Hybrid quantum-classical algorithm for variational coupled cluster method

We present a hybrid quantum algorithm for the variational coupled cluster (vCC) method in quantum chemistry. We show that for a problem instance of $n$ electrons described by $m$ spin orbitals and a constant level $\ell$ of excitations, the energy expectation value of the vCC trial wavefunction can be estimated to precision $\epsilon$ in time proportional to $\widetilde{\O}(\ell m^\ell n^\ell /\epsilon^2)$ on a quantum computer. Classically, computing the same expectation generally incurs a cost exponential in $n$ and $m$, implying that our quantum algorithm can yield a significant speedup over known classical methods. We envision that such capability combined with the framework of the variational quantum eigensolver (VQE) will add to recent quantum algorithms for unitary coupled cluster methods, enriching the toolset of quantum chemistry calculations beyond what is feasible on classical computers. We also illustrate a method for calculating analytical gradients for the vCC method, which can be used with gradient-free direct-search optimisation methods (such as the Nelder-Mead and COBYLA algorithms).