Dissertation Defense: Luogen Xu

Candidate: Luogen Xu

Major: Physics

Advisor: James Freericks, Ph.D.

Title: Efficient Implementations of the Factorized Unitary-Coupled Cluster Ansatz

The variational quantum eigensolver has been proposed as a low-depth quantum circuit that can be employed to examine strongly correlated systems on today’s noisy intermediate-scale quantum computers, and the factorized form of the unitary coupled-cluster (UCC) approximation is one of the most promising methodologies to prepare trial states for strongly correlated systems within the variational quantum eigensolver (VQE) framework. It has the potential to solve quantum chemistry problems involving strongly correlated electrons, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scales exponentially with the rank of the operator.

This thesis examines details associated with the factorized form of the unitary coupled-cluster variant. We applied it to a simple strongly correlated condensed-matter system with nontrivial behavior—the four-site Hubbard model at half-filling. This work shows some of the subtle issues one needs to take into account when applying this algorithm in practice, especially to condensed-matter systems. We then proposed a set of new schemes that trade-off using extra qubits for a reduced gate depth to decompose high-rank UCC excitation operators into significantly lower-depth circuits. These results will remain useful even when fault-tolerant machines are available to reduce the overall state-preparation circuit depth. Finally, we employed the linear combination of unitaries approach by exploiting a hidden SU($2$) symmetry to allow us to directly simulate the transformed UCC factors. Our algorithm scales like the cube of the rank of the operator $n^3$, a significant reduction in complexity for rank five or higher operators. This approach, when combined with other algorithms for lower-rank operators (when compared to the standard implementation), will make the factorized form of the unitary coupled-cluster approach much more efficient to implement on all types of quantum computers.