# Publications

## 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.

## Deep Variational Quantum Eigensolver: a divide-and-conquer method for solving a larger problem with smaller size quantum computers

We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers. Specifically, we concatenate variational quantum eigensolver (VQE) with reducing the dimensions of the system, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis. Then the effective Hamiltonian is further solved by VQE, which we call *deep VQE*. Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intra-subsystem interactions and weak inter-subsystem interactions, or strongly correlated spin models on large regular lattices. As proof-of-principle numerical demonstrations, we use the proposed method for Heisenberg anti-ferromagnetic models, including one-dimensionally coupled 12-qubit Heisenberg anti-ferromagnetic models on Kagome lattices. The largest problem size of 48 qubits is solved by simulating 12-qubit quantum computers. The proposed scheme enables us to handle the problems of >1000 qubits by concatenating VQE with a few tens of qubits. Deep VQE will provide us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.

## Optimal resource cost for error mitigation

We provide a general methodology for evaluating the optimal resource cost for an error mitigation employing methods developed in resource theories. We consider the probabilistic error cancellation as an error mitigation technique and show that the optimal sampling cost realizable using the full expressibility of near-term devices is related to a resource quantifier equipped with a framework in which noisy implementable operations are considered as the free resource, allowing us to obtain its universal bounds. As applications, we show that the cost for mitigating the depolarizing noise presented in [Temme, Bravyi, and Gambetta, Phys. Rev. Lett. 119, 180509 (2017)] is optimal, and extend the analysis to several other classes of noise model, as well as provide generic bounds applicable to general noise channels given in a certain form. Our results not only provide insights into the potential and limitations on feasible error mitigation on near-term devices but also display an application of resource theories as a useful theoretical toolkit.

## Calculating nonadiabatic couplings and Berry's phase by variational quantum eigensolvers

Investigating systems in quantum chemistry and quantum many-body physics with the variational quantum eigensolver (VQE) is one of the most promising applications of forthcoming near-term quantum computers. The VQE is a variational algorithm for finding eigenenergies and eigenstates of such quantum systems. In this paper, we propose VQE-based methods to calculate the nonadiabatic couplings of molecules in quantum chemical systems and Berry's phase in quantum many-body systems. Both quantities play an important role to understand various properties of a system (e.g., nonadiabatic dynamics and topological phase of matter) and are related to derivatives of eigenstates with respect to external parameters of the system. Here, we show that the evaluation of inner products between the eigenstate and the derivative of the same/different eigenstate reduces to the evaluation of expectation values of observables, and we propose quantum circuits and classical post-processings to calculate the nonadiabatic couplings and Berry's phase. In addition, we demonstrate our methods by numerical simulation of the nonadiabatic coupling of the hydrogen molecule and Berry's phase of a spin-1/2 model. Our proposal widens the applicability of the VQE and the possibility of near-term quantum computers to study molecules and quantum many-body systems.

## Predicting excited states from ground state wavefunction by supervised quantum machine learning

Excited states of molecules lie in the heart of photochemistry and chemical reactions. The recent development in quantum computational chemistry leads to inventions of a variety of algorithms which calculate the excited states of molecules on near-term quantum computers, but they require more computational burdens than the algorithms for the ground states. In this study, we propose a scheme of supervised quantum machine learning which predicts excited state properties of molecules only from its ground state wavefunction and results in reducing the computational cost for calculating the excited states. Our model is comprised of a quantum reservoir and a classical machine learning unit which processes the results of measurements of single-qubit Pauli operators. The quantum reservoir effectively transforms the single-qubit operators into complicated multi-qubit ones which contain essential information of the system, so that the classical machine learning unit may decode them appropriately. The number of runs for quantum computers is saved by training only the classical machine learning unit and the whole model requires modest resources of quantum hardwares which may be implemented in current experiments. We illustrate the predictive ability of our model by numerical simulations for small molecules with and without including noise inevitable in near-term quantum computers. The results show that our scheme well reproduces the first and second excitation energies as well as the transition dipole moment between the ground states and excited states only from the ground state as an input. Our contribution will enhance applications of quantum computers in the study of quantum chemistry and quantum materials.

## Calculating transition amplitudes by variational quantum deflation

Variational quantum eigensolver (VQE) is an appealing candidate for the application of near-term quantum computers. A technique introduced in [Higgot et al., Quantum 3, 156 (2019)], which is named variational quantum deflation (VQD), has extended the ability of the VQE framework for finding excited states of a Hamiltonian. However, no method to evaluate transition amplitudes between the eigenstates found by the VQD without using any costly Hadamard-test-like circuit has been proposed despite its importance for computing properties of the system such as oscillator strengths of molecules. Here we propose a method to evaluate transition amplitudes between the eigenstates obtained by the VQD avoiding any Hadamard-test-like circuit. Our method relies only on the ability to estimate overlap between two states, so it does not restrict to the VQD eigenstates and applies for general situations. To support the significance of our method, we provide a comprehensive comparison of three previously proposed methods to find excited states with numerical simulation of three molecules (lithium hydride, diazene, and azobenzene) in a noiseless situation and find that the VQD method exhibits the best performance among the three methods. Finally, we demonstrate the validity of our method by calculating the oscillator strength of lithium hydride in numerical simulations with shot noise. Our results illustrate the superiority of the VQD to find excited states and widen its applicability to various quantum systems.

## Orbital optimized unitary coupled cluster theory for quantum computer

We propose an orbital optimized method for unitary coupled cluster theory (OO-UCC) within the variational quantum eigensolver (VQE) framework for quantum computers. OO-UCC variationally determines the coupled cluster amplitudes and also molecular orbital coefficients. Owing to its fully variational nature, first-order properties are readily available. This feature allows the optimization of molecular structures in VQE without solving any additional equations. Furthermore, the method requires smaller active space and shallower quantum circuit than UCC to achieve the same accuracy. We present numerical examples of OO-UCC using quantum simulators, which include the geometry optimization of the water molecule.

## Calculation of the Green's function on near-term quantum computers

The Green's function plays a crucial role when studying the nature of quantum many-body systems, especially strongly-correlated systems. Although the development of quantum computers in the near future may enable us to compute energy spectra of classically-intractable systems, methods to simulate the Green's function with near-term quantum algorithms have not been proposed yet. Here, we propose two methods to calculate the Green's function of a given Hamiltonian on near-term quantum computers. The first one makes use of a variational dynamics simulation of quantum systems and computes the dynamics of the Green's function in real time directly. The second one utilizes the Lehmann representation of the Green's function and a method which calculates excited states of the Hamiltonian. Both methods require shallow quantum circuits and are compatible with near-term quantum computers. We numerically simulated the Green's function of the Fermi-Hubbard model and demonstrated the validity of our proposals.

## Variational Quantum Algorithm for Non-equilibrium Steady States

We propose a quantum-classical hybrid algorithm to simulate the non-equilibrium steady state of an open quantum many-body system, named the dissipative-system Variational Quantum Eigensolver (dVQE). To employ the variational optimization technique for a unitary quantum circuit, we map a mixed state into a pure state with a doubled number of qubits and design the unitary quantum circuit to fulfill the requirements for a density matrix. This allows us to define a cost function that consists of the time evolution generator of the quantum master equation. Evaluation of physical observables is, in turn, carried out by a quantum circuit with the original number of qubits. We demonstrate our dVQE scheme by both numerical simulation on a classical computer and actual quantum simulation that makes use of the device provided in Rigetti Quantum Cloud Service.