Deep Variational Quantum Eigensolver: A Divide-And-Conquer Method for Solving a Larger Problem with Smaller Size Quantum Computers

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 a variational quantum eigensolver (VQE) with a reduction in the system dimension, 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 the 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 intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices. As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices. The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few $\mathrm{\%}$. The proposed scheme enables us to handle the problems of $>1000$ qubits by concatenating VQEs with a few tens of qubits. While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement. Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.

Popular Summary

Currently, quantum-computing devices are making remarkable progress, and are becoming capable of performing tasks that are hard even for classical computers. One of the most notable applications of such a quantum computer is simulation of quantum many-body systems, such as molecules and materials, which are described by the same rules of quantum mechanics. However, the number of qubits in current quantum computers, i.e., noisy intermediate-scale quantum (NISQ) devices, is still limited. This is the most significant barrier to the application of quantum computers to the simulation of materials and molecules of practical importance.

Here we propose a divide-and-conquer method using NISQ devices to solve practically important large-scale problems. We divide the system of interest into subsystems, each of which is solved by a variational quantum eigensolver (VQE). The solution is further used to span a low-energy subspace, and VQE is further performed by taking the intersubsystem interactions. Namely, this is a real-space renormalization method using a quantum computer. In a numerical evaluation, it was shown that a 20-qubit quantum computer could calculate a problem requiring 64 qubits with an error of a few percent. As is the case with quantum chemistry and condensed-matter physics on conventional computers, the divide-and-conquer method or real-space renormalization method for a quantum computer will be an important key technology for solving large-scale quantum many-body problems using small-scale near-term quantum computers.

Figure 1

(a) The system consists of subsystems of Hamiltonian $H_i$, each of which interact with each other by intersubsystem interaction $V_{ij}$. (b) To solve the system depicted in (a), we first construct an approximate ground state $|{\psi }_0^{(i)}⟩$ of each $H_i$ with VQE (the first VQE). Then we form a basis set by applying excitation operators on $|{\psi }_0^{(i)}⟩$. Using the basis, we can construct an effective Hamiltonian, which gives better approximation of the ground state.

Figure 2

Concatenation of VQEs. Red, blue, green squares correspond to the first, second, and third VQEs, respectively. The red, blue, green, and orange edges indicate the intersubsystem interactions taken at the first, second, third, and fourth VQEs. At each level, the local basis is generated by the local excitations on each qubit at the boundary, that is, the qubits engaged in the intersubsystem interactions. At each level, the effective Hamiltonian is constructed from suitably chosen local basis.

Figure 3

(a) The system used for the numerical demonstration. It consists of four-qubit subsystems of Hamiltonian $H_i$, each of which interacts with each other by intersubsystem interaction $V_{ij}$. (b) A unit cell of Heisenberg antiferromagnetic model on a 12-qubit kagome lattice. The 12-qubit systems interact with each other in a nearest-neighbor way. (c) The energy obtained by the first VQE for the 12-qubit Heisenberg antiferromagnetic model. $d$ indicates the depth, i.e., the number of cycles, of the parameterized quantum circuits. See Table 1 for the converged energies with increasing the ansatz depth.

Figure 4

The protocol of concatenating of VQEs employed in the numerical demonstration of 2D systems. Qubits marked by white symbols are located at the boundary of subsystems for the first VQE. Hexagons (both blue and white) indicate the boundary sites of subsystems for the second VQE, which are used for constructing the local basis following Eq. (31). (a) $32$ sites with the concatenation up to the second VQE. (b) $32$ sites with the concatenation up to the third VQE. (c) $64$ sites with the concatenation up to the second VQE. (d) $64$ sites with the concatenation up to the third VQE.