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.

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.

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.

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.

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.

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.

Quantum simulation is one of the key applications of quantum computing, which can accelerate research and development in chemistry, material science, etc. Here, we propose an efficient method to simulate the time evolution driven by a static Hamiltonian, named subspace variational quantum simulator (SVQS). SVQS employs the subspace-search variational eigensolver (SSVQE) to find a low-energy subspace and further extends it to simulate dynamics within the low-energy subspace. More precisely, using a parameterized quantum circuit, the low-energy subspace of interest is encoded into a computational subspace spanned by a set of computational basis, where information processing can be easily done. After the information processing, the computational subspace is decoded to the original low-energy subspace. This allows us to simulate the dynamics of low-energy subspace with lower overhead compared to existing schemes. While the dimension is restricted for feasibility on near-term quantum devices, the idea is similar to quantum phase estimation and its applications such as quantum linear system solver and quantum metropolis sampling. Because of this simplicity, we can successfully demonstrate the proposed method on the actual quantum device using Regetti Quantum Cloud Service. Furthermore, we propose a variational initial state preparation for SVQS, where the initial states are searched from the simulatable eigensubspace. Finally, we demonstrate SVQS on Rigetti Quantum Cloud Service.

We propose a sequential minimal optimization method for quantum-classical hybrid algorithms, which converges faster, robust against statistical error, and hyperparameter-free. Specifically, the optimization problem of the parameterized quantum circuits is divided into solvable subproblems by considering only a subset of the parameters. In fact, if we choose a single parameter, the cost function becomes a simple sine curve with period 2π, and hence we can exactly minimize with respect to the chosen parameter. Furthermore, even in general cases, the cost function is given by a simple sum of trigonometric functions with certain periods and hence can be minimized by using a classical computer. By repeatedly performing this procedure, we can optimize the parameterized quantum circuits so that the cost function becomes as small as possible. We perform numerical simulations and compare the proposed method with existing gradient-free and gradient-based optimization algorithms. We find that the proposed method substantially outperforms the existing optimization algorithms and converges to a solution almost independent of the initial choice of the parameters. This accelerates almost all quantum-classical hybrid algorithms readily and would be a key tool for harnessing near-term quantum devices.