Relativistic quantum chemistry on quantum computers

The past few years have witnessed a remarkable interest in the application of quantum computing for solving problems in quantum chemistry more efficiently than classical computers allow. Very recently, proof-of-principle experimental realizations have been reported. However, so far only the nonrelativistic regime (i.e., the Schrödinger equation) has been explored, while it is well known that relativistic effects can be very important in chemistry. We present a 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 spin-orbit splitting in the SbH molecule. Finally, we propose quantum circuits with three qubits and nine or ten controlled-not (cnot) gates, which implement a proof-of-principle relativistic quantum chemical calculation for this molecule and might be suitable for an experimental realization.

Figure 1

The $k$th iteration of the iterative phase estimation algorithm (IPEA). The feedback angle ${\omega }_k$ depends on the previously measured bits.

Figure 2

Simulated potential energy curves of ground (0${}^{+}$) and excited (1) states of SbH, and spin-orbit energy splitting. Absolute energies are shifted by $6481E{}_H$.

Figure 3

SbH ground- (0${}^{+}$) and excited- (1) state qFCI success probabilities (SPs) corresponding to HF initial guesses.

Figure 4

Adiabatic state preparation (ASP) of the SbH ground state ($0^{+}$) for different internuclear distances. The solid lines correspond to qFCI success probabilities, and the$|\langle {\psi }_{\mathrm{ASP}}|{\psi }_{\mathrm{exact}}\rangle {|}^2·(0.81,1\rangle$ interval is colored (shaded). $1000\hslash E_H^{-1}\approx {10}^{-14}$ s.

Figure 5

Scheme of a circuit corresponding to CAS(4,3) calculations on SbH. The empty squares represent generic single-qubit gates. $R_z$ gates are without angle specification. For derivation, details, and all the parameters, see the Supplemental Material [23].