Simulation of quantum many-body systems is a promising application of quantum computers. However, implementing the time-evolution operator as a quantum circuit efficiently on near-term devices with limited resources is challenging. Standard approaches like Trotterization often require deep circuits, making them impractical. This paper proposes a hybrid quantum-classical algorithm called Local Subspace Variational Quantum Compilation (LSVQC) for compiling the time-evolution operator. The LSVQC uses variational optimization to reproduce the action of the target time-evolution operator within a physically reasonable subspace. Optimization is performed on small local subsystems based on the Lieb-Robinson bound, allowing for cost function evaluation using small-scale quantum devices or classical computers. Numerical simulations on a spin-lattice model and an ab initio effective model of strongly correlated material Sr2CuO3 demonstrate the algorithm's effectiveness. It is shown that the LSVQC achieves a 95% reduction in circuit depth compared to Trotterization while maintaining accuracy. The subspace restriction also reduces resource requirements and improves accuracy. Furthermore, we estimate the gate count needed to execute the quantum simulations using the LSVQC on near-term quantum computing architectures in the noisy intermediate-scale or early fault-tolerant quantum computing era. Our estimation suggests that the acceptable physical gate error rate for the LSVQC can be significantly larger than for Trotterization.

We study the deep multi-scale entanglement renormalization ansatz (DMERA) on quantum hardware and the causal cone of a subset of the qubits which make up the ansatz. This causal cone spans O(M+logN) physical qubits on a quantum device, where M and N are the subset size and the total number qubits in the ansatz respectively. This allows for the determination of the von Neumann entanglement entropy of the N qubit wave-function using O(M+logN) qubits by diagonalization of the reduced density matrix (RDM). We show this by randomly initializing a 16-qubit DMERA and diagonalizing the resulting RDM of the M-qubit subsystem using density matrix simulation. As an example of practical interest, we also encode the variational ground state of the quantum critical long-range transverse field Ising model (LRTIM) on 8 spins using DMERA. We perform density matrix simulation with and without noise to obtain entanglement entropies in separate experiments using only 4 qubits. Finally we repeat the experiment on the IBM Kyoto backend reproducing simulation results.

Evaluating the relative performance of different quantum algorithms for quantum computers is of great significance in the research of quantum algorithms. In this study, we consider the problem of quantum chemistry, which is considered one of the important applications of quantum algorithms. While evaluating these algorithms in systems with a large number of qubits is essential to see the scalability of the algorithms, the solvable models usually used for such evaluations typically have a small number of terms compared to the molecular Hamiltonians used in quantum chemistry. The large number of terms in molecular Hamiltonians is a major bottleneck when applying quantum algorithms to quantum chemistry. Various methods are being considered to address this problem, highlighting its importance in developing quantum algorithms for quantum chemistry. Based on these points, a solvable model with a number of terms comparable to the molecular Hamiltonian is essential to evaluate the performance of such algorithms. In this paper, we propose a set of exactly solvable Hamiltonians that has a comparable order of terms with molecular Hamiltonians by applying a spin-involving orbital rotation to the one-dimensional Fermi-Hubbard Hamiltonian. We verify its similarity to the molecular Hamiltonian from some prospectives and investigate whether the difficulty of calculating the ground-state energy changes before and after orbital rotation by applying the density matrix renormalization group up to 24 sites corresponding to 48 qubits. This proposal would enable proper evaluation of the performance of quantum algorithms for quantum chemistry, serving as a guiding framework for algorithm development.