The Green's function has been an indispensable tool to study many-body systems that remain one of the biggest challenges in modern quantum physics for decades. The complicated calculation of Green's function impedes the research of many-body systems. The appearance of the noisy intermediate-scale quantum devices and quantum-classical hybrid algorithm inspire a new method to calculate Green's function. Here we design a programmable quantum circuit for photons with utilizing the polarization and the path degrees of freedom to construct a highly-precise variational quantum state of a photon, and first report the experimental realization for calculating the Green's function of the two-site Fermionic Hubbard model, a prototypical model for strongly-correlated materials, in photonic systems. We run the variational quantum eigensolver to obtain the ground state and excited states of the model, and then evaluate the transition amplitudes among the eigenstates. The experimental results present the spectral function of Green's function, which agrees well with the exact results. Our demonstration provides the further possibility of the photonic system in quantum simulation and applications in solving complicated problems in many-body systems, biological science, and so on.

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.

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.

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.

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.

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.

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.

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.

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.

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.