Relativistic Internally Contracted Multireference Electron Correlation Methods

We report internally contracted relativistic multireference configuration interaction (ic-MRCI), complete active space second-order perturbation (CASPT2), and strongly contracted n-electron valence state perturbation theory (NEVPT2) on the basis of the four-component Dirac Hamiltonian, enabling accurate simulations of relativistic, quasi-degenerate electronic structure of molecules containing transition-metal and heavy elements. Our derivation and implementation of ic-MRCI and CASPT2 are based on an automatic code generator that translates second-quantized ansatze to tensor-based equations, and to efficient computer code. NEVPT2 is derived and implemented manually. The rovibrational transition energies and absorption spectra of HI and TlH are presented to demonstrate the accuracy of these methods

A Characterization of the Randomized Uniform Rule

We consider the problem of allocating several units of an indivisible object among the agents with single-peaked and risk-averse utility functions. We introduce equal probability for the best, and show that the randomized uniform rule is the only randomized rule satisfying strategy-proofness, Pareto optimality, and equal probability for the best. This is an alternative characterization of the result of Ehlers and Klaus (2004).The Randomized Uniform Rule, Single-Peaked Utility Functions, Equal Probability for the Best, Strategy-Proofness, Indivisibility

Equal probability for the best and the assignment of identical indivisible objects

We consider the problem of allocating several units of an indivisible object among agents with single-peaked utility functions. We introduce an axiom called equal probability for the best, and show that it is equivalent to both equal treatment of equals and symmetry in the presence of Pareto optimality. Moreover, we also show that the randomized uniform rule is the only randomized rule satisfying strategy-proofness, Pareto optimality, and equal probability for the best.

Equal probability for the best and the assignment of identical indivisible objects

We consider the problem of allocating several units of an indivisible object among agents with single-peaked utility functions. We introduce an axiom called equal probability for the best, and show that it is equivalent to both equal treatment of equals and symmetry in the presence of Pareto optimality. Moreover, we also show that the randomized uniform rule is the only randomized rule satisfying strategy-proofness, Pareto optimality, and equal probability for the best

Analytical formulation of the second-order derivative of energy for orbital-optimized variational quantum eigensolver: application to polarizability

We develop a quantum-classical hybrid algorithm to calculate the analytical second-order derivative of the energy for the orbital-optimized variational quantum eigensolver (OO-VQE), which is a method to calculate eigenenergies of a given molecular Hamiltonian by utilizing near-term quantum computers and classical computers. We show that all quantities required in the algorithm to calculate the derivative can be evaluated on quantum computers as standard quantum expectation values without using any ancillary qubits. We validate our formula by numerical simulations of quantum circuits for computing the polarizability of the water molecule, which is the second-order derivative of the energy with respect to the electric field. Moreover, the polarizabilities and refractive indices of thiophene and furan molecules are calculated as a testbed for possible industrial applications. We finally analyze the error-scaling of the estimated polarizabilities obtained by the proposed analytical derivative versus the numerical one obtained by the finite difference. Numerical calculations suggest that our analytical derivative may require fewer measurements (runs) on quantum computers than the numerical derivative to achieve the same fixed accuracy.Comment: 34 + 4 page

ADAPT-QSCI: Adaptive Construction of Input State for Quantum-Selected Configuration Interaction

We present a quantum-classical hybrid algorithm for calculating the ground state and its energy of the quantum many-body Hamiltonian by proposing an adaptive construction of a quantum state for the quantum-selected configuration interaction (QSCI) method. QSCI allows us to select important electronic configurations in the system to perform CI calculation (subspace diagonalization of the Hamiltonian) by sampling measurement for a proper input quantum state on a quantum computer, but how we prepare a desirable input state has remained a challenge. We propose an adaptive construction of the input state for QSCI in which we run QSCI repeatedly to grow the input state iteratively. We numerically illustrate that our method, dubbed \textit{ADAPT-QSCI}, can yield accurate ground-state energies for small molecules, including a noisy situation for eight qubits where error rates of two-qubit gates and the measurement are both as large as 1\%. ADAPT-QSCI serves as a promising method to take advantage of current noisy quantum devices and pushes forward its application to quantum chemistry.Comment: 14 page