Efficient classical simulation of large-scale unitary cluster Jastrow circuits

Recent quantum computing experiments in chemistry have frequently utilized the unitary cluster Jastrow (UCJ) ansatz, a quantum circuit architecture designed to be compatible with current hardware. While these experiments often claim to push the boundaries of classical simulation, this paper investigates the classical complexity of simulating these circuits. The authors develop a polynomial-time classical algorithm to compute the energy of any single-layer UCJ circuit, regardless of whether the circuit is constrained by hardware-specific locality requirements.

The algorithm operates in three primary steps. First, it uses Heisenberg evolution to backpropagate the fermionic Hamiltonian through the circuit's orbital rotations. Second, it modifies the Hamiltonian to account for the Jastrow operator, which preserves the polynomial scaling of the term count. Finally, it employs Löwdin’s formula to compute the energy as a matrix element between non-orthogonal Slater determinants. This approach allows for exact energy calculation in O(N^7) time, where N is the number of orbitals.

The authors demonstrate the power of their algorithm by reproducing the results of a prominent 77-qubit quantum chemistry experiment in less than a minute on a standard laptop. Furthermore, by using their fast simulation to optimize circuit parameters, they achieve a lower ground state energy than that reported in the original experiment, which relied on expensive supercomputing resources and quantum hardware samples. The study concludes that single-layer UCJ circuits are not sufficient for beyond-classical computation, as they can be efficiently simulated classically. The authors suggest that future efforts toward quantum advantage in chemistry must move toward multi-layer circuits, which are significantly more complex to simulate.