# Publications

# Theory of analytical energy derivatives for the variational quantum eigensolver

**by K.Mitarai, Y.Nakagawa, W.Mizukami**

The variational quantum eigensolver (VQE) and its variants, which is a method for finding eigenstates and eigenenergies of a given Hamiltonian, are appealing applications of near-term quantum computers. Although the eigenenergies are certainly important quantities which determines properties of a given system, their derivatives with respect to parameters of the system, such as positions of nuclei if we target a quantum chemistry problem, are also crucial to analyze the system. Here, we describe methods to evaluate analytical derivatives of the eigenenergy of a given Hamiltonian, including the excited state energy as well as the ground state energy, with respect to the system parameters in the framework of the VQE. We give explicit, low-depth quantum circuits which can measure essential quantities to evaluate energy derivatives, incorporating with proof-of-principle numerical simulations. This work extends the theory of the variational quantum eigensolver, by enabling it to measure more physical properties of a quantum system than before and to perform the geometry optimization of a molecule.

# Subspace Variational Quantum Simulator

**by K.Heya, K.Nakanishi, K.Mitarai and K.Fujii**

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.

# Subspace-search variational quantum eigensolver for excited states

**by K.M Nakanishi, K. Mitarai and K. Fuji**

The variational quantum eigensolver (VQE), a variational algorithm to obtain an approximated ground state of a given Hamiltonian, is an appealing application of near-term quantum computers. The original work [Peruzzo et al.; \textit{Nat. Commun.}; \textbf{5}, 4213 (2014)] focused only on finding a ground state, whereas the excited states can also induce interesting phenomena in molecules and materials. Calculating excited states is, in general, a more difficult task than finding ground states for classical computers. To extend the framework to excited states, we here propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE). This algorithm searches a low energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of output states. The k-th excited state is obtained as the highest energy state in the low energy subspace. The proposed algorithm consists only of two parameter optimization procedures and does not employ any ancilla qubits. The disuse of the ancilla qubits is a great improvement from the existing proposals for excited states, which have utilized the swap test, making our proposal a truly near-term quantum algorithm. We further generalize the SSVQE to obtain all excited states up to the k-th by only single optimization procedure. From numerical simulations, we verify the proposed algorithms. This work greatly extends the applicable domain of the VQE to excited states and their related properties like a transition amplitude without sacrificing any feasibility of it.

# Generalization of the output of variational quantum eigensolver by parameter interpolation with low-depth ansatz

**by K. Mitarai, T. Yan and K. Fuji**

The variational quantum eigensolver (VQE) is an attracting possible application of near-term quantum computers. Originally, the aim of the VQE is to find a ground state for a given specific Hamiltonian. It is achieved by minimizing the expectation value of the Hamiltonian with respect to an ansatz state by tuning parameters θ on a quantum circuit which constructs the ansatz. Here we consider an extended problem of the VQE, namely, our objective in this work is to “generalize” the optimized output of the VQE just like machine learning. We aim to find ground states for a given set of Hamiltonians {H(x)}, where x is a parameter which specifies the quantum system under consideration, such as geometries of atoms of a molecule. Our approach is to train the circuit on the small number of x’s. Specifically, we employ the interpolation of the optimal circuit parameter determined at different x’s, assuming that the circuit parameter θ has simple dependency on a hidden parameter x as θ(x). We show by numerical simulations that, using an ansatz which we call the Hamiltonian-alternating ansatz, the optimal circuit parameters can be interpolated to give near-optimal ground states in between the trained x’s. The proposed method can greatly reduce, on a rough estimation by a few orders of magnitude, the time required to obtain ground states for different Hamiltonians by the VQE. Once generalized, the ansatz circuit can predict the ground state without optimizing the circuit parameter θ in a certain range of x.

# Quantum Circuit Learning

**by K. Mitarai, M. Negoro, M. Kitagawa and K. Fujii**

We propose a classical-quantum hybrid algorithm for machine learning on near-term quantum processors, which we call quantum circuit learning. A quantum circuit driven by our framework learns a given task by tuning parameters implemented on it. The iterative optimization of the parameters allows us to circumvent the high-depth circuit. Theoretical investigation shows that a quantum circuit can approximate nonlinear functions, which is further confirmed by numerical simulations. Hybridizing a low-depth quantum circuit and a classical computer for machine learning, the proposed framework paves the way toward applications of near-term quantum devices for quantum machine learning.

# Boosting computational power through spatial multiplexing in quantum reservoir computing

**by K. Nakajima, K. Fujii, M. Negoro, K. Mitarai and M. Kitagawa**

Quantum reservoir computing provides a framework for exploiting the natural dynamics of quantum systems as a computational resource. It can implement real-time signal processing and solve temporal machine learning problems in general, which requires memory and nonlinear mapping of the recent input stream using the quantum dynamics in computational supremacy region, where the classical simulation of the system is intractable. A nuclear magnetic resonance spin-ensemble system is one of the realistic candidates for such physical implementations, which is currently available in laboratories. In this paper, considering these realistic experimental constraints for implementing the framework, we introduce a scheme, which we call a spatial multiplexing technique, to effectively boost the computational power of the platform. This technique exploits disjoint dynamics, which originate from multiple different quantum systems driven by common input streams in parallel. Accordingly, unlike designing a single large quantum system to increase the number of qubits for computational nodes, it is possible to prepare a huge number of qubits from multiple but small quantum systems, which are operationally easy to handle in laboratory experiments. We numerically demonstrate the effectiveness of the technique using several benchmark tasks and quantitatively investigate its specifications, range of validity, and limitations in detail.

# Harnessing disordered ensemble quantum dynamics for machine learning

**by K. Fuji and K. Nakajima**

The quantum computer has an amazing potential of fast information processing. However, the realization of a digital quantum computer is still a challenging problem requiring highly accurate controls and key application strategies. Here we propose a platform, quantum reservoir computing, to solve these issues successfully by exploiting the natural quantum dynamics of ensemble systems, which are ubiquitous in laboratories nowadays, for machine learning. This framework enables ensemble quantum systems to universally emulate nonlinear dynamical systems including classical chaos. A number of numerical experiments show that quantum systems consisting of 5–7 qubits possess computational capabilities comparable to conventional recurrent neural networks of 100–500 nodes. This discovery opens up a paradigm for information processing with artificial intelligence powered by quantum physics.