vector chiral orders appear for easy-axis anisotropy, while two distinct symmetry protected topological (SPT) phases appear for easy-plane anisotropy. The two SPT phases can be viewed as twisted variants of the Haldane phase. We also present an effective description in terms of (spinor) hard-core bosons, which reveals critical behavior on the self-dual line in the easy-axis and easy-plane regimes. We perform numerical simulations to confirm the predicted phase structure and critical properties. We further demonstrate that the two SPT phases and a trivial phase are distinguished by topological indices in the presence of certain symmetries. A similar phase structure is expected in a spin-$\frac12$ XXZ ladder with four-spin ring exchange.

Speaker: Matéo Olivier Jean-Marie Michel FONTAINE (Keio University)

27 Quantum circuit optimization based on tensor decomposition techniques

The construction of quantum circuits for classical data using tensor network methods is attracting attention as a scalable methodology when approximation is possible. However, in order to ensure accuracy equivalent to classical calculations, it is necessary to decompose a multi-qubit quantum gate, which has an amount of information equivalent to classical information, into gates that can be executed on an actual machine.　Therefore, we are developing a method to decompose multi-qubit unitary operators into basic gates using tensor decomposition methodology.　In this presentation, we will introduce a hierarchical decomposition method to improve learning performance and a decomposition method using machine learning techniques aimed at optimizing network structure. We also discuss the advantages and disadvantages of each method and the problem range that the methodology can address.

Speaker: Tomonori Shirakawa (RIKEN)

P22Shirakawa.pdf

28 Emergence of vortex state in the S = 1 Kitaev-Heisenberg model with single-ion anisotropy

The search for Kitaev spin liquid states has recently broadened to include a number of honeycomb materials
with integer spin moments. The qualitative difference with their spin-1/2 counterparts is the presence of single-
ion anisotropy (SIA). This motivates our investigation of the effects of SIA on the ground state of the spin-1
Kitaev-Heisenberg (KH) model using the density-matrix renormalization group which allows construction of
detailed phase diagrams around the Kitaev points. We demonstrate that positive out-of-plane SIA induces an
in-plane vortex state without the need for off-diagonal interactions. Conversely, negative SIA facilitates the
emergence of a ferromagnetic state in presence of antiferromagnetic Heisenberg interactions, while a Néel state
can emerge for ferromagnetic Heisenberg coupling. These findings, pertinent even for weak SIA, not only
enhance our theoretical understanding of the spin-1 KH model but also propose experimental prospects for
observing these novel magnetic states in material realizations.

Speaker: Ayushi Singhania

29 Multi-impurity method for bond-weighted tensor renormalization group

Tensor network (TN) methods are attracting much attention as powerful tools for computing strongly correlated many-body problems. The partition function of classical statistical systems can be represented by the TN form. However, the contraction of a large TN still requires an exponentially large computational effort. The concept of the real-space renormalization group resolves this problem. The tensor renormalization group (TRG) method and its variants calculate a coarse-grained tensor by information compression by the singular value decomposition. These methods can calculate the partition function approximately in the polynomial time. Recently, Adachi, et al. have proposed the bond-weighted TRG (BWTRG) method, which improves the accuracy of TRG by introducing a bond weight and distributing it appropriately.

In this presentation, we propose an algorithm to calculate higher-order moments of physical quantities based on BWTRG. We introduce a coarse-grained matrix on a bond representing a summation of all configurations of multiple impurities and derive its update rule. Our method is compared with conventional methods on the two-dimensional classical spin model. The proposed method achieves higher accuracy at a lower computational cost than the higher-order TRG algorithm. We also show that the finite-size scaling analysis of the squared magnetization provides critical exponents and distinguishes the weakly first-order and continuous phase transitions.

Speaker: Dr Satoshi Morita (Keio University)

30 General tensor network theory for frustrated classical spin models in two dimensions

Frustration is a ubiquitous phenomenon in many-body physics that influences the nature of the system in a profound way with exotic emergent behavior. Despite its long research history, the analytical or numerical investigations on frustrated spin models remain a formidable challenge due to their extensive ground-state degeneracy. In this paper, we propose a unified tensor network theory to numerically solve the frustrated classical spin models on various two-dimensional (2D) lattice geometry with high efficiency. We show that the appropriate encoding of emergent degrees of freedom in each local tensor is of crucial importance in the construction of the infinite tensor network representation of the partition function. The frustrations are thus relieved through the effective interactions between emergent local degrees of freedom. Then the partition function is written as a product of a one-dimensional (1D) transfer operator, whose eigenequation can be solved by the standard algorithm of matrix product states rigorously, and various phase transitions can be accurately determined from the singularities of the entanglement entropy of the 1D quantum correspondence. We demonstrated the power of our general theory by numerically solving 2D fully frustrated XY spin models on the kagome, square, and triangular lattices, giving rise to a variety of thermal phase transitions from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order transitions, to first-order phase transitions. Our approach holds the potential application to other types of frustrated classical systems like Heisenberg spin antiferromagnets.

References:
[1] F.-F. Song, T.-Y. Lin & G.-M. Zhang, Phys. Rev. B 108, 224404(2023)
[2] F.-F. Song & G.-M. Zhang, Phys. Rev. B 108, 014424(2023)
[3] F.-F. Song & G.-M. Zhang, Phys. Rev. B 105, 134516(2022)

Speaker: Dr Fengfeng Song (ISSP UTokyo)

31 Error-mitigated quantum computation of string order parameters across a topological phase transition

Recently, Smith et al. investigated a topological phase transition in a spin chain by measuring string order parameters on the IBM quantum computers [Phys. Rev. Research 4, L022020 (2022)]. The measured quantities showed appreciable reduction from theoretical values, owing to inherent noise in the devices. In this work, we reproduce their results in the noisy quantum circuit simulator (Qiskit Aer), and improve the accuracy of the measurements by exploiting the error mitigation technique called the virtual distillation. This technique enables an exponential error suppression given the preparation of n copies of the original circuits and some additional gates.

Speaker: Mr Ryuhei Kawase (Keio University)

32 Variational tensor network study of the anisotropic Kitaev model under magnetic field

We conducted both unconstrained and symmetric infinite projected entangled pair state (iPEPS)[1] simulations for the honeycomb Kitaev model along out-of-plane external magnetic field h and the anisotropic interactions $Kz$ axes. In particular, for the $Kz/K > 2$ region where the Majorana fermion being gapped out and the low energy excitation dominated by the $Z_2$ fluxes, based on the gauge symmetry and topological entanglement entropy measurements, we showed that the proposed symmetric ansatz - which implements the flux conservation and local symmetry under finite field and anisotropy - well captured the gapped $Z_2$ quantum spin liquid (QSL) phase and the phase transition point, comparable to the unconstrained iPEPS and agrees with the perturbation theory prediction[2]. The anyon dynamics in this QSL are further explored by the generating function[3] to calculate the excitation spectrum and dynamical spectral function. We include the projector derivative[4] and determine the correct truncation dimension by enforcing the sum rule.

References
[1] Yu-Hsueh Chen, Ke Hsu, Wei-Lin Tu, Hyun-Yong Lee, and Ying-Jer Kao. Variational tensor network operator. Phys. Rev. Res., 4:043153, Nov 2022.
[2] Feng, S., Agarwala, A., Bhattacharjee, S., Trivedi, N. (2023). Anyon dynamics in field-driven phases of the anisotropic Kitaev model. Phys. Rev. B, 108, 035149.
[3] Tu, W.-L. et al. Generating function for projected entangled-pair states. Preprint at http://arxiv.org/abs/2307.08083 (2023).
[4] Ponsioen, B., Hasik, J. Corboz, P. Improved summations of n-point correlation functions of projected entangled-pair states. Phys. Rev. B 108, 195111 (2023).

Speaker: Mr Chang-Teng Lin (National Taiwan University)

33 Extracting conformal data from frustrated classical spin models with nuclear norm regularization techniques on tensor network renormalization algorithm

We propose a loop optimization algorithm based on nuclear norm regularization for tensor network. The key ingredient of this scheme is to introduce a rank penalty term proposed in the context of data processing. Compared to standard variational periodic matrix product states method, this algorithm can circumvent the local minima related to short-ranged correlation in a simpler fashion. We demonstrate its performance when used as a part of the tensor network renormalization algorithms [S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)] for the critical 2D Ising model. The scale invariance of the renormalized tensors is attained with higher accuracy while the higher parts of the scaling dimension spectrum are obtained in a more stable fashion.

Using the nuclear norm regularization techniques on tensor network renormalization algorithm, we study the phase diagram, the critical behavior, and the duality property of the antiferromagnetic 6-state clock model on the Union Jack lattice. We find that this model undergoes multiple phase transitions; there is the Berezinskii-Kosterlitz-Thouless, $Z_{6}$ symmetry breaking, and chiral transition
with decreasing temperature. Furthermore, we provide convincing numerical evidence that its quasi-long range order is well explained by the compactified boson conformal field theory (CFT) and the chiral transition is in perfect agreement with the Ising CFT, including central charge, scaling dimension spectrum, and operator product expansion coefficients.

Speaker: Kenji Homma (The Institute for Solid State Physics, the University of Tokyo)

34 Learning tensor networks from noisy functions

Recently, tensor networks have expanded beyond their initial application in quantum state compression, finding versatile uses in other fields of physics including image compression [1], turbulence [2], and quantum field theory [3,4]. Among these applications, it has been revealed that tensor networks can efficiently compress functions with a low-rank structure into an operable format. Particularly, the Quantics Tensor Cross Interpolation (QTCI) [5] method has received significant attention. QTCI is a technique that discretizes a function as a tensor and interpolates it by selecting important evaluation points, allowing the efficient transformation of the function into an operable format, i.e., tensor train. However, when function evaluations are contaminated with noise, such as in quantum computing, QTCI can suffer from the problem of overfitting.
In this study, we propose a robust QTCI method for functions containing noise, enabling more accurate learning of noise-free functions. Furthermore, by applying this method to a ground-state energy solver using quantum computing [6], we demonstrate that the proposed method can improve accuracy in terms of the number of samples required compared to the simplified Monte Carlo method employed in the previous research.
[References]
1. J. I. Latorre, arXiv:quant-ph/0510031v1.
2. N. Gourianov et al., Nat. Comput. Sci., 2, 30 (2022).
3. Y. N. Fernández el al., Phys. Rev. X, 12, 4, 041018 (2022).
4. H. Shinaoka et al., Phys. Rev. X, 13, 021015 (2023).
5. M. K. Ritter, H. Shinaoka et al., Phys. Rev. Lett. 132, 056501 (2024).
6. M. Huo et al., Quantum, 7, 916 (2023).

Speaker: Kohtaroh Sakaue (Department of Physics, Saitama University)

35 Volatility time series analysis by quantum circuit learning

Volatility is of great importance for quantifying potential risk of financial assets. In empirical finance, usually volatility is estimated by a suitable model selected various existing volatility models. Here we model the volatility time series by quantum circuits. Using artificial volatility time series generated by the GARCH model often used in empirical finance, we perform the quantum circuit learning and verify that simple quantum circuits can reproduce the time series generated by the GARCH model.

Speaker: Tetsuya Takaishi

36 Estimating Evolution Time in Adiabatic Quantum Dynamics of Spin Chains

The XXZ spin chain model is a relevant benchmark for evaluating the effectiveness of adiabatic time evolution in quantum computers. In this study, we aim to estimate the evolution time required to reach the final ground state in adiabatic quantum dynamics of spin chains through classical matrix product state simulations. We configured the initial Hamiltonian as independent XX or Heisenberg dimers and the final Hamiltonian as XX or Heisenberg chain with open boundary conditions. We employed standard trotterized real-time evolution methods using finite matrix product states.

Speaker: Hidehiko Kohshiro (RIKEN)

37 Tensor network formulation of the three-dimensional SU(2) principal chiral model

Tensor network and quantum computing are providing novel numerical approaches for lattice gauge theories in high-energy physics. These methods allow us to investigate the models suffering from the sign problem and many attempts have been made recently toward their applications to the QCD at finite density. For these future applications, it is necessary to establish how to deal with non-Abelian fields with a certain discretization scheme.
In this study, we consider the character expansion as a regularization scheme for SU(2) fields based on the path integral formalism. As a benchmark, we formulate the SU(2) principal chiral model on a cubic lattice as a tensor network and evaluate its internal energy and magnetization by the tensor renormalization group methods. Comparing them with those obtained by the Monte Carlo, we discuss the truncation effect from the character expansion and the possible extension for other lattice models.

Speaker: Shinichiro Akiyama

38 Quantum many-body scars and their dynamics in the U(1) lattice gauge theory

Understanding the fundamental theory of preventing thermalization is crucial for practical quantum device development. In this work, we expand upon prior research on quantum many-body scars in the U(1) quantum link and quantum dimer models. By employing a graphical representation of the basis, we extend the analytical expressions for specific scars to encompass a broader family of such scars. As for the dynamics of these scars, we conduct numerical tests to evaluate their robustness in the presence of noise, alongside the investigation of other long-time physics. Further understanding of these scar states has the potential to provide insights into the broader context of thermalization.

Keywords: Quantum many-body scars, eigenstate thermalization hypothesis

Speaker: Mr Tao-Lin Tan (National Tsing Hua University)

39 Learning tensor networks with parameter dependencies for Fourier-based option pricing

The fast solution of option pricing is a critical issue in quantitative finance. In the case of multiple assets, the computational cost of numerical simulations increases with the number of assets. Recent research has shown the potential for speeding up Fourier-based option pricing [1] using a tensor network learning algorithm, namely, tensor cross interpolation [2]. Another advantage of the tensor network is its ability to compress functions, including their parameter dependencies. In this study, we propose a scheme that utilizes the tensor train embedding parameter dependencies, thereby enabling the rapid calculation of option prices for various parameter changes. To benchmark the proposed method, we focus on scenarios involving fluctuations in volatility ($\sigma$). We demonstrate through numerical analysis that the resulting error of option pricing stays within the statistical error margin of a Monte Carlo simulation with $10^5$ samples. Asl, we would like to discuss the speed advantage of the proposed method against the Monte Carlo approach.

[1] M. Kastoryano et al., arXiv:2203.02804 (2022).
[2] I. V. Oseledets, Linear Algebra and Its Applications 80 653 (2009).

Speaker: Rihito Sakurai (Department of Physics, Saitama University)

40 Magnon Spectra of Cuprate Materials beyond Spin Wave Theory

Spin wave theory (SWT), in particular at lowest order, is often used to extract spin exchanges from scattering experiments such as inelastic neutron scattering and resonant inelastic X-ray scattering. However, this approach has limitations in accounting for large quantum fluctuations and possible fractional excitations that go beyond a magnon description. To address this, we employ a large-scale matrix product state method to compute the dynamical spin structure factor for an effective spin model on the square-lattice that is relevant to various cuprate materials. When comparing our results with experimental data for La$_2$CuO$_4$ and CaCuO$_2$, we observe a significant disparity in the strength of Hubbard-parameterized interactions with respect to those inferred from SWT. An empirical relation is derived in terms of the interaction strength and data from experimental measurements to obtain better estimates of the Hubbard parameter. Our simulations are in good agreement with ab initio studies and a modified spin wave theory analysis for La$_2$CuO$_4$, while for CaCuO$_2$ we provide a new estimate. We extent our analysis by including first order corrections to SWT, which result in a wave-vector-dependent rescaling of the magnon bands and partially recover the discrepancy with experiment and numerics. Furthermore, we provide a detailed discussion of spin dynamics in La$_2$CuO$_4$.

Speaker: Mr Jiahui Bao (OIST)

41 Multi-replica swap optimization of replica exchange method

The replica exchange Monte Carlo method, or parallel tempering, is a widely used extended ensemble method to overcome the difficulty of sampling from a complex multi-modal target distribution typical in frustrated spin systems and protein folding. Replicas having different model parameters, such as the temperature of a system, are stochastically swapped using the Metropolis algorithm. Enhancing the replica swap probability and the round trip rate is crucial to the success of the replica exchange method. In the meantime, nonreversible Markov chains can generally outperform reversible chains and enhance sampling efficiency. The lifting technique and the probability optimization beyond detailed balance have been applied to various physical and chemical systems. In this work, we propose combining these approaches to enhance the replica swap efficiency and the round trip rate in the replica exchange method. The multi-replica swap probability is maximized beyond the Metropolis algorithm. Our approach can be combined with any local update method for each temperature.

Speaker: Hidemaro Suwa

42 Supersolid states in two-dimensional hard-core bosonic Hubbard model with dipole interactions

For realizing supersolid [1,2], many efforts have been made so far. Theoretically, short-range frustrated interactions in lattice Bose-Hubbard models play a key role to stabilize the supersolid states. Ultracold-atomic gases in optical lattices are promising experimental systems for the supersolid state. However, the presence of the dipole-dipole interactions is expected there. The effect of the dipole-dipole interactions in the two-dimensional bosonic Hubbard model has been investigated [3] and the authors have studied the ground state phase diagram by using the mean-field (MF) approach and infinite entangled-pair-state (iPEPS) calculations. The MF approach and iPEPS calculations have predicted that several supersolid phases appear in between solid phases with different commensurate fillings, when the dipole axis is tilted on the two-dimensional lattice plain and the range for the dipole-dipole interactions is finite.
In this study, we investigate the ground-state phase diagram of a two-dimensional bosonic Hubbard model with dipole-dipole interactions by quantum Monte Carlo (QMC) calculations. To characterize the nature of each phase appeared, we apply not only conventional finite-size-scaling approaches but also a machine-learning assisted approach. We confirm that QMC results reproduce the phase diagram obtained by the MF approach and iPEPS calculations [3]. Next, by changing the cut-off distance for the long-range dipole interactions, we further investigate the phase diagram. When the cut-off distance becomes longer, two quarter solid states ($\rho$=1/4 and 3/4) observed in ref. [3] becomes unstable and diagonal stripe solid states ($\rho$=1/3 or 2/3) are stabilized. This means that the corresponding quarter-supersolid states becomes unstable and the diagonal-stripe supersolid states are stabilized instead.

References:
[1] G. V. Chester, Phys. Rev. A 2, 256 (1970).
[2] A. J. Leggett, Phys. Rev. Lett. 25, 1543 (1970).
[3] H. -K. Wu and W. -L. Tu, Phys. Rev. A 102, 053306 (2020).

Speaker: Takafumi Suzuki (University of Hyogo)

43 Using Matrix Product State to Solve Scale Invariant Hamiltonian in Efimov Physics

Matrix Product State (MPS) and Density Matrix Renormalization Group (DMRG) serve as effective variational techniques for investigating the low-energy states within quantum many-body systems, utilizing the underlying entanglement structures. By broadening the scope of MPS as a data representation framework, it becomes more adept at capturing intricate correlations within the system. Recent advancements have expanded the utility of this approach to efficiently address hydrodynamic equations, including the complex dynamics of phenomena like turbulence, and can compress data well. This study endeavors to adapt these methodologies to Efimov physics, which is characterized by unique universal properties and discrete scale invariance. Within this context, two new distinct approaches for generating the inverse of the required $1/R^2$ potential into MPS, thereby reproducing discrete scaling behavior, have been identified, alongside a detailed exploration of associated numerical challenges.

Speaker: Che-Chia Hsu (National Tsing Hua University)

44 Quantum State Preparation for Probability Distributions with Mirror Symmetry Using Matrix Product States

Quantum circuits for loading probability distributions into quantum states are essential subroutines in quantum algorithms used in physics, finance engineering, and machine learning. The ability to implement these with high accuracy in shallow quantum circuits is a critical issue. We propose a novel quantum state preparation method for probability distribution with mirror symmetry using matrix product states. By considering mirror symmetry, our method reduces the entanglement of probability distributions and improves the accuracy of approximations by matrix product states. As a result, we improved the accuracy by two orders of magnitude over existing methods using matrix product states. Our approach, characterized by a shallow quantum circuit primarily comprising nearest-neighbor qubit gates and linear scalability with qubit count, is highly advantageous for noisy quantum devices. Also, our experimental findings reveal that the approximation accuracy in tensor networks depends heavily on the bond dimension, with minimal reliance on the number of qubits. We experimentally demonstrated our method for a normal distribution encoded into 10 and 20 qubits on a real quantum processor.

Speaker: Yuichi Sano (Kyoto University)

45 Spin Screening Cloud in Local Moment Phases

A magnetic impurity in a metal is screened by conduction electrons via the Kondo effect. However, if the local density of states (LDOS) of the electrons near the impurity exhibits a pseudogap or hard gap at the Fermi level, the screening becomes imperfect, leading to a local moment (LM) phase. It has been believed that in the LM phase, the impurity spin is not screened but decoupled from the conduction electrons. In contrast to this common belief, we show that in the LM phase, the impurity spin is screened by the conduction electrons by computing quantum entanglement between the impurity spin and conduction electrons. The electrons form a spin screening cloud spatially extended over a distance, which is a generalization of the Kondo cloud. For the pseudogap LDOS, we employ the numerical renormalization group method and show that the cloud spatially decays algebraically with increasing distance from the impurity. For the hard gap LDOS, we use the density matrix renormalization group method, and the spin cloud decays exponentially. In each case, the spatial distribution of the spin cloud follows a single universal function of a rescaled distance by a characteristic length of the system.

Speaker: Mr Minsoo Kim (Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea)

46 Variational tensor network operator

Combining the ideas of the imaginary-time evolution and the variational optimization of trial wave functions, we propose a generic construction of the variational tensor network operators[1] to study the quantum spin systems. We demonstrated that accurate variational ground state wave functions with extremely few tunable parameters can be obtained by applying these operators to some simple initial states. We further showed that this framework can be applied to study spontaneously symmetry breaking, symmetry protected topological, and intrinsic topologically ordered phases, it is found that symmetries of the local tensors associated with these phases can emerge directly after the optimization without any gauge fixing. This provides an universal way to identify quantum phase transitions without prior knowledge of the system.

[1] Yu-Hsueh Chen, Ke Hsu, Wei-Lin Tu, Hyun-Yong Lee, and Ying-Jer Kao.
Variational tensor network operator. Phys. Rev. Res., 4:043153, Nov 2022.

Speaker: Ke Hsu (Department of Physics, National Taiwan University, Taipei 10607, Taiwan)

47 Tailoring the Phonon Environment of Embedded Rydberg Aggregates

State-of-the-art experiments can controllably create Rydberg atoms inside a Bose-Einstein condensate (BEC) [1]. The large Rydberg electron orbital volume contains many neutral atoms, resulting in electron-atom scattering events. The number of atoms within the orbit, and hence the Rydberg-BEC interaction, can be tuned by choice of principal quantum number or condensate density. This makes the hybrid system a fascinating platform for quantum simulation. My poster presentation will discuss the physics of the interaction and corresponding dynamics of single or multiple Rydberg atoms in two internal electronic states embedded inside a BEC, to assess their utility for controlled studies of decoherence and quantum simulations of excitation transport similar to photosynthetic light-harvesting.

The poster will initially include the theoretical framework that we developed to calculate the open quantum system input parameters like the bath correlation function and the spectral density, initially for a single Rydberg atom, possibly in two internal states with angular momentum quantum numbers l = 0 (|s⟩) and l = 1 (|p⟩) [2], in BEC and then for a chain of Rydberg atoms, forming an aggregate. The electron-atom contact interactions lead to Rydberg-BEC coupling, which creates Bogoliubov excitations (phonons) in the BEC.

Using this spin-boson model with the calculated parameters, we examine the decoherence dynamics of a Rydberg atom in a superposition of |s⟩ and |p⟩ states, resulting from the interaction with its condensate environment. Further, the poster will discuss the emergence of non-Markovian features in the system in the presence of a microwave external drive of the Rydberg atom using a stochastic computational technique for non-Markovian open quantum systems [3].

Finally, the poster will preview the results for the aggregate case, where one of the atoms in the aggregate is in the state |p⟩, while the rest are in the state |s⟩, resulting in excitation transport via dipole-dipole interaction [4]. We investigate the effects of non-Markovianity and decoherence on the excitation transport based on an effective model described by a Holstein Hamiltonian, allowing us to set up the dynamics similar to those found in light-harvesting complexes, but at a different time and energy scales.

References:

[1] J. B. Balewski, et. al.; Nature 502 664 (2013).

[2] S. Rammohan, et. al.; Phys. Rev. A 103, 063307 (2021).

[3] S. Rammohan, et. al.; Phys. Rev. A 104, L060202 (2021).

[4] D. W. Schönleber, et. al.; Phys. Rev. Lett. 114 123005 (2015).

Speaker: Dr Sidharth Rammohan (1. Indian Institute of Science Education and Research Bhopal, India 2. Kindai University, Japan)

48 Tensor Network Finite-size Scaling for 2D Classical Models

Finite size scaling analysis (FSS) provides a method to approach critical behavior using quantities in finite size systems. For computation, we employ the tensor network method and higher-order tensor renormalization group (HOTRG) to calculate various quantities for finite size systems. We demonstrate that finite size analysis enables us to accurately extract critical temperature, critical exponents of the system.

Speaker: Sing-Hong Chan (National Tsing-Hua University)

49 Numerical analysis of the angular-time correlation functions for the S = 1 Heisenberg chain

The ground states of low-dimensional quantum many-body systems exhibit theoretically fascinating behaviors, such as quantum spin liquids and symmetry-protected topological orders. In their theoretical analysis, entanglement entropy(EE) and spectrum(ES) often play an essential role as a quantitative indicator of the quantum many-body entanglement. Meanwhile, verifying quantum many-body entanglement in realistic quantum spin systems is still a challenging problem because EE and ES are not directly observable quantities.
In this work, we discuss the angular-time evolution approach for spin operators to detect the ES in the ground state of the S=1 Heisenberg spin chain, which is well-known as a Haldane state. Such a protocol for the ES utilizing the angular-time correlation function was initially introduced for the XXZ chain based on the theoretical analogy between the Unruh effect in quantum gravity and the quantum entanglement structure for its bipartitioned ground state[1]. Moreover, recently, we successfully applied the protocol to the Affleck-Kennedy-Lieb-Tasaki (AKLT) chain, which has symmetry-protected topological entanglement associated with Z2 x Z2 symmetry, and found that the angular-time evolution can be interpreted as a real-time evolution of the edge state induced by a uniform magnetic field in the system part with the use of a gauge transformation for the matrix product state[2]. However, the AKLT chain is a rather mathematical model with no experimental counterpart. Using the density matrix renormalization group, we thus analyze the angular-time evolution for the ground state of the S=1 Heisenberg chain and then discuss whether we can correctly capture the ES in a realistic experimental situation.

[1] K. Okunishi and K. Seki, J.Phys. Soc. Jpn. 88, 114002 (2019)
[2] K. Nakajima and K. Okunishi, Phys. Rev. B 106, 134304 (2022)

Speaker: Takuma Kaise (Niigata University)

50 Generating Function for Projected Entangled-pair States

Speaker: Wei-Lin Tu (Keio University)

51 Quantum Simluations for Lattice Gauge Theories

Speaker: Jesse Osborne (University of Queensland)

Thursday, 28 March

Symposia talks

Convener: Rongyang Sun

52 Couteracting errors in early fault-tolerant quantum computing

Quantum error mitigation methods are designed to eliminate the effect of noise in quantum computation by introducing a trade-off between bias and variance, using modified quantum circuits and classical postprocessing. While various techniques have been proposed with their own advantages and disadvantages, there is still no universal criterion to choose the best method for a given application. In this talk, we focus on the sample complexity, or the measurement overhead, to perform unbiased quantum error mitigation, and discuss the performance bounds and their achievability. We show that, based on quantum estimation theory, the overhead generally grows exponentially with the circuit depth and also with the number of qubits in scrambling quantum circuits. We then show that the bounds on the overhead are provably tight under white noise, and that a simple rescaling technique achieves cost-optimality. Based on numerical simulations, we argue that a wide class of unital and nonunital noise are converted into white noise under sufficiently deep scrambling quantum circuits. This implies that, our findings become increasingly important when the error rate is reduced by hardware advancement or implementation of error correction. In this context, we also also discuss how to suppress algorithmic errors in a cost-optimal way under the framework of fault-tolerant quantum computing.

Speaker: Nobuyuki Yoshioka (Department of Applied Physics, University of Tokyo,)

53 Nonvariational imaginary-time evolution for first-quantized molecular systems

While the variational quantum eigensolver (VQE) is widely used today, the imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. We proposed recently an algorithm for finding the optimal molecular geometries [1] based on the the probabilistic ITE (PITE) [2] for a first-quantized molecular system. We discuss the applicability of the scheme by focusing on the computational cost. The scheme exhibits quantum advantage with respect to electronic and nuclear degrees of freedom when employing the multi-step PITE [3].

[1] Kosugi, Nishi, and Matsushita, npj Quantum Inf. 9, 112 (2023)
[2] Kosugi, Nishiya, Nishi, and Matsushita, Phys. Rev. Research 4, 033121 (2022)
[3] Nishi, Kosugi, Nishiya, and Matsushita, arXiv:2308.03605

Speaker: Taichi Kosugi (Quemix Inc.)

54 Adaptive measurement strategy for quantum subspace methods

Estimating physical properties for unknown quantum states is a crucial matter spanning various domains, including quantum information processing, quantum physics, and quantum chemistry. In the realm of quantum computation, existing research has predominantly focused on comprehensive state tomography or estimating specific observables with known classical descriptions. However, a notable gap exists in addressing problems where the target for estimation depends on the measurement outcome. In this study, we introduce an adaptive optimization approach for measurements, specifically useful for quantum subspace methods, which are variational simulation techniques that involve classical postprocessing of measurement outcomes. Our proposed method initially establishes the measurement protocol for classically simulatable states. Subsequently, it adaptively updates the protocol based on the Quantum Subspace Expansion (QSE) method using the outcomes of quantum measurements. Through numerical experiments, we demonstrate that our approach achieves two significant outcomes: (i) a substantial reduction in the number of required measurements, by constructing an effective measurement strategy; (ii) successful convergence of the adaptive iteration, even for strongly correlated molecules like H$_4$ during excited-state simulations. This work emphasizes the potential enhancement of the QSE method through sophisticated measurement protocols, paving the way for further exploration of efficient quantum measurement techniques in practical computations.

Speaker: Yuma Nakamura (Healthcare & Life Science, IBM Japan)

Symposia talks

Convener: Masahiko G. Yamada (SQAI, UTokyo)

55 Defect, generalized symmetry and edge modes

Protected edge modes are one of the most exotic phenomena in contemporary condensed matter physics. In this presentation, I show a general quantum Hamiltonian formalism to the protected edge modes based on the recent development of bulk and boundary renormalization groups. Our formalism gives a way to express a series of boundary phenomena in contemporary physics in a concise way. Generalized symmetry and its defect realization which have been formulated and studied in the language of Tensor-Network plays a significant role. I hope the new formalism may be useful both for theoretical and numerical researchers and give a guideline for future research directions. This presentation is based on arXiv:2312.12887.

Speaker: Yoshiki Fukusumi (Nanyang Technological University)

56 Entanglement measures in massless Lifshitz field theory with arbitrary anisotropy

We study salient information and correlation measures, namely, entanglement entropy, reflected entropy, Markov gap and timelike entanglement entropy in the 1+1D massless Lifshitz field theory that follows anisotropic scaling along temporal and spatial directions. We introduce a continuous family of Lifshitz scale invariant degenerate Rokhsar-Kivelson ground states for our chosen theory with any real arbitrary anisotropy index. By using the notion of fractional derivatives, we employ the associated kernels to study different entanglement measures for both adjacent and disjoint subsystems. Non-trivial dependencies of each of the measures on the arbitrary anisotropy are found. Based on our observations, we propose a holographically consistent 2+1D Lifshitz bulk with an anisotropy-dependent radius of curvature. Subsequently, we use the holographic picture to compute the timelike entanglement entropy for Lifshitz theories.

Speaker: Dr Adrita Chakraborty (Department of Physics and Center of Theory and Computation, National Tsing Hua University)

57 A statistical mechanics approach to holographic renormalization group

We discuss a holographic aspect of the Bethe lattice Ising model, which is a classic model of the phase transition in statistical mechanics and, at the same time, the simplest example of the tree tensor network. We analytically formulate a holographic renormalization group for the model and explain how the power-law decay of the boundary spin correlations emerges based on the network geometry. We also discuss its connection to the p-adic AdS/CFT.

Speaker: Kouichi Okunishi (Niigata University)

Symposia talks

Convener: Kouichi Okunishi (Niigata University)

58 Tensor-network algorithms for nonequilibrium dynamics

Laser technology has made remarkable progress recently, opening new possibilities for nonequilibrium physics experiments. One of the exciting phenomena is quantum phase transitions induced by pulse irradiation, which lead to nonequilibrium metal-superconductor transitions. However, the theoretical analysis of these phenomena is challenging, as it requires computing the dynamic correlation function over two-time domains.
Exploiting the translational symmetry in infinite matrix-product-state representations, we present a novel simulation method based on the (infinite) time-evolved block decimation technique. We apply this technique to Mott insulators and simulate the time-dependent single-particle photoemission spectra, which can be measured by time- and angle-resolved photoemission spectroscopy experiments. We demonstrate that applying pump pulse irradiation to the Mott insulator in the simple Hubbard model triggers photoinduced insulator-to-metal transitions, associated with the formation of eta pairs.

References
[1] SE, F. Lange and H. Fehske, Phys. Rev. Res. 4, L012012 (2022).
[2] SE, T. Kaneko, F. Lange, S. Yunoki and H. Fehske, Phys. Rev. Res. 2, 032008(R) (2020).
[3] SE, F. Lange and H. Fehske, Eur. Phys. J. Spec. Top. (2023).

Speaker: Dr Satoshi Ejima (German Aerospace Center)

59 DC electric field-driven discretized excitation spectra in Mott insulators: an infinite matrix-product state approach

Phenomena induced by dc electric fields in strongly correlated electron systems, such as Mott breakdown and field-induced magnetism, have been widely discussed both experimentally and theoretically. Recently, intense terahertz light pulses generated from synchrotron radiation have attracted particular attention as a method for observing these phenomena [1] since the energy of terahertz light is quite small compared to the energy gap of Mott insulators and it can be regarded as a low-frequency limit, i.e., almost a dc electric field. Using this terahertz light as the pump light and examining the dynamical response from the probe light, the properties of strongly correlated materials in a dc electric field can be clarified.
In this study, we investigate the optical conductivity and single-particle excitation spectra in one-dimensional Mott insulators under a dc electric field, employing an infinite matrix-product state approach [2,3]. In Mott insulators, the energy level of lower and upper Hubbard bands due to electronic correlations are discretized by the electric field, resulting in the multiple peaks in the spectra. Our results are associated with the Wannier-Stark ladder [4] in a tilted potential.
[1] D. Nicoletti and A. Cavalleri, Adv. Opt. Photonics 8, 401 (2016).
[2] M. Udono, T. Kaneko, and K. Sugimoto, Phys. Rev. B 108, L081304 (2023).
[3] K. Sugimoto, arXiv:2401.17466.
[4] Y. Murakami and P. Werner, Phys. Rev. B 98, 075102 (2018).

Speaker: Koudai Sugimoto (Keio University)

60 Thermal and Spatial Entanglement of Quantum Impurity Systems

The quantum coherent screening of a local spin is an essential concept of quantum impurity problems. In this work, we theoretically analyze the quantum coherent screening by using quantum entanglement. We develop a method to compute the entanglement negativity between the impurity spin and electrons in spin-1/2 impurity problems, based on the boundary conformal field theory and numerical renormalization group [1]. We analyze thermal entanglement and its spatial profile, which is called the screening cloud, in multichannel Kondo systems [1,2] and two-impurity Kondo systems [3]. At low temperature, we show the universal behavior that the entanglement exhibits a power-law thermal decay with fractional exponent. The exponent is determined by the scaling dimension of the boundary operator representing the impurity spin. We show that the spatial distribution of entanglement also has the universal power law structure with the same exponent. We find that distinct (non-)Fermi liquids coexist in the distribution, forming concentric shells centered at the impurity. Outer shells are suppressed one by one as temperature increases, and the remaining outermost shell determines the entire phase of the system at that temperature.

[1] D. Kim, J. Shim, and H.-S. Sim, Phys. Rev. Lett. 127, 226801 (2021)
[2] J. Shim, D. Kim, and H.-S. Sim, Nat. Commun. 14, 3521 (2023).
[3] D. Kim, M. Kim, J. Shim, and H.-S. Sim, in preparation.

Speaker: Donghoon Kim (RIKEN)

Symposia talks

Convener: Koudai Sugimoto (Keio University)

61 Anderson impurity solver integrating tensor network methods with quantum computing

Solving the Anderson impurity model typically involves a two-step process, where one first calculates the ground state of the Hamiltonian, and then computes its dynamical properties to obtain the Green's function. Here we propose a hybrid classical/quantum algorithm where the first step is performed using a classical computer to obtain the tensor network ground state as well as its quantum circuit representation, and the second step is executed on the quantum computer to obtain the Green's function. Our algorithm exploits the efficiency of tensor networks for preparing ground states on classical computers, and takes advantage of quantum processors for the evaluation of the time evolution, which can become intractable on classical computers. We demonstrate the algorithm using 20 qubits on a quantum computing emulator for SrVO3 with a multi-orbital Anderson impurity model within the dynamical mean field theory. The tensor network based ground state quantum circuit preparation algorithm can also be performed for up to 40 qubits with our available computing resources, while the state vector emulation of the quantum algorithm for time evolution is beyond what is accessible with such resources. We show that, provided the tensor network calculation is able to accurately obtain the ground state energy, this scheme does not require a perfect reproduction of the ground state wave function on the quantum circuit to give an accurate Green's function. This hybrid approach may lead to quantum advantage in materials simulations where the ground state can be computed classically, but where the dynamical properties cannot.

Speaker: Francois Jamet (IQM France)

62 Developing SUNDMRG.jl

Based on the previous idea of implementing SU(N) symmetries in the density matrix renormalization group (DMRG) [1], we invented a new algorithm, which has been helpful in extending the previous standard-Young-tableaux approach to generic two-dimensional models, for SU(N)-symmetric DMRG [2]. This new algorithm intensively uses the so-called 9ν coefficients of SU(N) irreducible representations, which are the simplest generalization of the 6j/9j-symbols implementation of SU(2).

SUNDMRG.jl [3] is a Julia implementation of our new algorithm. Not only by strictly implementing such a complicated simulation but also by supporting MPI/CUDA high-level parallelization have we achieved the world-record-level DMRG code. More than a million effective bond dimensions can be used in the calculation on the GPU system with two NVIDIA A100s, for example. Parallel GPU simulations are highly suitable for the next-generation DMRG simulation for two-dimensional correlated matters, revealing a hidden side of the quantum correlation in condensed matter systems.
[1] P. Nataf and F. Mila, Phys. Rev. B 97, 134420 (2018).
[2] M. G. Yamada, K. Penc, and F. Pollmann, to appear.
[3] https://github.com/MGYamada/SUNDMRG.jl

Speaker: Masahiko G. Yamada (SQAI, UTokyo)

63 Scalable Quantum Circuits Simulation with Real-Space Parallelizable Tensor Network Methods

Classical simulation of current noisy intermediate-scale quantum (NISQ) devices forwards the development of all the research directions in near-term quantum computing. While the simulation algorithms based on tensor network states can efficiently simulate a NISQ device with around 100 qubits, restricted by the real-space sequential nature of these algorithms, efficiently simulating a NISQ device with hundreds, or even thousands, of qubits remains elusive. One of the approaches to releasing this limitation is developing real-space parallelizable algorithms.

In this presentation, I will introduce a newly developed real-space parallelizable matrix-product state (MPS) compression method [1] that can efficiently compress all the virtual dimensions of the MPS in a constant time against increasing the system size and simultaneously stabilize the wavefunction norm [see Fig. 1(a)] without triggering sequential renormalization procedures. Moreover, the deviated canonical form is partially recovered by appended parallel regauging steps. Based on this method, we propose the parallel time-evolving block-decimation (pTEBD) algorithm for the simulation of unitary quantum circuits. After benchmarking the pTEBD algorithm with extensive simulations of typical one- and two-dimensional quantum circuits containing up to over 1000 qubits on Supercomputer Fugaku, we demonstrate that the pTEBD algorithm achieves the same simulation precision as compared with the current state-of-the-art MPS algorithm using a polynomially shorter time [see Fig. 1(b)], exhibiting a nearly perfect performance weak scaling [see Fig. 1(c)].

Reference
[1] Rong-Yang Sun, Tomonori Shirakawa, and Seiji Yunoki. "Improved real-space parallelizable matrix-product state compression and its application to unitary quantum dynamics simulation." arXiv preprint arXiv:2312.02667 (2023).

Speaker: Dr Rongyang Sun (RIKEN)

Friday, 29 March

Invited talks: Mingpu Qin

Convener: Wei-Lin Tu (Keio University)

64 Fully-augmented Matrix Product States: algorithm, application, and parent Hamiltonian

I will introduce a new tensor network states ansatz called Fully-augmented Matrix Product States (FAMPS), in which MPS is augmented with disentanglers to encode area-law-like entanglement entropy (entanglement entropy supported in FAMPS scales as $l$ for an $l$ × $l$ system). I will discuss the optimization algorithm of FAMPS in the study of 2D quantum system. With FAMPS, we reexamine the $J_1-J_2$ Heisenberg model on square lattice and find the absence of the spin liquid phase in the phase diagram. I will also discuss the 2D parent Hamiltonian of FAMPS.

[1] Xiangjian Qian, Mingpu Qin, Chin. Phys. Lett. 40, 057102 (2023) (Express Letter).
[2] Xiangjian Qian, Mingpu Qin, arXiv:2309.13630 (2023).
[3] Xiangjian Qian, Mingpu Qin, arXiv:2401.07659 (2023).

Speaker: Mingpu Qin (Shanghai Jiao Tong University)

Invited talks: Didier Poilblanc

Convener: Wei-Lin Tu (Keio University)

65 Chiral spin liquids with PEPS: should we fear the no-go theorem

Speaker: Didier Poilblanc (Univ. of Toulouse)

Invited talks: Keisuke Fujii

Convener: Dr Satoshi Morita (Keio University)

66 Quantum Machine Learning: interplay between implicit and explicit quantum models

Quantum machine learning, promising for quantum computers, explores implicit models utilizing quantum kernel methods or explicit models, known as quantum circuit learning. While implicit models often yield lower training errors, they face linear prediction time scaling with data size, potentially overfitting. Explicit models predict in constant time but encounter challenges with optimization, notably the barren plateau phenomenon. This study introduces a quantum-classical hybrid algorithm to convert implicit models efficiently to explicit ones. The resulting explicit model matches implicit model performance but requires fewer quantum circuit executions for inference. In classification tasks using MNISQ and VQE-generated datasets, our explicit model shows comparable generalization to implicit models with reduced computational costs. Our algorithm accelerates prediction for implicit models and aids in constructing high-performance explicit models, notably addressing the barren plateau phenomenon. We also discuss our efforts in developing infrastructure for quantum machine learning, including datasets, libraries, and simulators.

Speaker: Prof. Keisuke Fujii (Osaka University)

Invited talks: Miles Stoudenmire

Convener: Dr Satoshi Morita (Keio University)

67 Complex Time Evolution for Time-Dependent Correlations

Speaker: Miles Stoudenmire (Flatiron Institute)

Epilogue: Farewell