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arXiv:2205.03539 · The Journal of Physical Chemistry A

Correlated Reference-Assisted Variational Quantum Eigensolver

Nhan Trong Le, Lan Nguyen Tran

Published Jun 9, 2023• 10 citations
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Abstract

We propose an active-space approximation to reduce the quantum resources required for variational quantum eigensolver (VQE). Starting from the double exponential unitary coupled-cluster ansatz and employing the downfolding technique, we arrive at an effective Hamiltonian for active space composed of the bare Hamiltonian and a correlated potential caused by the internal-external interaction. The correlated potential is obtained from the one-body second-order Møller-Plesset perturbation theory (OBMP2), which is derived using the canonical transformation and cumulant approximation. Considering different systems with singlet and doublet ground states, we examine the accuracy in predicting both energy and density matrix (by evaluating dipole moment). We show that our approach can dramatically outperform the active-space VQE with an uncorrelated Hartree-Fock reference.

DOI
10.1021/acs.jpca.3c00993

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We propose an active-space approximation to reduce the quantum resources required for variational quantum eigensolver (VQE). Starting from the double exponential unitary coupled-cluster ansatz and employing the downfolding technique, we arrive at an effective Hamiltonian for active space composed of the bare Hamiltonian and a correlated potential caused by the internal-external interaction. The correlated potential is obtained from the one-body second-order Møller-Plesset perturbation theory (OBMP2), which is derived using the canonical transformation and cumulant approximation. Considering different systems with singlet and doublet ground states, we examine the accuracy in predicting both energy and density matrix (by evaluating dipole moment). We show that our approach can dramatically outperform the active-space VQE with an uncorrelated Hartree-Fock reference.
Applications of quantum simulation algorithms to obtain electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices require careful consideration of resources describing the complex electron correlation effects. In modeling second-quantized problems, the biggest challenge confronted is that the number of qubits scales linearly with the size of the molecular basis. This poses a significant limitation on the size of the basis sets and the number of correlated electrons included in quantum simulations of chemical processes. To address this issue and enable more realistic simulations on NISQ computers, we employ the double unitary coupled-cluster (DUCC) method to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space. Using downfolding techniques, we demonstrate that properly constructed effective Hamiltonians can capture the effect of the whole orbital space in small-size active spaces. Combining the downfolding preprocessing technique with the variational quantum eigensolver, we solve for the ground-state energy of H 2, Li 2, and BeH 2 in the cc-pVTZ basis using the DUCC-reduced active spaces. We compare these results to full configuration-interaction and high-level coupled-cluster reference calculations.
In this work, we combine the recently developed double unitary coupled cluster (DUCC) theory with the adaptive, problem-tailored variational quantum eigensolver (ADAPT-VQE) to explore the accuracy of unitary downfolded Hamiltonians for quantum simulation of chemistry. We benchmark the ability of DUCC effective Hamiltonians to recover dynamical correlation energy outside of an active space. We consider the effects of strong correlation, commutator truncation, higher-body terms, and approximate external amplitudes on the accuracy of these effective Hamiltonians. When combining these DUCC Hamiltonians with ADAPT-VQE, we observe similar convergence of the ground state as compared with bare active space Hamiltonians, demonstrating that DUCC Hamiltonians provide increased accuracy without increasing the load on the quantum processor.
In this paper we discuss the utilization of Variational Quantum Solver (VQE) and recently introduced Generalized Unitary Coupled Cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N2_22​, H2_22​O, and C2_22​H4_44​, and benchmark systems to illustrate the performance of the combined framework.
We present a novel approach for efficient preparation of electronic ground states, leveraging the optimizer ExcitationSolve [Jäger et al., Comm. Phys. (2025)] and established variational quantum eigensolver-based operator selection methods, such as Energy Sorting (ES). By combining these tools, we demonstrate a computationally efficient protocol that enables the construction of an approximate ground state from a unitary coupled cluster ansatz via a single sweep over the operator pool. Utilizing efficient classical pre-processing to select the majority of relevant operators, this approach reduces the computational complexity associated with traditional variational quantum eigensolver (VQE) optimization methods. We further show that second-order Epstein-Nesbet (EN2) perturbation theory emerges as the first-order Taylor expansion of our protocol in terms of a correlation measure, clarifying why our approach provides a more robust initial guess for the ground state in strongly correlated regimes. We also find that second-order Møller-Plesset perturbation (MP2) theory, which is widely used for unitary coupled cluster (UCC) initialization, performs worse than both EN2 and our protocol. Furthermore, we show that our method can be seamlessly integrated with one-variational-parameter couple exchange operators, thereby further reducing the number of required CNOT operations. Overall, we empirically observe a quadratic convergence speedup beyond state-of-the-art methods, advancing the preparation of high-fidelity electronic ground states - one of the cornerstones of meaningful electronic structure calculations in the noisy intermediate-scale quantum computing (NISQ) era, and a prerequisite for fault-tolerant quantum computing (FTQC) algorithms such as quantum phase estimation.
Abstract In this paper, we discuss the utilization of variational quantum solver (VQE) and recently introduced generalized unitary coupled cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N 2 , H 2 O, and C 2 H 4 , as benchmark systems to illustrate the performance of the combined framework.
Current noisy intermediate-scale quantum (NISQ) devices remain limited in their ability to perform accurate quantum chemistry simulations because of restricted numbers of high-fidelity qubits and short coherence times. To overcome these challenges, we introduce a reduced-density-matrix (RDM)-based perturbative variational quantum eigensolver (VQE) framework that augments active-space VQE with perturbation theory to recover electron correlation beyond the active space without increasing the qubit count or variational circuit depth. We formulated a fully coupled approach (VQE-PTs) and a diagonal approximation (VQE-PT). The former retains couplings among orthonormalized perturbers, whereas the latter neglects these couplings to simplify the classical postprocessing. Numerical simulations of HF, N2, and F2 show that VQE-PTs provides a robust formulation across different molecular systems, while VQE-PT offers an efficient approximation. We further experimentally implement VQE-PT on the Quafu superconducting quantum processor for F2, achieving a mean absolute error of 1.2 mHa along the potential energy surface after error mitigation. These results demonstrate perturbative VQE as a practical framework for incorporating dynamic correlations in quantum chemistry simulations.