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arXiv:2508.21679 · Journal of Chemical Theory and Computation

A Simple Method for Seniority-Zero Quantum State Preparation

Michał Krompiec, Josh J. M. Kirsopp, A. M. Romero, Vicente P. Soloviev

Published Dec 26, 2025• 2 citations
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Abstract

Quantum phase estimation (QPE), the quantum algorithm for estimating eigenvalues of a given Hermitian matrix and preparing its eigenvectors, is considered the most promising approach to finding the ground states and their energies of electronic systems using a quantum computer. It requires, however, to be warm, starting from an initial state with a sufficiently high overlap with the ground state. For strongly correlated states, where QPE is expected to have an advantage over classical methods, preparation of such initial states requires deep quantum circuits and/or expensive hybrid quantum-classical optimization. It is well-known that the orbital-optimized paired coupled cluster doubles (oo-pCCD) method can describe the static correlation features of many strongly correlated singlet states. We show that pCCD and its unitary counterpart, UpCCD, become equivalent in the limit of small amplitudes or if the number of large amplitudes is below 5. We demonstrate that substituting leading oo-pCCD amplitudes into the UpCCD ansatz allows one to prepare high-fidelity singlet states for models of multiple-bond dissociation in ethene, ethyne, and dinitrogen, as well as for 1D Hubbard models at half-filling, with very shallow circuits. We envisage our method to be of general use for the approximate preparation of singlet states for QPE and related algorithms.

DOI
10.1021/acs.jctc.5c01452

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Quantum phase estimation (QPE), the quantum algorithm for estimating eigenvalues of a given Hermitian matrix and preparing its eigenvectors, is considered the most promising approach to finding the ground states and their energies of electronic systems using a quantum computer. It requires, however, to be warm, starting from an initial state with a sufficiently high overlap with the ground state. For strongly correlated states, where QPE is expected to have an advantage over classical methods, preparation of such initial states requires deep quantum circuits and/or expensive hybrid quantum-classical optimization. It is well-known that the orbital-optimized paired coupled cluster doubles (oo-pCCD) method can describe the static correlation features of many strongly correlated singlet states. We show that pCCD and its unitary counterpart, UpCCD, become equivalent in the limit of small amplitudes or if the number of large amplitudes is below 5. We demonstrate that substituting leading oo-pCCD amplitudes into the UpCCD ansatz allows one to prepare high-fidelity singlet states for models of multiple-bond dissociation in ethene, ethyne, and dinitrogen, as well as for 1D Hubbard models at half-filling, with very shallow circuits. We envisage our method to be of general use for the approximate preparation of singlet states for QPE and related algorithms.
The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi-reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum unitary variant of CC (UCC) with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the variational quantum eigensolver algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular, for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems, such as H4, H2O, and N2 molecules, as well as the one-dimensional periodic Fermi–Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry.
Many quantum algorithms rely on a quality initial state for optimal performance. Preparing an initial state for specific applications can considerably reduce the cost of probabilistic algorithms such as the well studied quantum phase estimation (QPE). Fortunately, in the application space of quantum chemistry, generating approximate wave functions for molecular systems is well studied, and quantum computing algorithms stand to benefit from importing these classical methods directly into a quantum circuit. In this work, we propose a state preparation method based on coupled cluster (CC) theory, which is a pillar of quantum chemistry on classical computers, by incorporating midcircuit measurements into the circuit construction. Currently, the most well studied state preparation method for quantum chemistry on quantum computers is the variational quantum eigensolver (VQE) with a unitary-CC with single- and double-electron excitation terms (UCCSD) ansatz whose operations are limited to unitary gates. We verify the accuracy of our state preparation protocol using midcircuit measurements by performing energy evaluation and state overlap computation for a set of small chemical systems. We further demonstrate that our approach leads to a reduction of the classical computation overhead, and the number of CNOT and T gates by 28 and 57% on average when compared against the standard VQE-UCCSD protocol.
We consider the question of how correlated the system hardness is between classical algorithms of electronic structure theory in ground state estimation and quantum algorithms. To define the system hardness for classical algorithms, we employ empirical criterion based on the deviation of electronic energies produced by coupled cluster and configuration interaction methods from the exact ones along multiple bonds dissociation in a set of molecular systems. For quantum algorithms, we have selected the Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE) methods. As characteristics of the system hardness for quantum methods, we analyzed circuit depths for the state preparation, the number of quantum measurements needed for the energy expectation value, and various cost characteristics for the Hamiltonian encodings via Trotter approximation and linear combination of unitaries (LCU). Our results show that the quantum resource requirements are mostly unaffected by classical hardness, with the only exception being the state preparation part, which contributes to both VQE and QPE algorithm costs. However, there are clear indications that constructing the initial state with a significant overlap with the true ground state is easier than obtaining the state with an energy expectation value within chemical precision. These results support optimism regarding the identification of a molecular system where a quantum algorithm excels over its classical counterpart, as quantum methods can maintain efficiency in classically challenging systems.
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.
Quantum algorithms for ground-state energy estimation of chemical systems require a high-quality initial state. However, initial state preparation is commonly either neglected entirely, or assumed to be solved by a simple product state like Hartree-Fock. Even if a nontrivial state is prepared, strong correlations render ground-state overlap inadequate for quality assessment. In this work, we address the initial state preparation problem with an end-to-end algorithm that and the quality of initial states, accomplishing the latter with a new metric—the energy distribution. To be able to prepare more complicated initial states, we introduce an implementation technique for states in the form of a sum of Slater determinants that exhibits significantly better scaling than all prior approaches. We also propose low-precision quantum phase estimation (QPE) for further state quality refinement. The complete algorithm is capable of generating high-quality states for energy estimation, and is shown in select cases to lower the overall estimation cost by several orders of magnitude when compared with the best single product state ansatz. More broadly, the energy distribution picture suggests that the goal of QPE should be reinterpreted as generating improvements compared to the energy of the initial state and other classical estimates: such an improvement can still be achieved even if QPE does not project directly onto the ground state. Finally, we show how the energy distribution can help in identifying potential quantum advantage. Published by the American Physical Society 2024
Quantum Phase Estimation (QPE) is the quantum algorithmic workhorse for computing ground state energies of quantum Hamiltonians with quantum computers. Ground state energy calculation of physical systems is perhaps the most promising use case for quantum computing in terms of scientific and commercial value with a plausible path to outperformance of classical alternatives. This path, however, hinges on the availability of initial states for QPE with significant overlap with the true ground state. Using extensive (classical) numerical computations, we study whether the NISQ-era algorithm VQE (Variational Quantum Eigensolver) could be used to efficiently prepare high-overlap states of disordered fully-connected anisotropic Heisenberg spin glass quantum Hamiltonians with up to 151515 qubits. We find that (i) -- consistent with widely held, but rarely numerically illustrated beliefs -- VQE is generally unable to efficiently converge to the ground state for our Hamiltonians, which is a well-known issue with VQE due to a variety of factors including vanishing gradients and local minima; (ii) low energy states do not necessarily have large ground-state overlap, but there is typically a correlation between the two measures; (iii) adding more than three layers to the VQE ansatz neither improves overlap nor the energies found; and (iv) the best-found overlap scaling as a function of the Hamiltonian system size is not strongly exponentially decreasing, suggesting potential for VQE to be a heuristic state preparation algorithm for QPE.