# Publications

## Quantum expectation value estimation by computational basis sampling

Measuring expectation values of observables is an essential ingredient in variational quantum algorithms. A practical obstacle is the necessity of a large number of measurements for statistical convergence to meet requirements of precision, such as chemical accuracy in the application to quantum chemistry computations. Here we propose an algorithm to estimate the expectation value based on its approximate expression as a weighted sum of classically-tractable matrix elements with some modulation, where the weight and modulation factors are evaluated by sampling appropriately prepared quantum states in the computational basis on quantum computers. Our algorithm is expected to require fewer measurements than conventional methods for a required statistical precision of the expectation value when a target quantum state is concentrated in particular computational basis states. We provide numerical comparisons of our method with existing ones for measuring electronic ground state energies (expectation values of electronic Hamiltonians for the lowest-energy states) of various small molecules. Numerical results show that our method can reduce the numbers of measurements to obtain the ground state energies for a targeted precision by several orders of magnitudes for molecules whose ground states are concentrated. Our results provide another route to measure expectation values of observables, which could accelerate the variational quantum algorithms.

## Non-adiabatic Quantum Wavepacket Dynamics Simulation Based on Electronic Structure Calculations using the Variational Quantum Eigensolver

A non-adiabatic nuclear wavepacket dynamics simulation of the H_{2}O^{+} de-excitation process is performed based on electronic structure calculations using the variational quantum eigensolver. The adiabatic potential energy surfaces and non-adiabatic coupling vectors are computed with algorithms for noisy intermediate-scale quantum devices, and time propagation is simulated with conventional methods for classical computers. The results of non-adiabatic transition dynamics from the B^{~} state to A^{~ }state reproduce the trend reported in previous studies, which suggests that this quantum-classical hybrid scheme may be a useful application for noisy intermediate-scale quantum devices.

## Analytic energy gradient for state-averaged orbital-optimized variational quantum eigensolvers and its application to a photochemical reaction

Elucidating photochemical reactions is vital to understand various biochemical phenomena and develop functional materials such as artificial photosynthesis and organic solar cells, albeit its notorious difficulty by both experiments and theories. The best theoretical way so far to analyze photochemical reactions at the level of ab initio electronic structure is the state-averaged multi-configurational self-consistent field (SA-MCSCF) method. However, the exponential computational cost of classical computers with the increasing number of molecular orbitals hinders applications of SA-MCSCF for large systems we are interested in. Utilizing quantum computers was recently proposed as a promising approach to overcome such computational cost, dubbed as SA orbital-optimized variational quantum eigensolver (SA-OO-VQE). Here we extend a theory of SA-OO-VQE so that analytical gradients of energy can be evaluated by standard techniques that are feasible with near-term quantum computers. The analytical gradients, known only for the state-specific OO-VQE in previous studies, allow us to determine various characteristics of photochemical reactions such as the minimal energy (ME) points and the conical intersection (CI) points. We perform a proof-of-principle calculation of our methods by applying it to the photochemical *cis-trans* isomerization of 1,3,3,3-tetrafluoropropene. Numerical simulations of quantum circuits and measurements can correctly capture the photochemical reaction pathway of this model system, including the ME and CI points. Our results illustrate the possibility of leveraging quantum computers for studying photochemical reactions.

## Molecular Structure Optimization based on Electrons-Nuclei Quantum Dynamics Computation

A new concept of the molecular structure optimization method based on quantum dynamics computations is presented. Nuclei are treated as quantum mechanical particles, as are electrons, and the many-body wave function of the system is optimized by the imaginary time evolution method. A demonstration with a 2-dimensional H_{2}^{+ }molecule shows that the optimized nuclear positions can be specified with a small number of observations. This method is considered to be suitable for quantum computers, the development of which will realize its application as a powerful method.

## Deep variational quantum eigensolver for excited states and its application to quantum chemistry calculation of periodic materials

A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational Quantum Eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called "Deep VQE" which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the technique of coarse-graining [K. Fujii, et al, arXiv:2007.10917]. In this paper, we extend the original proposal of Deep VQE to obtain the excited states and apply it to quantum chemistry calculation of a periodic material, which is one of the most impactful applications of the VQE. We first propose a modified scheme to construct quantum states for coarse-graining in Deep VQE to obtain the excited states. We also present a method to avoid a problem of meaningless eigenvalues in the original Deep VQE without restricting variational quantum states. Finally, we classically simulate our modified Deep VQE for quantum chemistry calculation of a periodic hydrogen chain as a typical periodic material. Our method reproduces the ground-state energy and the first-excited-state energy with the errors up to O(1)% despite the decrease in the number of qubits required for the calculation by two or four compared with the naive VQE. Our result will serve as a beacon for tackling quantum chemistry problems with classically-intractable sizes by smaller quantum devices in the near future.

## Variational quantum simulations of stochastic differential equations

Stochastic differential equations (SDE), which models uncertain phenomena as the time evolution of random variables, are exploited in various fields of natural and social sciences such as finance. Since SDEs rarely admit analytical solutions and must usually be solved numerically with huge classical-computational resources in practical applications, there is strong motivation to use quantum computation to accelerate the calculation. Here, we propose a quantum-classical hybrid algorithm that solves SDEs based on variational quantum simulation (VQS). We first approximate the target SDE by a trinomial tree structure with discretization and then formulate it as the time-evolution of a quantum state embedding the probability distributions of the SDE variables. We embed the probability distribution directly in the amplitudes of the quantum state while the previous studies did the square-root of the probability distribution in the amplitudes. Our embedding enables us to construct simple quantum circuits that simulate the time-evolution of the state for general SDEs. We also develop a scheme to compute the expectation values of the SDE variables and discuss whether our scheme can achieve quantum speed-up for the expectation-value evaluations of the SDE variables. Finally, we numerically validate our algorithm by simulating several types of stochastic processes. Our proposal provides a new direction for simulating SDEs on quantum computers.

## Qulacs: a fast and versatile quantum circuit simulator for research purpose

We introduce Qulacs, a fast simulator for quantum circuits intended for research purpose. To explore the possibilities of a near-term intermediate-scale quantum algorithm and long-term fault-tolerant quantum computing, a fast and versatile quantum circuit simulator is needed. Herein we show the main concepts of Qulacs, explain how to use its features via examples, and demonstrate its performance with numerical benchmarks.

## Quadratic Clifford expansion for efficient benchmarking and initialization of variational quantum algorithms

Variational quantum algorithms are appealing applications of near-term quantum computers. However, there are two major issues to be solved, that is, we need an efficient initialization strategy for parametrized quantum circuit and to know the limitation of the algorithms by benchmarking it on large scale problems. Here, we propose a perturbative approach for efficient benchmarking and initialization of variational quantum algorithms. The proposed technique performs perturbative expansion of a circuit consisting of Clifford and Pauli rotation gates, which enables us to determine approximate optimal parameters and an optimal value of a cost function simultaneously. The classical simulatability of Clifford circuits is utilized to achieve this goal. Our method can be applied to a wide family of parameterized quantum circuits, which consist of Clifford gates and single-qubit rotation gates. Since the introduced technique provides us a perturbative energy of a quantum system when applied to the variational quantum eigensolver (VQE), our proposal can also be viewed as a quantum-inspired classical method for perturbative energy calculation. As the first application of the method, we perform a benchmark of so-called hardware-efficient-type ansatzes when they are applied to the VQE of one-dimensional hydrogen chains up to H24, which corresponds to 48-qubit system, using a standard workstation.

## Penalty methods for variational quantum eigensolver

The variational quantum eigensolver (VQE) is a promising algorithm to compute eigenstates and eigenenergies of a given quantum system that can be performed on a near-term quantum computer. Obtaining eigenstates and eigenenergies in a specific symmetry sector of the system is often necessary for practical applications of the VQE in various fields ranging from high energy physics to quantum chemistry. It is common to add a penalty term in the cost function of the VQE to calculate such a symmetry-resolving energy spectrum, but systematic analysis on the effect of the penalty term has been lacking, and the use of the penalty term in the VQE has not been justified rigorously. In this work, we investigate two major types of penalty terms for the VQE that were proposed in the previous studies. We show a penalty term in one of the two types works properly in that eigenstates obtained by the VQE with the penalty term reside in the desired symmetry sector. We further give a convenient formula to determine the magnitude of the penalty term, which may lead to the faster convergence of the VQE. Meanwhile, we prove that the other type of penalty terms does not work for obtaining the target state with the desired symmetry in a rigorous sense and even gives completely wrong results in some cases. We finally provide numerical simulations to validate our analysis. Our results apply to general quantum systems and lay the theoretical foundation for the use of the VQE with the penalty terms to obtain the symmetry-resolving energy spectrum of the system, which fuels the application of a near-term quantum computer.

## Variational Quantum Simulation for Periodic Materials

We present a quantum-classical hybrid algorithm that simulates electronic structures of periodic systems such as ground states and quasiparticle band structures. By extending the unitary coupled cluster (UCC) theory to describe crystals in arbitrary dimensions, we numerically demonstrate in hydrogen chain that the UCC ansatz implemented on a quantum circuit can be successfully optimized with a small deviation from the exact diagonalization over the entire range of the potential energy curves. Furthermore, with the aid of the quantum subspace expansion method, in which we truncate the Hilbert space within the linear response regime from the ground state, the quasiparticle band structure is computed as charged excited states. Our work establishes a powerful interface between the rapidly developing quantum technology and modern material science.