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Journal of Chemical Theory and Computation

Efficient Energy Measurement of Chemical Systems via One-Particle Reduced Density Matrix: A NOF-VQE Approach for Optimized Sampling

Juan Felipe Huan Lew-Yee, Mario Piris

Published Feb 21, 2025• 2 citations
Publisher

Abstract

In this work, we explore the use of the one-particle reduced density matrix (1RDM) to streamline energy measurements of chemical systems on quantum computers, particularly within the variational quantum eigensolver (VQE) framework. This approach leverages the existence of an exact energy functional of the 1RDM, enabling a reduction in both the number of expectation values to measure and the number of circuits to execute, thereby optimizing quantum resource usage. Specifically, sampling the 1RDM involves measuring only elements, which is significantly fewer than the required for the Hamiltonian’s expectation value ⟨ Ĥ ⟩. We demonstrate our approach by harnessing the well-established natural orbital functional (NOF) theory, using the natural orbitals and occupation numbers derived from the diagonalization of the 1RDM measured from the quantum computer. Starting with the H 2 system, we validate the accuracy of our method by comparing the energy derived from NOF approximations applied to the exact wave function with the value obtained from ⟨ Ĥ ⟩. This is followed by an optimization of the gate parameters by minimizing the energy using the NOF approximations as the objective function. The analysis is extended to larger systems, such as LiH, Li 2, OH –, FH, NeH +, and F 2 using a wave function ansatz with single and double excitation gates. This NOF-based method reduces the scaling cost of circuit executions compared to standard VQE implementations, achieving around 90% savings in the systems used in this work. Overall, by using a well-performing NOF as the objective function, the proposed NOF-VQE demonstrates the viability of NOF approximations for obtaining accurate energies in the noisy intermediate-scale quantum era and underscores the potential for developing new functionals tailored to quantum computing applications.

DOI
10.1021/acs.jctc.4c01734

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In this work, we explore the use of the one-particle reduced density matrix (1RDM) to streamline energy measurements of chemical systems on quantum computers, particularly within the variational quantum eigensolver (VQE) framework. This approach leverages the existence of an exact energy functional of the 1RDM, enabling a reduction in both the number of expectation values to measure and the number of circuits to execute, thereby optimizing quantum resource usage. Specifically, sampling the 1RDM involves measuring only elements, which is significantly fewer than the required for the Hamiltonian’s expectation value ⟨ Ĥ ⟩. We demonstrate our approach by harnessing the well-established natural orbital functional (NOF) theory, using the natural orbitals and occupation numbers derived from the diagonalization of the 1RDM measured from the quantum computer. Starting with the H 2 system, we validate the accuracy of our method by comparing the energy derived from NOF approximations applied to the exact wave function with the value obtained from ⟨ Ĥ ⟩. This is followed by an optimization of the gate parameters by minimizing the energy using the NOF approximations as the objective function. The analysis is extended to larger systems, such as LiH, Li 2, OH –, FH, NeH +, and F 2 using a wave function ansatz with single and double excitation gates. This NOF-based method reduces the scaling cost of circuit executions compared to standard VQE implementations, achieving around 90% savings in the systems used in this work. Overall, by using a well-performing NOF as the objective function, the proposed NOF-VQE demonstrates the viability of NOF approximations for obtaining accurate energies in the noisy intermediate-scale quantum era and underscores the potential for developing new functionals tailored to quantum computing applications.
We present a quantum-classical algorithm for electronic structure calculations that dramatically reduces the quantum measurement cost of variational quantum eigensolver (VQE) approaches. While conventional VQE methods require measurements scaling as O(M^4) with system size M, the proposed occupation-number VQE (ON-VQE) reduces this cost to O(M/2) by avoiding reduced density matrix (RDM) measurements and relying exclusively on ONs. The method exploits only the diagonal elements of the one-particle RDM in the natural orbital representation, where occupations are obtained directly from computational-basis measurement outcomes. By restricting the variational ansatz to double excitations within orbital subspaces associated with electron pairs, the required measurements can be grouped into a small number of qubit-wise commuting observables, yielding an efficient and scalable measurement strategy. The approach is validated through simulations and executions on quantum hardware for the cubic H8_88​ cluster, demonstrating the feasibility of extracting accurate ONs from quantum measurements and evaluating electronic energies within the natural orbital functional (NOF) framework. Across representative molecular systems, the extracted ONs enable accurate energy evaluation with state-of-the-art NOFs while maintaining a dramatically reduced measurement cost. These results establish a scalable route toward quantum simulation of strongly correlated electronic systems, demonstrating that accurate electronic energies can be obtained from quantum measurements of ONs alone.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) computers. It is promising for quantum chemical calculations (QCC) because it can calculate the ground-state energy of a target molecule. Although VQE has a potential to achieve a higher accuracy than classical approximation methods in QCC, it is challenging to achieve it on current NISQ computers due to the significant impact of noises. Density matrix embedding theory (DMET) is a well-known technique to divide a molecule into multiple fragments, which is available to mitigate the noise impact on VQE. However, our preliminary evaluation shows that the naive combination of DMET and VQE does not outperform a gold standard classical method. In this work, we present three approaches to mitigate the noise impact for the DMET+VQE combination. (1) The size of quantum circuits used by VQE is decreased by reducing the number of bath orbitals which represent interactions between multiple fragments in DMET. (2) Reduced density matrices (RDMs), which are used to calculate a molecular energy in DMET, are calculated accurately based on expectation values obtained by executing quantum circuits using a noise-less quantum computer simulator. (3) The parameters of a quantum circuit optimized by VQE are refined with mathematical post-processing. The evaluation using a noisy quantum computer simulator shows that our approaches significantly improve the accuracy of the DMET+VQE combination. Moreover, we demonstrate that on a real NISQ device, the DMET+VQE combination applying our three approaches achieves a higher accuracy than the gold standard classical method.
The variational quantum eigensolver (VQE) is one of the most promising quantum algorithms for the near-term noisy intermediate-scale quantum (NISQ) devices. The VQE typically involves finding the minimum energy of a quantum Hamiltonian through classical optimization of a parametrized quantum ansatz. One of the bottlenecks in VQEs is the number of quantum circuits to be measured. In this work, we propose a physically intuitive truncation technique that starts the optimization procedure with a truncated Hamiltonian and then gradually transitions to the optimization for the original Hamiltonian via an operator classification method. This strategy allows us to reduce the required number of evaluations for the expectation value of Hamiltonian on a quantum computer. The reduction in required quantum resources for our strategy is substantial and likely scales with the system size. With numerical simulations, we demonstrate our method for various molecular systems.
The variational quantum eigensolver (VQE) has emerged as one of the leading quantum algorithms for solving electronic structure problems on near-term noisy intermediate-scale quantum devices. However, its practical application to quantum chemistry remains challenging due to the limited coherence time, imperfect quantum gate fidelity, and the large number of measurements required, which together confine current electronic structure simulations to relatively small active spaces. In this work, we present an orbital-optimized VQE framework based on orbital compression, designed to improve the accuracy of electronic structure calculations while maintaining relatively small active spaces. Frozen natural orbitals (FNO) and split virtual orbitals (SVO) are first employed to construct compact active spaces for VQE simulations, leading to the FNO/SVO-VQE approach. Orbital optimization is then incorporated to further recover electron correlation effects, resulting in the FNO/SVO-OO-VQE methods. We apply the proposed method to simulate potential energy surfaces for molecular dissociation and the activation energy of formaldehyde decomposition. Numerical results demonstrate that both FNO-OO-VQE and SVO-OO-VQE improve the variational accuracy while substantially reducing measurement cost.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) computers. It is promising for quantum chemical calculations (QCC) because it can calculate the ground-state energy of molecules. Although VQE has a theoretical potential to achieve a higher accuracy than classical approximation methods, it is challenging to achieve it on current NISQ computers due to the significant impact of noise. Density matrix embedding theory (DMET) is a well-known technique that can be used to decompose a molecule into multiple fragments, which is available to mitigate the noise impact on VQE. However, our preliminary evaluation shows that the naive combination of DMET and VQE hardly mitigates it. In this work, we present two proposals to mitigate the noise impact on the DMET+VQE combination. (1) The size of quantum circuits used by VQE is decreased by reducing the number of bath orbitals which represent interactions between multiple fragments in DMET. (2) While VQE is executed on a NISQ computer, reduced density matrices (RDMs), which are used to calculate the energy of fragments in DMET, are calculated accurately by executing quantum circuits using a noise-less quantum computer simulator. The evaluation using a noisy quantum computer simulator shows that our proposals improve the accuracy of the DMET+VQE combination by an order of magnitude.
Quantum computing is finding increasingly more applications in quantum chemistry, particularly to simulate electronic structure and molecular properties of simple systems. The transformation of a molecular Hamiltonian from the fermionic space to the qubit space results in a series of Pauli strings. Calculating the energy then involves evaluating the expectation values of each of these strings, which presents a significant bottleneck for applying variational quantum eigensolvers (VQEs) in quantum chemistry. Unlike fermionic Hamiltonians, the terms in a qubit Hamiltonian are additive. This work leverages this property to introduce a novel method for extracting information from the partial qubit Hamiltonian, thereby enhancing the efficiency of VQEs. This work introduces the SHARC-VQE (Simplified Hamiltonian Approximation, Refinement, and Correction-VQE) method, where the full molecular Hamiltonian is partitioned into two parts based on the ease of quantum execution. The easy-to-execute part constitutes the partial Hamiltonian, and the remaining part, while more complex to execute, is generally less significant. The latter is approximated by a refined operator and added up as a correction into the partial Hamiltonian. SHARC-VQE significantly reduces computational costs for molecular simulations. The cost of a single energy measurement can be reduced from O(N4ϵ2) to O(1ϵ2) for a system of N qubits and accuracy ϵ, while the overall cost of VQE can be reduced from O(N7ϵ2) to O(N3ϵ2). Furthermore, measurement outcomes using SHARC-VQE are less prone to errors induced by noise from quantum circuits, reducing the errors from 20%-40% to 5%-10% without any additional error correction or mitigation technique. In addition, the SHARC-VQE is demonstrated as an initialization technique, where the simplified partial Hamiltonian is used to identify an optimal starting point for a complex problem. Overall, this method improves the efficiency of VQEs and enhances the accuracy and reliability of quantum simulations by mitigating noise and overcoming computational challenges.