Research

One of the factors that enabled machine learning techniques based on the deep-layered neural networks (NNs) is their representation power as non-linear function.

The probability distributions in physics are naturally expected to be different from those of datasets used in machine learning. Whatsoever, recent research suggests that neural networks are capable of expressing the physical states.

My goal is to extend the frontier of such state representations and to establish new efficient numerical methods applicable to many-body systems.

The third law of thermodynamics states that arbitrary physical system is prohibited to reach the absolute zero temperature. It is therefore indispensable to advance theoretical and numerical methods to construct accurate predictions and verifications of physical phenomena that are to be observed. However, finite-temperature studies in quantum many-body systems are even more puzzling than ground states; no effective method has yet been established for exotic realm such as strongly-correlated two-dimensional systems.

In this work, we propose two new methods for finite temperature calculations using a class of neural networks called Deep Boltzmann Machines (DBMs). In the first method, we discuss how to construct a DBM that exactly reproduces the imaginary time evolution of a quantum many-body system, fully taking the advantage of the quantum-classical correspondence. In the second method, we proposed an algorithm for efficiently approximating the finite-temperature state, by iteratively changing the parameters of the DBM based on a variational principle. With only polynomial computational cost, both proposed methods can handle a very wide temperature range from infinite to very low temperatures. The remarkable demonstration of the latter method, in particular, shows that we can accurately calculate the physical properties of frustrated spin systems.

Establishing a predictive ab initio method for crystalline systems with periodic atom configuration is one of the most fundamental goals in condensed matter physics and computational materials science. Despite the intensive effort over decades, scientists have not still yet succeeded in developing decisive methods that is valid for both weakly and strongly correlated regimes. In variational simulations which are indispensable for wavefunction-based approaches, the bottleneck is the lack of representative power in the existing quantum state ansatze.

Here, we propose to overcome this problem by employing neural networks, which are a highly expressive non-linear function known in the field of machine learning. It has been pointed out that even the networks with simlest structure, i.e., the Restricted Boltzmann Machine (RBM), is capable of capturing quantum entanglement that scales extensively. We show that
1. thermodynamic limit in weakly correlated system
2. strongly correlated regime where gold-standard method (CCSD(T)) breaks down
3. quasiparticle band spectrum
can be simulated up to chemical accuracy. For the ground state, we have demonstrated that the RBM state correctly describes the electronic structures in d(=1, 2, 3)-dimensional crystals, including real solids such as graphene and LiH. In particular, as shown in the bottom panel of Figure, the RBM state accurately represents the quantum state even in regions where the cluster coupling theory fails due to strong electronic correlation. Furthermore, we have shown that a technique called quantum subspace expansion can be used to calculate quasiparticle bands of trans-polyacetylene. We expect that further development of this study will lead to a better understanding of systems such as surface reactions and superconductors.

Classification of quantum phases in topological superconductor (Phys. Rev. B 97, 205110 (2018), Editor's suggestion)

Insulators and superconductors with spectral gap may exhibit nontrivial topological invariant corresponding to the discrete symmetry of the system. Such property has attracted much interest due to 1. robustness to disorders, 2. detectability in real materials via edge states, 3. relationship with the quantization of transport coefficients, and so on.

While the formulae of the topological invariant is well-established under the presence of translational symmetry, our knowledge is limited when the symmetry is absent due to disorder such as inhomogeneous potential.

In this work, we propose a method that "extrapolates" phase diagrams. In other words, we develop a scheme that apply neural network to learn the quasiparticle distribution, or the "image," in the clean limit and predict the phase in the disordered system. While the prediction for single-shot images under disorder result in a complete mess, the classification for averaged quasiparticle distribution is in good agreement with another independent method. We argue that this is due to the statistical recovery of translation symmetry. Such a method that detects the edge localization is applicable to a system in arbitrary dimension, symmetry, or geometry.

Variational Quantum Algorithm for Non-equilibrium Steady State (arXiv:1908.09836)

The effect of the dissipation stemming from energy exchange with the external environment is crucial in non-equilibrium phenomena such as electric transport. Although the understanding of the non-equilibrium steady state that is reached after a sufficient long time is essential, method to simulate using the NISQ devices has remained an open question.

In this collaborative work with QunaSys Inc., we proposed an algorithm named "the dissipative-system Variational Quantum Eigensolver (dVQE)" to compute the non-equilibrium steady state. The variational quantum algorithm for ground-state-search of closed systems, the VQE, has been extended and thus become capable of open quantum systems. Also, we demonstrated our results of numerical simulation and quantum simulation which is performed on the NISQ device provided by the Rigetti Quantum Cloud Service.

Ab initio calculation of electronic structures in periodic systems has been recognized as one of the most significant problems since the completion of quantum mechanics in the 20th century. The notorious difficulty lies in solving the many-body Schroedinger equation. While a great deal of efforts have been made in the context of condensed matter community, in which the periodic system is simplified to capture the essence of the physical and chemical property, the methods are not directly transferrable.

In this collaborative work with QunaSys Inc. and JSR Corporation, we proposed a classical-quantum hybrid method that simulates ground states and quasiparticle band structures of periodic systems. We demonstrated that, using the unitary coupled-cluster ansatz, ground state of a linear hydrogen chain can be calculated even when conventional methods such as the golden-standard CCSD(T) breaks down. Also, we showed that quasiparticle bands can be computed as excited states by using the quantum subspace expansion method, in which the Hamiltonian is diagonalized in a truncated subspace that is related with the ground state via single particle excitation.

The key to accomplish quantum advantage using quantum devices without full tolerance is to control the effect of quantum noise that rise from e.g., finite lifetime of qubits or imperfect gate operations. This has been recognized as one of the most important problems. In particular, there has been some proposal of algorithms that compensate for unwanted errors in exchange with additional classical resource. Among such techniques, or the quantum error mitigation methods, there is a surging interest in methods that are error-agnoistic; it would boost the power of quantum device if we could suppress the effect of error without any characterization. However, there is no efficient algorithm that can handle both coherent and stochastic errors at once.

In this collaborative work with NTT Inc. and AIST, we have proposed a novel error-agnostic method named the "Generalized Quantum Subspace Expansion (GSE) method." A quantum subspace is an effective variational ansatz created by interfering quantum states via post-processing. In our work, we have significantly extended the quantum subspace by employing multiple quantum states, and also have shown that the existing error-agnostic methods can be unified under our protocol. The high practicality of our GSE methods has been demonstrated under two specific cases where the base of the subspace consists of 1. the power of identical noisy state and 2. non-identical quantum states corresponding to different noise levels.

Construction of Perfect Scar State using Onsager Algebra (Phys. Rev. Lett. 124, 180604 (2020)., Editor's suggestion)

The condition of thermalization (or equilibration) in nonintegrable isolated quantum many-body systems has been recognized as a significant issue that bridges the quantum mechanics for the microscopic world and the statistical mechanics for the macroscopic world. There has been interest, in particular, in physical systems with only a small peculiar fraction of the excited states that do not thermalizes -- the quantum many-body scar states.

In our work, we explicitly constructed a family of "perfect scar states" whose orbital is completely trapped in a subspace of the Hilbert space to show periodic dynamics and hence never undergo thermalization. Moreover, this is the first explicitly constructed example of scar states that are robust to perturbation without translational symmetry.

Approximate descriptions of gapped subsystems via appropriate restriction of the Hilbert space have been considered ubiquitously in various fields including condensed-matter physics, quantum physics, quantum chemistry, and particle physics. While the effect of perturbation on the static properties such as the eigenenergy can be well-captured via the well-known perturbation theory, the quantitative argument on the error of the dynamics induced by constraining the Hilbert space remains lacking.

In our work, we provided an error bound on the observable-based dynamics under perturbation that introduces transition between the gapped subspaces. Our main result is the rigorous justification of the linear growth of the error given a sufficiently large gap. Furthermore, we found and numerically demonstrated that the error scales as

• • • O(t^{d+1}) in a d-dimensional quantum many-body isolated system with frustration-free non-perturbative Hamiltonian

    • O(t) in Markovian open quantum system if the relaxation time is sufficiently large

Intrinsic thermal Hall effect in 3D chiral superconductor with nodal gaps (J. Phys. Soc. Jpn. 87, 124602 (2018). arXiv:1804.03843)

In chiral superconductors, Cooper pairs have nonzero angular momenta due to spontaneous breaking of time reversal symmetry. Among various probes to detect such exotic coexistence of the superconductivity and finite magnetization, spontaneous thermal Hall effect is expected to play a crucial role.

The candidates of chiral superconductivity have "nodes," which refer to the gap-closing point or lines in the quasiparticle spectrum. While a node change the behavior of the low-lying states drastically, its effect on the thermal Hall effect had not been discussed thoroughly.

In our work, we showed that the low-lying states would give a higher-order correction to the thermal Hall efficient with respect to the temperature. This order is determined from the energy dispersion in the vicinity of the nodes. Hence, combined with the knowledge of the lowest order, which directly reflects the topological property, we may impose a strong restriction to the symmetry of the pairing.