We present an efficient quantum algorithm for beyond-Born-Oppenheimer molecular energy computations. Our approach combines the quantum full configuration interaction method with the nuclear orbital plus molecular orbital (NOMO) method. We give the details of the algorithm and demonstrate its performance by classical simulations. Two isotopomers of the hydrogen molecule (H 2 , HT) were chosen as representative examples and calculations of the lowest rotationless vibrational transition energies were simulated. * These authors contributed equally. †
Last years witnessed a remarkable interest in application of quantum computing for solving problems in quantum chemistry more efficiently than classical computers allow. Very recently, even first proof-of-principle experimental realizations have been reported. However, so far only the nonrelativistic regime (i.e. Schroedinger equation) has been explored, while it is well known that relativistic effects can be very important in chemistry. In this communication we present the first quantum algorithm for relativistic computations of molecular energies. We show how to efficiently solve the eigenproblem of the Dirac-Coulomb Hamiltonian on a quantum computer and demonstrate the functionality of the proposed procedure by numerical simulations of computations of the spinorbit splitting in the SbH molecule. Finally, we propose quantum circuits with 3 qubits and 9 or 10 CNOTs, which implement a proof-of-principle relativistic quantum chemical calculation for this molecule and might be suitable for an experimental realization.Quantum computing [1] is one of the fastest growing fields of computer science nowadays. Recent huge interest in this interdisciplinary field has been fostered by the prospects of solving certain types of problems more effectively than in the classical setting [2,3]. The prominent example is the integer factorization problem where quantum computing offers an exponential speedup over its classical counterpart [2]. But it is not only cryptography that can benefit from quantum computers. As was first proposed by R. Feynman [4], quantum computers could in principle be used for efficient simulation of another quantum system. This idea, which employs mapping of the Hilbert space of a studied system onto the Hilbert space of a register of quantum bits (qubits), both of them being exponentially large, can in fact be adopted also in quantum chemistry.Several papers using this idea and dealing with the interconnection of quantum chemistry and quantum computing have appeared in recent years. An efficient (polynomially scaling) algorithm for calculations of non-relativistic molecular energies, that employs the phase estimation algorithm (PEA) of Abrams and Lloyd [19], was proposed in the pioneering work by Aspuru-Guzik, et al. [6]. When the ideas of measurement based quantum computing are adopted [20], the phase estimation algorithm can be formulated in an iterative manner [iterative phase estimation (IPEA)] with only one read-out qubit [8,9]. If the phase φ (0 ≤ φ < 1), which is directly related to the desired energy [9], is expressed in the binary form: φ = 0.φ 1 φ 2 . . ., φ i = {0, 1}, one bit of φ is measured on the read-out qubit at each iteration step. The algorithm is iterated backwards from the least significant bits of φ to the most significant ones, where the k-th iteration is shown in Figure 1. Not to confuse the reader, H in the exponential denotes the Hamiltonian operator, whereas H (in a box) denotes the standard single-qubit Hadamard gate. |ψ system represents the part of a quantum register that e...
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