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Shallow-circuit variational quantum eigensolver based on symmetry-inspired Hilbert space partitioning for quantum chemical calculations

Feng Zhang, Niladri Gomes, Noah Berthusen, Peter P. Orth, Cai‐Zhuang Wang, Kai‐Ming Ho, Yongxin Yao

Published Jan 1, 2021• 27 citations
OpenAlex

Abstract

Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4–8. With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H4-square, H4-chain, and H6-hexagon molecules, while the number of cnot gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient “score” parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using the H4 square and H4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.

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Point group symmetry has been exploited for designing compact ansätze in the Variational Quantum Eigensolver (VQE), thereby facilitating the computation of molecular energy levels on current Noisy Intermediate-Scale Quantum (NISQ) devices. However, the widely used Symmetry-reduced Unitary Coupled Cluster Singles and Doubles (SymUCCSD) is restricted to molecules with Abelian point groups and often yields deficient ansätze for non-Abelian molecular systems. In this paper, we propose Hamiltonian-informed UCCSD (HiUCCSD), a novel shallow ansatz engineered based on the intrinsic information encoded in the molecular Hamiltonian. We theoretically prove the effectiveness of HiUCCSD for molecules belonging to Abelian point groups. Furthermore, numerical results for 10 molecular systems with distinct symmetries demonstrate that HiUCCSD may also be applicable to non-Abelian point group systems. Compared with the standard UCCSD ansatz, HiUCCSD reduces the parameter count and circuit size of VQE by 18-83% and 26-83%, respectively, and shrinks the size of the excitation operator pool for Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT)-VQE by 27-84% across the studied molecules. Given its superior performance and broad applicability, we expect that HiUCCSD will facilitate the realization of large-scale molecular VQE implementations.

Journal of Chemical Theory and Computation•Jun 2, 2026•0 citations
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4–8. With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H4-square, H4-chain, and H6-hexagon molecules, while the number of cnot gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient “score” parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using the H4 square and H4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing.Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace.In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4-8.With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H 4 -square, H 4 -chain, and H 6 -hexagon molecules, while the number of CNOT gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35.Furthermore, we introduced an efficient "score" parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first.Using the H 4 square and H 4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.
A variational quantum eigensolver (VQE) designed for noisy intermediate-scale quantum (NISQ) computers is a promising hybrid quantum-classical algorithm for quantum chemical calculations (QCC). Its accuracy and computational cost strongly depend on an ansatz, and unitary coupled-cluster (UCC) ansatze are well-known chemistry-inspired ansatze that can achieve a high accuracy for QCC. However, UCC-based quantum circuits have a high depth and a lot of 2-qubit controlled NOT (CNOT) gates which are sensitive to noises, leading to the low accuracy of VQE running on NISQ computers. To address this issue, we propose a novel offline quantum circuit pruning technique that reduces the number of CNOT gates and depth of UCC-based quantum circuits by reducing the number of excitation operators per parameter to a user-specified value. It is applied only once before VQE is executed (i.e., offline), which is the advantage over state-of-the-art quantum circuit pruning techniques. The evaluation using a quantum computer simulator with a noise model based on a real NISQ device shows that our offline pruning technique improves the accuracy of UCCSD-VQE while sustaining the potential accuracy of the UCCSD ansatz and has a synergy with a well-known noise mitigation method called zero noise extrapolation (ZNE).
High Resolution Image Download MS PowerPoint Slide Quantum computers have the potential to efficiently solve the electronic structure problem but are currently limited by noise and shallow circuits. We present an enhanced Variational Quantum Eigensolver (VQE) ansatz based on the Qubit Coupled Cluster (QCC) approach that requires optimization of only n parameters, where n is the number of Pauli string generators, rather than the typical n + 2 m parameters, where m is the number of qubits. We evaluate the ground state energies and molecular dissociation curves of strongly correlated molecules, namely O 3 and Li 4, using active spaces of varying sizes in conjunction with our enhanced QCC ansatz, Unitary Coupled Cluster Single–Double (UCCSD) ansatz, and the classical Coupled Cluster Singles and Doubles (CCSD) method. Compared to UCCSD, our approach significantly reduces the number of parameters while maintaining high accuracy. Numerical simulations demonstrate the effectiveness of our approach, and experiments on superconducting and trapped-ion quantum computers showcase its practicality on real hardware. By eliminating the need for symmetry-restoring gates and reducing the number of parameters, our enhanced QCC ansatz enables accurate quantum chemistry calculations on near-term quantum devices for strongly correlated systems.
The exact evaluation of the molecular ground state in quantum chemistry requires an exponentially increasing computational cost. Quantum computation is a promising way to overcome the exponential problem using polynomial-time quantum algorithms. A quantum-classical hybrid optimization scheme known as the variational quantum eigensolver is preferred for noisy intermediate-scale quantum devices. However, the circuit depth becomes one of the bottlenecks of its application to large molecules of more than 20 qubits. In this work, we employ point-group symmetry to reduce the number of operators in constructing ansatz so as to achieve a more compact quantum circuit. We illustrate this methodology with a series of molecules ranging from LiH (12 qubits) to C2H4{\mathrm{C}}_{2}{\mathrm{H}}_{4}C2​H4​ (28 qubits). A significant reduction of up to 82% of the operator numbers is reached on C2H4{\mathrm{C}}_{2}{\mathrm{H}}_{4}C2​H4​. This also sheds light onto further work in this direction to construct even shallower ansatz with enough expressive power and simulate even larger scale systems.
Quantum computation represents a revolutionary approach to solving problems in quantum chemistry. However, due to the limited quantum resources in the current noisy intermediate-scale quantum (NISQ) devices, quantum algorithms for large chemical systems remain a major task. In this work, we demonstrate that the circuit depth of the unitary coupled cluster (UCC) and UCC-based ansatzes in the algorithm of the variational quantum eigensolver can be significantly reduced by an energy-sorting strategy. Specifically, subsets of excitation operators are first prescreened from the operator pool according to its contribution to the total energy. The quantum circuit ansatz is then iteratively constructed until convergence of the final energy to a typical accuracy. For demonstration, this method has been successfully applied to molecular and periodic systems. Particularly, a reduction of 50%-98% in the number of operators is observed while retaining the accuracy of the original UCCSD operator pools. This method can be straightforwardly extended to general parametric variational ansatzes.
Point group symmetry has been exploited for designing compact ansätze in the Variational Quantum Eigensolver (VQE), thereby facilitating the computation of molecular energy levels on current Noisy Intermediate-Scale Quantum (NISQ) devices. However, the widely used Symmetry-reduced Unitary Coupled Cluster Singles and Doubles (SymUCCSD) is restricted to molecules with Abelian point groups and often yields deficient ansätze for non-Abelian molecular systems. In this paper, we propose Hamiltonian-informed UCCSD (HiUCCSD), a novel shallow ansatz engineered based on the intrinsic information encoded in the molecular Hamiltonian. We theoretically prove the effectiveness of HiUCCSD for molecules belonging to Abelian point groups. Furthermore, numerical results for 10 molecular systems with distinct symmetries demonstrate that HiUCCSD may also be applicable to non-Abelian point group systems. Compared with the standard UCCSD ansatz, HiUCCSD reduces the parameter count and circuit size of VQE by 18-83% and 26-83%, respectively, and shrinks the size of the excitation operator pool for Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT)-VQE by 27-84% across the studied molecules. Given its superior performance and broad applicability, we expect that HiUCCSD will facilitate the realization of large-scale molecular VQE implementations.