We proposed near-term quantum algorithms for calculating the Green’s function.

Endo (intern), Kurata (intern), and Nakagawa from QunaSys Inc. proposed a near-term quantum algorithm for computing the Green’s function, which is crucial for analyzing properties of molecules and materials.  Now you can read the preprint of the paper on arXiv.

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



Near-term quantum computers with hundreds to thousands of qubits which are not fault-tolerant are approaching realization. Such computers are called noisy intermediate-scale quantum (NISQ) devices, and application of them to calculations in quantum chemistry and condensed matter physics is expected. In particular, the variational quantum eigensolver (VQE) algorithm which can find energy spectrum of molecules and materials is the most featured algorithm on NISQ devices.


Meanwhile, another important physical quantity other than energy spectrum has left a bit disregard in the recent development of NISQ algorithms: the Green’s function. It tells us crucial information on properties of molecules and materials, and allows us to calculate their responses to external field (e.g. electric resistivity). Although its ubiquitous character in the theory of computational quantum chemistry and condensed matter physics, no algorithm for evaluating the Green’ s function on NISQ devices has yet been proposed. 

Methods & Results 

Endo (intern), Kurat a (intern), and Nakagawa from QunaSys Inc. proposed two methods to compute the Green’s function efficiently on NISQ devices. One of the methods is based on the variational quantum simulation (VQS) algorithm which calculates the time evolution of quantum systems on NISQ devices. We extend the original VQS algorithm and make a quantum circuit for evaluating the Green's function drastically simple. The other method takes advantage of the previous VQE algorithms that compute energy spectrum and transition amplitude, and calculates the Lehmann representation of the Green’s function. Both methods require shallow quantum circuits compatible with NISQ devices. We perform numerical simulation of our proposed methods by using fast quantum simulator Qulacs, and successfully reproduce the Green’s function of the Hubbard model, a prototypical model of electrons in solids.


It is expected that our new quantum algorithms will significantly broaden the possibility to utilize NISQ devices in chemistry and material research, because they allow the computation of the Green’s function for large systems which could not be tackled with classical computers.

We published an algorithm to calculate the non-equilibrium steady states using a quantum computer.

Yoshioka, Nakagawa, and Mitarai from QunaSys Inc., and our advisor, Prof. Fujii, published a paper (preprint) that proposed an algorithm named “dissipative-system Variational Quantum Eigensolver (dVQE)” to simulate the non-equilibrium steady state on a quantum computer.

Variational Quantum Algorithm for Non-equilibrium Steady States


The recent technological developments in quantum technology have now reached a stage to realize Noisy Intermediate-Scale Quantum (NISQ) devices with tens to hundreds of qubits that are not fault-tolerant. In particular, the quantum algorithm named the variational quantum eigensolver (VQE) is expected to enable larger-scale calculations in various fields including quantum chemistry, condensed matter physics and material science.


The targets of calculation, such as the molecules and materials, are investigated mainly under two setups: the isolated system which is decoupled from others and the open system which exchanges energy with its external environment. The effect of such energy dissipation, or non-equilibrium phenomena, could be observed quite ubiquitously in, e.g., electric transport. Despite its significance in industrial application including device design, no method has been proposed to compute the non-equilibrium steady state using the NISQ devices.

Our Method and Result

Yoshioka, Nakagawa, and Mitarai from QunaSys Inc., and our advisor, Prof. Fujii, proposed an algorithm named “the dissipative-system Variational Quantum Eigensolver (dVQE)” to simulate the non-equilibrium steady state on a quantum computer. First, we apply an mapping from the open system to an appropriate equivalent system in which the VQE can be applied. The structure of the variational quantum circuit is restricted so that the solution is assured to be a valid physical state. Secondly, the VQE is executed to optimize the parameters of the quantum circuit. A measurement scheme for the physical observables that is far more efficient than a naive approach is also proposed. Finally, our algorithm has been demonstrated by both numerical simulations on a classical computer and also actual quantum simulations that are performed on the NISQ device provided by the Rigetti Quantum Cloud Service.

Future Prospects

The dVQE algorithm opens up a path to simulate microscopic systems under more realistic setups by considering the effect of the dissipation. This enables us to carry out calculations that take the environmental effect from, e.g., the solvent, electric voltage, or heat bath into account. We have established a method to compute the non-equilibrium steady state using the NISQ devices, and thus expect to invoke further research that applies quantum computers for industrial application.

Quantaggle: Your Platform for Quantum Algorithm

We launched quantum algorithm competition website, Quantaggle.

In Quantaggle, you can compare the performance of various quantum algorithms/methods of the specific problem in the competition. You are also welcome to join it to become an inventor of the world's state-of-the-art method to facilitate quantum computers!

Some of our benchmark results of NISQ algorithm are also shown.

We continue focusing on developing practical applications of quantum computers to solve problems in the real world.