# Outline of Project E01

## Exploration of quantum gravity and quantum matter based on symmetry and information geometry

The University of Electro-Communications,

Graduate School of Informatics and Engineering

In recent years, Resource Theory of Asymmetry (RTA) has been rapidly developing a comprehensive analysis of the effects of symmetry on quantum mechanics. RTA treats quantum operations which are covariant with respect to given symmetry as free operations, and defines the quantum fluctuation of conserved quantity corresponds to the given symmetry as resource. Recently, I found several trade-off relations in RTA (PRL2018, QIP2020, PRL2021, etc.). The goal of this project is to unify and extend the trade-off relations and apply them to find several useful theorems in quantum gravity and condensed matter theory.

The results obtained to date include a universal trade-off structure between symmetry, irreversibility, and quantum coherence. This trade-off particularly reveals that (1) under a global symmetry, any attempt to induce local dynamics that change the conserved quantity will cause inevitable irreversibility, and (2) such irreversibility could be mitigated by quantum coherence. This relation admits broad applications including measurement, error-correction, gate implementation, thermodynamics, and black hole physics. For example, we apply our relation to black hole physics and obtain a universal lower bound on how many bits of classical information thrown into a black hole become unreadable under the Hayden-Preskill model with the energy conservation law. This particularly shows that when the black hole is large enough, under suitable encoding, at least about $m/4$ bits of the thrown $m$ bits will be irrecoverable until 99 percent of the black hole evaporates. In the future, I aim to extend and apply these results more and more.

## Study of physically natural method to generate t-design in quantum many-body systems based on quantum control theory

Department of Computer Science,

Faculty of Informatics, Shizuoka University

Random unitary transformation is one of the central techniques in quantum information processing. In early stages of quantum information processing research, random unitary transformation based on Haar measure is used. Here, Haar measure is a regular invariant measure on the unitary group. However, it was recognized that the implementation of the random unitary transformation based on the Haar measure requires exponential time. Therefore, unitary t-design, a measure that approximates the Haar measure, has attracted much attention. A t-design are not only essential for various quantum information processing, but have recently been found to be important as a method for approximating quantum chaotic systems and for Markovizations of environmental systems.

Existing t-design generation protocols are designed as polynomial-time algorithms that work efficiently on a quantum computer. However, it difficult to implement them on a quantum many-body system. In this project, we are focus on finding a physically more natural way to generate t-designs in quantum many-body systems than existing methods, with applications such as Markovianization and quantum chaos in mind. First, by applying the techniques of quantum control theory, we prove the existence of a random control method such that the random dynamics of the whole system is t-design in a fully controllable quantum system. In a quantum many-body system, it is natural to assume a situation in which a random action is externally applied only to the restricted degrees of freedom. Therefore, by using the random control method proved above, we show that t-design of the whole system can be achieved by random unitary transformations to small subsystems in well-known one-dimensional and two-dimensional spin systems. Furthermore, we aim to apply the obtained new t-design generation method to quantum information processing. In particular, by creating parametric quantum circuits based on the new method, we aim to devise a new quantum many-body device that can perform variational quantum algorithms by accessing only a part of the system.

## Information-theoretical analysis of one-dimensional tensor network

Department of Mathematical Informatics,

Graduate School of Informatics, Nagoya University

Tensor networks are an efficient representation to describe quantum many-body states or operators with a small number of parameters. They are widely used in ExU researches too, to analyze the dynamics of quantum many-body systems and the holography principle. The use of tensor networks has been spreading to various fields of physics in recent years, but even in relatively simple cases such as one-dimensional (1D) networks, there are still many unresolved aspects in the theoretical foundation.

In this publicly offered research, we will clarify what kind of quantum states can be described by 1D tensor networks, called MPO. In 1D systems, two major types are known: MPS and MPO (there is also another type called MERA, but we do not discuss it here). It is theoretically known that the quantum states that can be efficiently represented by MPS coincide with the low-temperature states (ground states) of 1D systems. This indicates not only that MPS is suitable for describing low-temperature states, but also that MPS is not suitable for describing other states, which is a limitation of MPS's expressive capability. In contrast, MPO is a tensor network suitable for describing thermal states in 1D systems. However, it is known that the states described by MPO include states beyond that, and it is not clear to what exactly the physical states described by MPO. The problem of the description range of tensor networks is not well understood except for MPS, and this research will tackle this unsolved problem for MPO as a first step.

Interestingly, MPO is also related to the physics of 2D spin systems. It has been pointed out that the physical properties of 2D noncritical systems at low temperatures are reflected in the 1D system at the boundary of the system. The 1D state at the boundary is described by MPO, so we plan to address the "holographic" correspondence between such 2D low-temperature states and 1D thermal states in the latter half of this proposal.

## Development of quantum many body system out-of-equilibrium and quantum information control with NMR

Department of Physics, Nagoya University

Nuclear spins weakly interact with electron spins through hyperfine interactions in the energy scale of MHz-GHz. The hyperfine interactions include a variety of information associated with the electronic structure and symmetry. Our research group utilizes nuclear magnetic resonance (NMR) as the local probe of symmetry breaking and low energy excitation in condensed quantum matters by determining and solving the hyperfine interaction. Our target materials are the system having macroscopic quantum entanglements such as quantum Hall effects and quantum spin liquids without showing conventional long-range orders. These states can carry anyons with fractional statistics, which are important for the fault-tolerant quantum computation. Our aim is to observe such an emergent quasiparticle by NMR spectroscopy. Although the quantum Hall effect has been realized only on two-dimensional systems, the recent discovery of three-dimensional Dirac semimetals having linear energy dispersion opens the door for magnetic studies by NMR etc. Indeed, we have observed the quantized motion of Dirac fermions in the NMR relaxation rates at high fields and low temperatures. In quantum spin liquids, there is no electric charge carrier but spin excitation. It has a topological order with the long-range quantum entanglement for a gapped spin liquid case analogous to the quantum Hall regime. We have found some candidate materials exhibiting as quantum spin liquid in frustrated spin systems. Can nuclear spins mediate the long-range entanglement of electrons? In this project, we will develop the new open quantum many-body system by combining the macroscopic quantum states and nuclear spins, which is useful to read out the quantum information.

## Generation of graph states using cold Rydberg atoms and the application to measurement-based quantum computation

Institute for Molecular Science

In this research, we will perform an experiment to generate many-body entanglement states, called "graph states," which are important resources for quantum error correction and measurement-based quantum computation, via natural time evolution with the interaction between cold Rydberg atoms.

In the cold-atom experimental platform, it is possible to construct a qubit array with the optical tweezers array technique, where locally focused laser beams are used to trap each atom individually and arrange them in an arbitrary shape. Long-range interactions can also be induced between atoms by exciting them to a highly-excited electronic state called a Rydberg state. In this experiment, we produce a large-scale entanglement over the atomic qubit array and generate the graph state via a time evolution under the long-range Ising interaction between atoms.

We excite cold Rubidium atoms trapped in an optical tweezers array to the Rydberg state by pulsed lasers, which induce Ising-type long-range interactions between the atoms and time evolution to generate large-scale entanglement over the atomic qubit array, creating the graph state. By using pulsed lasers to excite the atoms in a short time compared to the time scale of the atomic interactions, we can ignore the Rydberg interactions during the excitation, subsequently inducing interactions between the atoms simultaneously, while continuous wave lasers are generally used in the Rydberg excitation. This is different from the conventional method of performing two-qubit gates for each individual qubit pair and forming entanglement sequentially. We investigate the optimal Rydberg electronic states, state manipulations, and graph geometries for graph state generation. We characterize the fidelity of the generated states by combining single-site addressing single-qubit gate operations.

In addition, we implement individual local measurement that does not affect other atoms in the neighborhood, and explore the possibility of using the generated graph states as a resource for measurement-based quantum computation.

# Outline of Project E02

## Search of topological phenomena in correlated open quantum systems based on quantum entanglement

Department of Physics, Kyoto University

In this decade, topological aspects of condensed matters have extensively studied. In topological phases, such as topological insulators, topologically nontrivial properties induce gapless boundary modes robust against perturbations. In addition to the above significant developments, non-Hermitian systems, such as open quantum systems, attract much interest recently. For such systems, a variety of novel phenomena (e.g., the emergence of exceptional points and the non-Hermitian skin effect) are reported due to the interplay between the non-Hermiticity and the topology. In my project, I address analysis of topological open quantum systems in terms of extreme universe.

In particular, recent developments of technology allow us to experimentally tune dissipation and correlations for cold atoms. In these systems, interplay between strong correlations and non-Hermitian topology would trigger off novel phenomena unique to strongly correlated open quantum systems. I particularly analyze these open quantum systems in search of exotic non-Hermitian topological phenomena. Furthermore, I address theoretical proposals how to experimentally observe such novel phenomena for cold atoms.

By communicating other researcher of extreme universe, I also address the above project from the interdisciplinary viewpoint.

## An approach to quantum gravity from quantum information theory and condensed matter theory

Institute for Solid State Physics, the University of Tokyo

General relativity and quantum mechanics are the most important developments in physics in the 20th century. However, quantum gravity, which is the combination of relativity and quantum mechanics, is still not fully understood yet. In this research, we study quantum gravity from the perspective of quantum information and condensed matter physics. Through that, we study how quantum mechanics and general relativity are combined and how spacetime emerges from quantum mechanical systems. Furthermore, we study the implication of quantum gravity on low energy effective theory through the lens of quantum information and condensed matter theory. Based on that, we study how quantum gravity gives predictions for our spacetime and universe as well as elementary particles. We also analyze the quantum information structure in condensed matter and quantum phases out of equilibrium and apply these results to problems in quantum gravity.

More specifically, we study 「[A]application of quantum channels and open quantum systems to quantum gravity」, 「[B]Quantum information and the Swampland program」, 「[C]Quantum information aspects of emergence and phases in condensed matter systems」.

In [A], we study open dynamics in the SYK model and then generalize to the top-down AdS/CFT. Based on those, we understand the emergence of time and the dual of non-unitary theories, which are not well understood. In [B], we aim to understand the swampland program in the language of quantum information theory. Beyond the black hole thermodynamics, we derive the swampland conditions by quantum information approach and quantum entanglement, which are more basic properties in quantum gravity. In [C], we study quantum information structures in emergent gauge fields from condensed matter (spin systems) like Lifsihtz gravity, fracton topological order. Emergent gauge fields are ubiquitous in condensed matter systems. Since gravity is also a gauge field, they are similar in that sense. Through the understanding of quantum information properties in the system with emergent gauge fields, we try to understand quantum gravity.

## Application of tensor networks to elementary particle physics

College of Science and Engineering, Kanazawa University

Tensor network (TN) method was originally proposed as a variational method to solve quantum many-body system in the condensed matter physics. At present, various improvements have been made on the method, and its applications have spread to many fields such as elementary particle physics, quantum information, and quantum chemistry. I am particularly interested in applying the TN method to quantum field theory, which is a theoretical foundation of elementary particle physics. My final goal is to obtain the dynamics of the theory, which are difficult to capture with the Monte Carlo method due to the sign problem, from first principle computation using the TN method. Interesting physical systems inherent the sign problem include QCD with finite baryon density, real-time path integral, lattice chiral gauge theory, lattice SUSY, and system having a topological theta term. Currently I am working on revealing a phase structure of CP(1) model with the topological term, which is known as a toy-model of QCD. Although it is expected that the knowledge of QCD can be obtained by analyzing the model, at present the phase structure of CP(1) itself is under debate. Thus, I would like to draw a final conclusion by using a more sophisticated TN method.

While the TN method has no sign problem, it still has a problem of drastic increase in computational cost when applied to higher dimensional system. A key to solve the problem is how to compress the information of tensor. Various ideas have been proposed so far, such as incorporating environment effects and local correlations, or a stochastic method. In this project, I aim to develop a highly efficient coarse-graining algorithm for high dimensional system by interweaving such knowledge.

## Study of emergence of spacetime in tensor network based on the renormalization group

Department of Physics, Shizuoka University

The emergence of space-time is considered to be an essential feature of quantum gravity. It has been recognized that quantum entanglement plays a crucial role there. The gauge/gravity correspondence is a typical example in which the emergence of space-time is realized. The purpose of this project is to make progress in the construction of quantum gravity by studying continuum tensor networks based on the renormalization group in the gauge/gravity correspondence.

Tensor networks models such as the MERA, the HaPPY code and the random tensor network are important in the studies of the emergence of space-time, since the models give networks that can be interpreted as discrete spaces emerging from boundary theories through entanglement entropy. Here it is needed to construct continuum tensor network models to obtain the emergence of continuum space.

The cMERA, a continuum analog of the MERA, is constructed in free field theories by using the variational method, while the variational method seems difficult to apply to interacting theories. Since the layers in the network of the MERA can be interpreted as the scale of renormalization group, which corresponds to the bulk direction, constructing the cMERA is considered to be equivalent to obtaining the scale dependence of wave functionals. In this project, we study the construction of continuum tensor networks based on renormalization group. In particular, we focus on the exact renormalization group and the tensor renormalization group, which can be viewed as a path-integral formulation of the MERA.

We construct the exact renormalization group for wave functionals in quantum scalar field theories, which enables nonperturbative analyses of the wave functionals. We also develop a numerical method to examine the scale dependence of the wave functionals by taking the continuum limit of the tensor renormalization group for lattice field theories. We construct the continuum tensor networks by using these methods and extract the dynamics of gravity from the networks to study the emergence of space through quantum entanglement.

## Quantum entanglement and information scrambling in inflationary models

Department of Physics, Naogya University

Stochastic approach of inflation treats the quantum inflaton field as a classical stochastic variable. The basic equation of this approach is the Langevin equation for the long wavelength modes of the inflaton field with the stochastic noise from the short wavelength inflaton fluctuations. Hence the evolution of the inflation field has the similar behavior as a Brownian particle. This approach can effectively include large-scale non-linear quantum fluctuations and its validity is discussed from the IR structure of quantum fluctuations in de Sitter spacetime.

Based on the stochastic picture of inflation, there appears the eternal phase in which inflation never terminates due to quantum fluctuations. The realization of the eternal phase depends on parameters of the inflaton potential, and in the eternal phase, the global structure of inflationary spacetime becomes inhomogeneous and very complicated.

Although the stochastic approach of inflation includes quantum effect as the short wavelength random noise, it does not take into account of quantum entanglement, which is a key property of quantum systems. The purpose of this research project is to investigate effect of quantum entanglement on the stochastic picture of inflation. For this purpose, we will employ a qubit model of inflation with quantum entanglement and investigate evolution of entanglement structure of this model. We will also investigate “information” of this model to understand emergence of “classical property” in the stochastic inflation.

## Study of quantum theory of gravity based on the Jackiw-Teitelboim gravity theory

Graduate School of Science and Engineering,

Saitama University

My main research area is superstring theory in elementary particle theory. Superstring theory is considered to be a leading candidate for the ultimate theory that can uniformly describe the four forces that exist in nature: electromagnetic force, weak force, strong force, and gravity. There are various research themes in string theory research, but I am interested in the duality (equivalence) between gauge theory and gravity (string) theory. This duality is realized by the gravitational (string) theory on the Anti-de Sitter (AdS) space and the field theory with conformal symmetry (Conformal Field Theory, CFT) equivalence. This duality was originally proposed by Maldacena. This equivalence is one of the realization examples of the holographic principle, but the basic mechanism of how this principle is realized has not yet been elucidated.

For this mechanism, I have applied mathematical and physical analysis based on the integrable structure that exists behind the duality, chaotic motion and turbulent behavior of strings, and chaotic scattering in gauge theory. We are conducting research from this perspective. It is a very important research topic to understand the relationship between quantum information theory, which is the subject of this research area, and these themes.

The subject of my publicly offered research is to analyze the Jackiw-Teitelboim gravity, a toy model in which dilaton is coupled to two-dimensional gravity, in order to clarify quantum information theory and the holographic principle. There are various choices for the dilaton potential, and our previous research has shown that the sinh-type potential leads to the well-known Liouville gravity. By promoting this research, I would like to explore the relationship with quantum information theory and the clues to clarify the fundamental degrees of freedom in quantum gravity.

## Theoretical modeling of optomechanical pendulum for testing quantumness of gravity

Department of Physics, Kyushu University

My research focuses on themes related to the verification of the quantum mechanical properties of gravity. This theme, which revisits the question of whether the gravity follows the framework of quantum mechanics or not, has a long history of deep consideration by Feynman and Penrose, and is a very interesting theme that is related to recent developments in quantum information and quantum technology. Quantum entanglement caused by gravitational interaction is one of the main interests, and I am also conducting theoretical research to clarify its significance for quantum gravity theory and whether it can be verified experimentally. We focus on how a quantum mechanical system interacts through gravity, but since gravity is very weak, it is necessary to realize the quantum state of a macroscopic object. Optomechanical systems play an important role in realizing the quantum states of such macroscopic objects. An optomechanical system is a mechanical system with one end of an optical cavity as a mirror oscillator that interacts with the cavity light. The Michelson interferometer, which combines them, is used in gravitational wave detection, and related research and technology are developing. In the proposal of my research, we are constructing a theoretical model of an optomechanical system with a suspended mirror oscillator called a beam model for the purpose of realizing the quantum state of macroscopic objects and verifying the quantum mechanical properties of gravity in the future. Based on the beam model, we plan to make theoretical predictions of the noise and dissipation to characterize optomechanical mirror oscillators, and of the quantum state when quantum control is performed by feedback control with continuous measurement.

## Mesoscopic picture of black holes from quantum information

Department of Physics, College of Humanities and Sciences,

Nihon University

The goal of this publicly offered research is to understand the quantum (microscopic) aspects of black holes and wormholes in terms of holographic principles and quantum information theory.

Black holes have been playing a crucial role to understand the quantum aspects of gravity. In particular, the recent discovery of microscopic wormholes (so-called replica wormholes) associated with black holes has made significant progress toward solving the famous black hole information loss puzzle. This progress is mainly based on the semi-classical (macroscopic) picture, while the quantum (microscopic) picture is still unclear and problematic.

I am now studying “What microscopic properties of the wormholes play an important role?” and “Is there any way to complement the relationship between the micro and macroscopic pictures of wormholes?”. These questions are directly related to the research of Group B.

More specifically, my current research plans are to (1) clarify the microscopic and macroscopic properties of candidate quantum states that describe the microstructure of wormholes in terms of quantum information theoretic quantities and (2) study the quantum operations that “decompose” macroscopic wormholes into microscopic ones and their effects on the wormhole geometries. Interestingly, these topics are closely related to the research of Group C and D, so I am very excited to join this Extreme Universe collaboration which will lead to new research directions. I am also interested in studying “(3) implications of such microscopic wormholes to physics such as symmetries”.

## Entanglement Witness in Quantum Frustrated Magnets

Okinawa Institute of Science and Technology Graduate University

Quantum entanglement has attracted a great deal of attention in recent years. In particular, theoretical studies using entanglement entropy (EE) have revealed that quantum entanglement is important for understanding unique and exotic states in quantum systems, such as quantum spin liquids and topological order. In addition, the concept of quantum entanglement has been shown to be intrinsically important for the development of new numerical methods, such as tensor network methods. On the other hand, it is known that it is challenging to measure EE directly experimentally, except for a small number of systems such as cold atoms or optical systems. Our study focuses on so-called entanglement witnesses (EWs) such as one tangle, concurrence, and quantum Fisher information. These can be written in terms of spin-spin correlation functions and dynamical spin structure factors, which are familiar in the field of magnetism, and are known to be accessible with some experimental techniques such as neutron scattering experiments [1-4]. We focus on quantum states such as quantum spin liquid states and random singlet states, which do not exhibit magnetic orders even at very low temperatures, and for which positive experimental evidence characterizing them is difficult to obtain. Our main purpose of this study is to clarify numerically whether the above EWs can characterize these unique quantum states. We also aim to establish theoretical guidelines that will lead to experimental detection and control methods of the exotic quantum states in candidate materials.

[1] L. Amico et al, Phys. Rev. A 69, 022304 (2004).

[2] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

[3] P. Hauke et al, Nat. Phys. 12, 778 (2016).

[4] P. Laurell, et al, Phys. Rev. Lett. 127, 037201 (2021).

## Study of novel quantum many-body phenomena originating from non-ergodicity in isolated quantum systems

Department of Physics, Tokyo University of Science

In this publicly offered research, I will study nonergodicity in isolated quantum systems.

The issue of thermalization in isolated quantum many-body systems has become a realistic problem thanks to the developments of various quantum simulators such as cold atoms or molecules in optical lattices, Rydberg atoms in optical tweezers, trapped ions, and superconducting qubits. According to previous studies, if a system satisfies the eigenstate thermalization hypothesis (ETH), the system will thermalize after a long-time evolution. However, to prove the ETH is an unsolved problem. It is also known that some systems do not satisfy the ETH. These systems are called nonergodic systems. Typical examples of nonergodic systems are quantum integrable systems, Anderson localized systems, and many-body localized systems. Recently, quantum many-body scarred systems and Hilbert space fragmented systems have been found as novel ergodicity-broken systems. Inspired by these findings, I will explore novel quantum-many-body phenomena originating from the nonergodicity in isolated quantum systems. In particular, I will study the Hilbert space fragmentation in momentum space, which is different from the previous works. If the Hilbert space fragmentation occurs in the momentum space, we can naively expect that persistent-current states emerge. I will study the nature of the persistent current induced by the Hilbert space fragmentation in momentum space. I also would like to find novel nonequilibrium quantum many-body phenomena and discover a new nonergodic mechanism. In addition to this research, I plan to develop and propose schemes for the experimental implementation of my results obtained in this project.

## Theoretical research on methods to measure quantum natures of evaporating primordial black holes by using future observations

IPNS, KEK

Inflation, which is believed to have occurred in the early universe, created quantum curvature (density) perturbations with a magnitude of approximately one part per hundred thousand on a large scale. At the same time, it also produces the cosmic gravitational wave background radiation. As temperature fluctuations caused by curvature perturbations were discovered by previous CMB observations, it is expected to observe B-mode polarization caused by gravitational waves in the future to verify inflation. Recently, the NANOGrav Collaboration reported the possibility of observing gravitational waves that have existed since the beginning of the universe: the NANOGrav experiment has observed significant modulation of radio waves in the nanohertz band over a 12.5 year period by correlating the signals of periodically emitted radio waves by multiple pulsars. If background gravitational waves exist, they stretch and contract space-time, thus modifying the periodicity of the radio waves of those precise pulsars. The observed value was more than seven orders of magnitude larger than the gravitational waves directly created by inflation in the early universe. On the other hand, if the observationally allowed curvature (density) fluctuations δ on small scales produced by inflation can be large (e.g.,δ~0.3. Then, it is possible that an extremely large number of gravitational waves are produced on small scales through second-order nonlinear effects. In our paper (Kohri and Terada, arXiv:2009.11853), we pointed out that this second-order gravitational wave background radiation can explain the observations. Furthermore, the above large density fluctuations are expected to collapse in the early universe to form a black hole of solar mass. This black hole is called a primordial black hole (PBH) to distinguish it from the one of astronomical origin. In fact, gravitational waves are also used as a method to verify this model. These PBHs formed a binary star in the early universe and are now merging. The LIGO-Virgo-Indigo-KAGRA collaboration may find the gravitational waves emitted at that time near future. Thus, the future gravitational wave experiments will soon be used to verify inflation in the early universe and discover PBHs.