https://arxiv.org/abs/2007.10917

BACKGROUND

Noisy Intermediate-Scale Quantum (NISQ) devices, or analog quantum computers with noise, have recently been developed and shown to outperform classical computers on a certain computational task (Nature 574, 505–510 (2019). Although this "quantum supremacy" has been demonstrated for the special computational problem of little practical significance, it is expected that quantum computers will achieve a "quantum advantage", which means that they outperform classical computers for practical problems, in the near future.

PROBLEMS

However, in order to solve problems of practical sizes in condensed matter physics and quantum chemistry, where the "quantum advantage" of NISQ devices is expected, the size (number of qubits) and accuracy of the currently realized NISQ devices are not sufficient yet. Therefore, there is a strong need for methods to solve large size problems using small size quantum computers.

METHOD

In the field of quantum chemistry computation using classical computers, there are plenty of methods based on the "divide-and-conquer" approach which enables us to treat large systems with small computational resources. In those methods, systems to be treated are divided into smaller subsystems, and calculations are done for the subsystems first. Then their results are combined to construct the effective Hamiltonian for the whole system, and the effective Hamiltonian is solved to obtain approximate solutions to the whole system. In this study, we combine this idea of "divide-and-conquer" with the variational quantum eigensolver (VQE), which is one of the most important examples of quantum-classical hybrid algorithms running on NISQ devices.

RESULT

In this work, we have developed a method called "Deep VQE" that takes advantage of the VQE calculations only on a small-sized quantum computer to solve the effective Hamiltonian of the whole system and enables calculations on very large systems compared to the size of the quantum computer. Compared to previous studies that applied an existing divide-and-conquer method in quantum chemistry to quantum computers [1], Deep VQE gives a general formulation that can be applied to various condensed matter physics and quantum chemistry problems. Furthermore, the hierarchical use of the "subsystem and whole system" structure allows us to deal with very large systems compared with actual sizes of quantum computers used for VQE. For example, the quantum spin model with nearest-neighbor interactions on 32*32=1024 sites of a two-dimensional square lattice (equivalent to 1024 qubits problem) can be treated with a quantum computer with up to 32 qubits by using Deep VQE. In the paper (preprint), we performed numerical simulations for frustrated spin systems coupled in a one-dimensional fashion to evaluate the accuracy of the method and confirmed that Deep VQE can obtain results close to the exact solution of the whole system.

OUTLOOK

Deep VQE allows us to solve problems of practical sizes (e.g., hundreds of qubits) to be solved on smaller quantum computers of tens of qubits. Although the results from Deep VQE can be trusted only when the interactions between the subsystems are not too large, various interesting systems, including quantum spin systems with nearest-neighbor interactions and molecular systems with potential industrial applications such as organic EL, solar cells, and photosynthesis, satisfy that condition. Deep VQE is a promising approach to realize the "quantum advantage" of NISQ devices in the near future.

[1] Yamazaki et al., "Towards the Practical Application of Near-Term Quantum Computers in Quantum Chemistry Simulations: A Problem Decomposition Approach", arXiv:1806.01305

BACKGROUND

Noisy Intermediate-Scale Quantum (NISQ) devices, or analog quantum computers with noise, have recently been developed and shown to outperform classical computers on a certain computational task (Nature 574, 505–510 (2019). Although this "quantum supremacy" has been demonstrated for the special computational problem of little practical significance, it is expected that quantum computers will achieve a "quantum advantage", which means that they outperform classical computers for practical problems, in the near future.

PROBLEMS

However, in order to solve problems of practical sizes in condensed matter physics and quantum chemistry, where the "quantum advantage" of NISQ devices is expected, the size (number of qubits) and accuracy of the currently realized NISQ devices are not sufficient yet. Therefore, there is a strong need for methods to solve large size problems using small size quantum computers.

METHOD

In the field of quantum chemistry computation using classical computers, there are plenty of methods based on the "divide-and-conquer" approach which enables us to treat large systems with small computational resources. In those methods, systems to be treated are divided into smaller subsystems, and calculations are done for the subsystems first. Then their results are combined to construct the effective Hamiltonian for the whole system, and the effective Hamiltonian is solved to obtain approximate solutions to the whole system. In this study, we combine this idea of "divide-and-conquer" with the variational quantum eigensolver (VQE), which is one of the most important examples of quantum-classical hybrid algorithms running on NISQ devices.

RESULT

In this work, we have developed a method called "Deep VQE" that takes advantage of the VQE calculations only on a small-sized quantum computer to solve the effective Hamiltonian of the whole system and enables calculations on very large systems compared to the size of the quantum computer. Compared to previous studies that applied an existing divide-and-conquer method in quantum chemistry to quantum computers [1], Deep VQE gives a general formulation that can be applied to various condensed matter physics and quantum chemistry problems. Furthermore, the hierarchical use of the "subsystem and whole system" structure allows us to deal with very large systems compared with actual sizes of quantum computers used for VQE. For example, the quantum spin model with nearest-neighbor interactions on 32*32=1024 sites of a two-dimensional square lattice (equivalent to 1024 qubits problem) can be treated with a quantum computer with up to 32 qubits by using Deep VQE. In the paper (preprint), we performed numerical simulations for frustrated spin systems coupled in a one-dimensional fashion to evaluate the accuracy of the method and confirmed that Deep VQE can obtain results close to the exact solution of the whole system.

OUTLOOK

Deep VQE allows us to solve problems of practical sizes (e.g., hundreds of qubits) to be solved on smaller quantum computers of tens of qubits. Although the results from Deep VQE can be trusted only when the interactions between the subsystems are not too large, various interesting systems, including quantum spin systems with nearest-neighbor interactions and molecular systems with potential industrial applications such as organic EL, solar cells, and photosynthesis, satisfy that condition. Deep VQE is a promising approach to realize the "quantum advantage" of NISQ devices in the near future.

[1] Yamazaki et al., "Towards the Practical Application of Near-Term Quantum Computers in Quantum Chemistry Simulations: A Problem Decomposition Approach", arXiv:1806.01305