Graph-|Q⟩⟨C|, a Graph-Based Quantum/Classical Algorithm for Efficient Electronic Structure on Hybrid Quantum/Classical Hardware Systems: Improved Quantum Circuit Depth Performance

Abstract

We present a procedure to reduce the depth of quantum circuits and improve the accuracy of results in computing post-Hartree–Fock electronic structure energies in large molecular systems. The method is based on molecular fragmentation where a molecular system is divided into overlapping fragments through a graph-theoretic procedure. This allows us to create a set of projection operators that decompose the unitary evolution of the full system into separate sets of processes, some of which can be treated on quantum hardware and others on classical hardware. Thus, we develop a procedure for an electronic structure that can be asynchronously spawned onto a potentially large ensemble of classical and quantum hardware systems. We demonstrate this method by computing Unitary Coupled Cluster Singles and Doubles (UCCSD) energies for a set of [H2] n clusters, with n ranging from 4 to 128. We implement our methodology using quantum circuits, and when these quantum circuits are processed on a quantum simulator, we obtain energies in agreement with the UCCSD energies in the milli-hartree energy range. We also show that our circuit decomposition approach yields up to 9 orders of magnitude reduction in the number of CNOT gates and quantum circuit depth for the large-sized clusters when compared to a standard quantum circuit implementation available on IBM’s Quantum Information Science kit, known as Qiskit.

Graph Theoretic Molecular Fragmentation for Multidimensional Potential Energy Surfaces Yield an Adaptive and General Transfer Machine Learning Protocol

“…In a series of publications − , graph theory-based techniques have been discussed to compute efficient and adaptive many-body expansions ,− that have strong connections to molecular fragmentation − and ONIOM. − The salient features of this approach are as follows: the molecular assembly is partitioned into a set of nodes, or vertices. These nodes are then connected through edges based on a chosen edge length threshold that captures the extent to which two-body bonded and nonbonded interactions are to be captured.…”

“…The above graphical description allows a dynamic and flexible representation of local many-body interactions. The energetic measure considered in refs − and consists of a perturbative, ONIOM-type, correction to a result obtained at a lower level of theory, where the perturbative correction is the difference between two many-body expansions (replacing the standard “model-high” minus “model-low” portion in ONIOM) given by the graphical representation above. Consistent with the notions behind ONIOM, the energy expression − , that conveys this general idea is where the left side, E MBE,gt ONIOM ( x̅ ), denotes the graph-theoretically obtained many-body correction to ONIOM, and the term E MBE level,I ( x̅ ) on the right side may encompass the full system or a chosen “active site”.…”

“…The energetic measure considered in refs − and consists of a perturbative, ONIOM-type, correction to a result obtained at a lower level of theory, where the perturbative correction is the difference between two many-body expansions (replacing the standard “model-high” minus “model-low” portion in ONIOM) given by the graphical representation above. Consistent with the notions behind ONIOM, the energy expression − , that conveys this general idea is where the left side, E MBE,gt ONIOM ( x̅ ), denotes the graph-theoretically obtained many-body correction to ONIOM, and the term E MBE level,I ( x̅ ) on the right side may encompass the full system or a chosen “active site”. Ab initio molecular dynamics (AIMD) trajectories have been studied with both options; − , furthermore, quantum nuclear effects arising from detailed potential energy surface treatments have also been introduced based on generalizations , to eq .…”

“…In a series of publications [48][49][50][51][52][53][54][55][56][57]73 graph theory-based techniques have been discussed to compute efficient and adaptive manybody expansions 1,58−66 that have strong connections to molecular fragmentation 28−47 and ONIOM. 67−72 The salient features of this approach are as follows: the molecular assembly is partitioned into a set of nodes, or vertices.…”

“…In essence, in refs and , each molecular fragmentation protocol leads to a different graph, and the overall potential surface is assembled through a linear combination of the associated potential surface functions. An efficient approach for quantum computation has also been introduced from these methods in ref . Together these descriptions allow us to expand potential energy surfaces as functions of lower-dimensional subspaces of the nuclear degrees of freedom .…”