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 variational quantum eigensolver (VQE) with reducing the dimensions of the system, 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 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 intra-subsystem interactions and weak inter-subsystem interactions, or strongly correlated spin models on large regular lattices. As proof-of-principle numerical demonstrations, we use the proposed method for Heisenberg anti-ferromagnetic models, including one-dimensionally coupled 12-qubit Heisenberg anti-ferromagnetic models on Kagome lattices. The largest problem size of 48 qubits is solved by simulating 12-qubit quantum computers. The proposed scheme enables us to handle the problems of > 1000 qubits by concatenating VQE with a few tens of qubits. Deep VQE will provide us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.

A discrete time crystal is a phase unique to nonequilibrium systems, where discrete time translation symmetry is spontaneously broken. Most conventional time crystals proposed so far rely on the spontaneous breaking of on-site symmetries and their corresponding on-site symmetry operations. In this Letter, we propose a new time crystal dubbed the "spatial-translation-induced discrete time crystal," which is realized by spatial translation and its symmetry breaking. Owing to the properties of spatial translation, in this new time crystal, various time crystal orders can only emerge by changing the filling but not changing the driving protocol. We demonstrate that the local transport of charges or spins shows a nontrivial oscillation, enabling detection and applications of time crystal orders, and also provide promising platforms including quantum circuits. Our proposal opens up a new avenue of realizing time crystal orders by spatial translation in various quantum simulators.

No abstract

Quantum many-body scars have attracted much interest as a violation of the eigenstate thermalization hypothesis (ETH) due to recent experimental observation in Rydberg atoms and related theoretical studies. In this paper, we construct a model hosting exact Floquet quantum many-body scars, which violate the Floquet version of ETH. We consider two uniformly driven static Hamiltonians prohibiting neighboring up spins (Rydberg blockade) like the PXP model, and construct a binary drive composed of them. We show that there exists a four-dimensional subspace which completely avoids thermalization to infinite temperature and that any other states, including some special scar states reported in the static PXP model, are vulnerable to heating and relax to infinite temperature. We also construct a more generalized periodic drive composed of time-dependent PXP-type Hamiltonians showing exact Floquet quantum many-body scars and discuss possible experimental realization of the model in Rydberg atoms.

No abstract