We propose quantum-selected configuration interaction (QSCI), a class of hybrid quantum-classical algorithms for calculating the ground- and excited-state energies of many-electron Hamiltonians on noisy quantum devices. Suppose that an approximate ground state can be prepared on a quantum computer either by variational quantum eigensolver or by some other method. Then, by sampling the state in the computational basis, which is hard for classical computation in general, one can identify the electron configurations that are important for reproducing the ground state. The Hamiltonian in the subspace spanned by those important configurations is diagonalized on classical computers to output the ground-state energy and the corresponding eigenvector. The excited-state energies can be obtained similarly. The result is robust against statistical and physical errors because the noisy quantum devices are used only to define the subspace, and the resulting ground-state energy strictly satisfies the variational principle even in the presence of such errors. The expectation values of various other operators can also be estimated for obtained eigenstates with no additional quantum cost, since the explicit eigenvectors in the subspaces are known. We verified our proposal by numerical simulations, and demonstrated it on a quantum device for an 8-qubit molecular Hamiltonian. The proposed algorithms are potentially feasible to tackle some challenging molecules by exploiting quantum devices with several tens of qubits, assisted by high-performance classical computing resources for diagonalization.

We propose an efficient and almost optimal scheme for measuring molecular Hamiltonians in quantum chemistry on quantum computers, which requires 2N² distinct measurements in the leading order with N being the number of molecular orbitals. It achieves the state-of-the-art by improving a previous proposal by Bonet-Monroig et al. [Phys. Rev. X 10, 031064 (2020)] which exhibits 17N²/6 scaling in the leading order. We develop a novel method based on a finite projective plane to construct sets of simultaneously-measurable operators contained in molecular Hamiltonians. Each measurement only requires a depth-O(N) circuit consisting of O(N²) one- and two-qubit gates under the Jordan-Wigner and parity mapping, assuming the linear connectivity of qubits on quantum hardwares. Because evaluating expectation values of molecular Hamiltonians is one of the major bottlenecks in the applications of quantum devices to quantum chemistry, our finding is expected to accelerate such applications.

Fault-tolerant quantum computation (FTQC) is essential to robustly implement quantum algorithms and thus to maximize advantages of quantum computers. In this context, a quantum circuit is decomposed into universal gates that can be fault-tolerantly implemented, for example, Clifford+T gates. Here, T gate is usually regarded as an essential resource for quantum computation because its action cannot be simulated efficiently on classical computers. Practically, it is highly likely that only a limited number of T gates are available in the near future due to its experimental difficulty of fault-tolerant implementation. In this paper, considering this Pre-FTQC era, we investigate Clifford+T decomposition with a limited budget of T gates and propose a new model of the error of such decomposition. More concretely, we propose an algorithm to generate the most accurate Clifford+T-gate decomposition of a given single-qubit rotation gate with a fixed number of T gates. We also propose to model the error of Clifford+T decomposition using well-known depolarizing noise by considering the average of the effects of the error. We numerically verified our model successfully explains the decomposition error for a wide variety of molecules using our decomposition algorithm. Thus, we shed light on a first-stage application of quantum computers from a practical point of view and fuel further research towards what quantum computation can achieve in the upcoming future.