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arXiv:2408.14289 · Quantum Science and Technology

Non-iterative disentangled unitary coupled-cluster based on lie-algebraic structure

Mohammad Haidar, Olivier Adjoua, Siwar Badreddine, Alberto Peruzzo, Jean‐Philip Piquemal

Published Feb 7, 2025• 4 citations
arXivHTMLPDF

Abstract

Abstract Due to their non-iterative nature, fixed unitary coupled cluster (UCC) ansätze are attractive for performing quantum chemistry variational quantum eigensolver (VQE) computations as they avoid pre-circuit measurements on a quantum computer. However, achieving chemical accuracy for strongly correlated systems with UCC requires further inclusion of higher-order fermionic excitations beyond triples increasing circuit depth. We introduce k -non-iterative disentangled unitary coupled cluster (NI-DUCC), a fixed and non-iterative disentangled UCC compact ansatz, based on specific ‘ k ’ sets of ‘qubit’ excitations, eliminating the needs for fermionic-type excitations. These elements scale linearly ( O ( n ) ) by leveraging Lie algebraic structures, with n being the number of qubits. The key excitations are screened through specific selection criteria, including the enforcement of all symmetries, to ensure the construction of a robust set of generators. NI-DUCC employs ‘ k ’ products of the exponential of O ( n ) - anti-Hermitian Pauli operators, where each single Pauli string has a length p . This results in a fewer two-qubit CNOT gates circuit scaling, O ( k n p ) , suitable for hardware implementations. Tested on LiH, H 6 and BeH 2 , NI-DUCC-VQE achieves both chemical accuracy and rapid convergence even for molecules deviating significantly from equilibrium. It is hardware-efficient, reaching the exact full configuration interaction energy solution at specific layers, while reducing significantly the VQE optimization steps. NI-DUCC-VQE effectively addresses the gradient measurement bottleneck of ADAPT-VQE-like iterative algorithms, yet the classical computational cost of constructing the O ( n ) set of excitations increases exponentially with the number of qubits. We provide a first implementation for constructing the generators’ set, able to handle up to 20 qubits and discuss the efficiency perspectives.

DOI
10.1088/2058-9565/adb3c5

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Abstract Due to their non-iterative nature, fixed unitary coupled cluster (UCC) ansätze are attractive for performing quantum chemistry variational quantum eigensolver (VQE) computations as they avoid pre-circuit measurements on a quantum computer. However, achieving chemical accuracy for strongly correlated systems with UCC requires further inclusion of higher-order fermionic excitations beyond triples increasing circuit depth. We introduce k -non-iterative disentangled unitary coupled cluster (NI-DUCC), a fixed and non-iterative disentangled UCC compact ansatz, based on specific ‘ k ’ sets of ‘qubit’ excitations, eliminating the needs for fermionic-type excitations. These elements scale linearly ( O ( n ) ) by leveraging Lie algebraic structures, with n being the number of qubits. The key excitations are screened through specific selection criteria, including the enforcement of all symmetries, to ensure the construction of a robust set of generators. NI-DUCC employs ‘ k ’ products of the exponential of O ( n ) - anti-Hermitian Pauli operators, where each single Pauli string has a length p . This results in a fewer two-qubit CNOT gates circuit scaling, O ( k n p ) , suitable for hardware implementations. Tested on LiH, H 6 and BeH 2 , NI-DUCC-VQE achieves both chemical accuracy and rapid convergence even for molecules deviating significantly from equilibrium. It is hardware-efficient, reaching the exact full configuration interaction energy solution at specific layers, while reducing significantly the VQE optimization steps. NI-DUCC-VQE effectively addresses the gradient measurement bottleneck of ADAPT-VQE-like iterative algorithms, yet the classical computational cost of constructing the O ( n ) set of excitations increases exponentially with the number of qubits. We provide a first implementation for constructing the generators’ set, able to handle up to 20 qubits and discuss the efficiency perspectives.
We present high-precision quantum computing simulations of three-body atoms (He, H−^-−) and molecules (H2+_2^+2+​, HD+^++), the latter being studied beyond the Born-Oppenheimer approximation. The Non-Iterative Disentangled Unitary Coupled Cluster Variational Quantum Eigensolver (NI-DUCC-VQE) [M. Haidar et al., Quantum Sci. Technol. 10, 025031 (2025)] is used. By combining a first-quantized Hamiltonian with a Minimal Complete Pool (MCP) of Lie-algebraic excitations, we construct a compact ansatz with a gradient-independent construction, avoiding costly gradient evaluations and yielding efficient computational scaling with both basis size and electron number. It avoids barren plateaus and enables rapid convergence, achieving energy errors as low as 10−11^{-11}−11 a.u. with state fidelities only limited by arithmetic precision in only a few thousand function evaluations in all four systems. These results make three-body atoms and molecules excellent candidates for benchmarking and testing on current Noisy Intermediate-Scale Quantum (NISQ) devices. Further, our approach can be extended to more complex systems with larger basis sets, taking advantage of the efficient scaling of qubit requirements to study electronic correlations and non-adiabatic effects with high precision. We also demonstrate the applicability of NI-DUCC-VQE for simulating higher-order effects such as relativistic corrections and hyperfine interactions.
The Trotterized Unitary Coupled Cluster Single and Double (UCCSD) ansatz has recently attracted interest due to its use in Variation Quantum Eigensolver (VQE) molecular simulations on quantum computers. However, when the size of molecules increases, UCCSD becomes less interesting as it cannot achieve sufficient accuracy. Including higher-order excitations is therefore mandatory to recover the UCC's missing correlation effects. In this Letter, we extend the Trotterized UCC approach via the addition of (true) Triple T excitations introducing UCCSDT. We also include both spin and orbital symmetries. Indeed, in practice, these later help to reduce unnecessarily circuit excitations and thus accelerate the optimization process enabling to tackle larger molecules. Our initial numerical tests (12-14 qubits) show that UCCSDT improves the overall accuracy by at least two-orders of magnitudes with respect to standard UCCSD. Overall, the UCCSDT ansatz is shown to reach chemical accuracy and to be competitive with the CCSD(T) gold-standard classical method of quantum chemistry.
The Trotterized Unitary Coupled Cluster Single and Double (UCCSD) ansatz has recently attracted interest due to its use in Variation Quantum Eigensolver (VQE) molecular simulations on quantum computers. However, when the size of molecules increases, UCCSD becomes less interesting as it cannot achieve sufficient accuracy. Including higher-order excitations is therefore mandatory to recover the UCC's missing correlation effects. Here, we extend the Trotterized UCC approach via the addition of (true) Triple T excitations introducing UCCSDT. We also include both spin and orbital symmetries. Indeed, in practice, the latter help to reduce unnecessary circuit excitations and thus accelerate the optimization process enabling researchers to tackle larger molecules. Our initial numerical tests (12-14 qubits) show that UCCSDT improves the overall accuracy by at least two orders of magnitude with respect to standard UCCSD. Overall, the UCCSDT ansatz is shown to reach chemical accuracy and to be competitive with the CCSD(T) gold-standard classical method of quantum chemistry.
Unitary Coupled Cluster (UCC) approaches are an appealing route to utilising quantum hardware to perform quantum chemistry calculations, as quantum computers can in principle perform UCC calculations in a polynomially scaling fashion, as compared to the exponential scaling required on classical computers. Current noisy intermediate scale quantum (NISQ) computers are limited by both hardware capacity in number of logical qubits and the noise introduced by the deep circuits required for UCC calculations using the Variational Quantum Eigensolver (VQE) approach. We present a combined classical--quantum approach where a stochastic classical UCC pre-processing step is used to determine the important excitations in the UCC ansatz. The reduced number of selected excitations are then used in a UCC-based VQE calculation. This approach gives a systematically improvable approximation, and we show that significant reductions in quantum resources can be achieved, with simulations on the CH2_22​, N2_22​ and N2_22​H2_22​ molecules giving sub-milliHartree errors.
The unitary coupled cluster (UCC) algorithm is one of the most promising implementations of the variational quantum eigensolver for quantum computers. However, for large systems, the number of UCC factors leads to deep quantum circuits, which are prohibitive for execution on quantum hardware. To address this, circuit depth can be reduced at the cost of more measurements with a Taylor series expansion of UCC factors with small angles, while treating the large-angle factors exactly. We implement this approach to quadratic order (qUCC) for systems with strong correlations and systems where conventional methods like coupled cluster (CC) with low excitation levels fail, but UCC and qUCC perform well. We study hydrogen chains and the BeH2 molecule that allow us to change the degree of strong correlation due to geometrical distortions. We show, via a dramatic increase in number of factors able to handle exactly, a systematic convergence of these results as more exact UCC factors are included in the calculations -- the hardest to converge regime is in the crossover from weak to strong coupling. In all cases the total number of UCC factors needed to be treated exactly is much less than the total number of UCC factors available (typically about one-third to one-half of the total number of factors).
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.