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arXiv:2502.17587 · Journal of Chemical Theory and Computation

Optimization of the Qubit Coupled Cluster Ansatz on Classical Computers

Ilya G. Ryabinkin, S. M. Hosseini Jenab, Scott N. Genin

Published Jun 17, 2025• 1 citation
arXivHTMLPDF

Abstract

Immense interest in quantum computing has prompted the development of electronic structure methods that are suitable for quantum hardware. However, the slow pace at which quantum hardware progresses forces researchers to implement their ideas on classical computers despite the obvious loss of any “quantum advantage.” As a result, the so-called q uantum-inspired methods emerge. They allow one to look at the electronic structure problem from a different angle, yet to fully exploit their capacity, efficient implementations are highly desirable. Here, we report two schemes for improving the amplitude optimization in the iterative qubit coupled cluster (iQCC) method: a variational quantum eigensolver-type approach, which is based on the qubit coupled cluster (QCC) Ansatz. Our first scheme approximates the QCC unitary as a sum of symmetrical polynomials of generators up to a given order. The resulting energy expression allows for flexible control of computational complexity via the order parameter. It also guarantees smoothness of trial energies and their derivatives, which is important for gradient-based optimization strategies. The second scheme limits the size of the expansion space in which the QCC unitary is generated. It provides better control of memory requirements but, in general, may lead to the nonsmooth variation of energy estimates upon changes in amplitudes. It can be used, however, to extrapolate energies for a given set of amplitudes toward the exact QCC value. Both schemes allow for a larger number of generators to be included in the QCC form compared to the exact formulation. This reduces the number of iterations in the iQCC method and/or leads to higher accuracy. We assess the capabilities of the new schemes to perform QCC amplitudes optimization for a few molecular systems: dinitrogen (N 2, 16 qubits), water (H 2 O, 36 qubits), and tris(2-(2,4-difluorophenyl)pyridine) iridium(III), (Ir(F 2 ppy) 3, 80 qubits).

DOI
10.1021/acs.jctc.5c00345

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Immense interest in quantum computing has prompted the development of electronic structure methods that are suitable for quantum hardware. However, the slow pace at which quantum hardware progresses forces researchers to implement their ideas on classical computers despite the obvious loss of any “quantum advantage.” As a result, the so-called q uantum-inspired methods emerge. They allow one to look at the electronic structure problem from a different angle, yet to fully exploit their capacity, efficient implementations are highly desirable. Here, we report two schemes for improving the amplitude optimization in the iterative qubit coupled cluster (iQCC) method: a variational quantum eigensolver-type approach, which is based on the qubit coupled cluster (QCC) Ansatz. Our first scheme approximates the QCC unitary as a sum of symmetrical polynomials of generators up to a given order. The resulting energy expression allows for flexible control of computational complexity via the order parameter. It also guarantees smoothness of trial energies and their derivatives, which is important for gradient-based optimization strategies. The second scheme limits the size of the expansion space in which the QCC unitary is generated. It provides better control of memory requirements but, in general, may lead to the nonsmooth variation of energy estimates upon changes in amplitudes. It can be used, however, to extrapolate energies for a given set of amplitudes toward the exact QCC value. Both schemes allow for a larger number of generators to be included in the QCC form compared to the exact formulation. This reduces the number of iterations in the iQCC method and/or leads to higher accuracy. We assess the capabilities of the new schemes to perform QCC amplitudes optimization for a few molecular systems: dinitrogen (N 2, 16 qubits), water (H 2 O, 36 qubits), and tris(2-(2,4-difluorophenyl)pyridine) iridium(III), (Ir(F 2 ppy) 3, 80 qubits).
Abstract The iterative qubit coupled cluster (iQCC) method is a systematic variational approach to solve the electronic structure problem on universal quantum computers. It is able to use arbitrarily shallow quantum circuits at expense of iterative canonical transformation of the Hamiltonian and rebuilding a circuit. Here we present a variety of a posteriori corrections to the iQCC energies to reduce the number of iterations to achieve the desired accuracy. Our energy corrections are based on a low-order perturbation theory series that can be efficiently evaluated on a classical computer. Moreover, capturing a part of the total energy perturbatively, allows us to formulate the qubit active-space concept, in which only a subset of all qubits is treated variationally. As a result, further reduction of quantum resource requirements is achieved. We demonstrate the utility and efficiency of our approach numerically on the examples of 10-qubit N 2 molecule dissociation, the 24-qubit H 2 O symmetric stretch, and 56-qubit singlet-triplet gap calculations for the technologically important complex, tris-(2-phenylpyridine)iridium(III) Ir(ppy) 3 .
A unitary coupled cluster (UCC) form for the wave function in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled cluster excitations, practically, the accuracy of this approach depends on the number and type of terms are included in the wave function parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed Fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems, we introduce a qubit coupled cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H 2 and LiH molecules with chemical accuracy, ≤1 kcal/mol, and a symmetric water dissociation curve.
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.
Abstract The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
We introduce the multistate iterative qubit coupled cluster (MS-iQCC) method, a quantum-inspired algorithm that runs efficiently on classical hardware and is designed to predict both ground and excited electronic states of molecules. Accurate excited-state energetics are essential for interpreting spectroscopy and chemical reactivity, but standard electronic structure methods are either too computationally expensive for larger systems or lose reliability in the presence of strong electron correlation. MS-iQCC addresses this challenge by simultaneously optimizing multiple electronic states in a single, state-averaged procedure that treats ground and excited states on equal footing. This removes the energetic bias that is introduced when excited states are computed one at a time and constrained to remain orthogonal to previously optimized states. The approach supports multireference electronic structure by allowing multideterminantal initial guesses and by adaptively building a compact exponential ansatz from a pool of qubit excitation generators. We apply MS-iQCC to H4_44​, H2_22​O, N2_22​, and C2_22​, including strongly correlated geometries, and observe robust convergence of all targeted state energies to chemically meaningful accuracy across their potential energy surfaces.
Ab initio quantum chemistry simulations are crucially useful for the accurate prediction andunderstanding of chemical phenomena. The quality of such simulations rely heavily on precise solutions to the electronic structure problem. To facilitate the approximate yet efficient solving of the electronic structure problem, many methods rely on a set of problem assumptions, such as the validity of truncations of the fermionic excitation hierarchy, or active space selections. In this work, we describe black-box algorithms for the electronic structure problem, namely the iterative qubit coupled cluster (iQCC) algorithm along with a few of its variants and extensions. The iQCC method is a variational adaptive technique, where an iteratively updated effective Hamiltonian directly guides the selection of transformation generators of the next iteration. We describe the algorithm in detail, and introduce variations of the algorithm to alleviate the proliferation of terms in the effective Hamiltonians. Finally, we describe how the algorithm can be modified to perform simultaneuous determination of ground and excited states. These methods are amenable to implementation using near-term quantum computing devices, and with certain restrictions on meta-parameters, can be implemented as fully classical “quantum-inspired” algorithms. ii