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Journal of Chemical Theory and Computation

Accuracy and Resource Advantages of Quantum Eigenvalue Estimation with Non-Hermitian Transcorrelated Electronic Hamiltonians

Alexey Uvarov, Artur F. Izmaylov

Published Jun 16, 2026• 0 citations
Publisher

Abstract

In electronic structure calculations, the transcorrelated method consists of transforming the Hamiltonian so as to remove the Coulomb cusp in its eigenfunctions. As a result, the wave function can be described more accurately without increasing the size of the basis set. However, the transcorrelated Hamiltonian is non-Hermitian and nonnormal, which makes many common quantum algorithms inapplicable. Recently, a quantum eigenvalue estimation (QEVE) algorithm was proposed for non-Hermitian Hamiltonians with real spectra [FOCS 65, 1051 (2024)]. Although the asymptotic scaling of this algorithm with the desired accuracy is shown to be optimal, the constant factor in its complexity scaling has not yet been analyzed. Here, we investigate the cost of QEVE applied to transcorrelated electronic Hamiltonians of Li, Be, B, C, and N atoms and compare it to the cost of applying standard qubitization to nontranscorrelated Hamiltonians. We find that with the x TC approximation, the T gate count of QEVE in the minimal STO-6G basis is between those of standard qubitization in the cc-pVTZ and cc-pVQZ bases. The accuracy of the transcorrelated energy differs between systems: for Li and Be, it is more accurate than the cc-pVQZ energy, while for larger atoms, the error is between those of the cc-pVDZ and cc-pVTZ energies.

DOI
10.1021/acs.jctc.6c00274

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In electronic structure calculations, the transcorrelated method consists of transforming the Hamiltonian so as to remove the Coulomb cusp in its eigenfunctions. As a result, the wave function can be described more accurately without increasing the size of the basis set. However, the transcorrelated Hamiltonian is non-Hermitian and nonnormal, which makes many common quantum algorithms inapplicable. Recently, a quantum eigenvalue estimation (QEVE) algorithm was proposed for non-Hermitian Hamiltonians with real spectra [FOCS 65, 1051 (2024)]. Although the asymptotic scaling of this algorithm with the desired accuracy is shown to be optimal, the constant factor in its complexity scaling has not yet been analyzed. Here, we investigate the cost of QEVE applied to transcorrelated electronic Hamiltonians of Li, Be, B, C, and N atoms and compare it to the cost of applying standard qubitization to nontranscorrelated Hamiltonians. We find that with the x TC approximation, the T gate count of QEVE in the minimal STO-6G basis is between those of standard qubitization in the cc-pVTZ and cc-pVQZ bases. The accuracy of the transcorrelated energy differs between systems: for Li and Be, it is more accurate than the cc-pVQZ energy, while for larger atoms, the error is between those of the cc-pVDZ and cc-pVTZ energies.
In electronic structure calculations, the transcorrelated method consists in transforming the Hamiltonian so as to remove the Coulomb cusp in its eigenfunctions. As a result, the wavefunction can be described more accurately without increasing the size of the basis set. However, the transcorrelated Hamiltonian is non-Hermitian and non-normal, which makes many common quantum algorithms inapplicable. Recently, a quantum eigenvalue estimation algorithm (QEVE) was proposed for non-Hermitian Hamiltonians with real spectra [FOCS 65, 1051 (2024)]. Although the asymptotic scaling of this algorithm with the desired accuracy is shown to be optimal, the constant factor in its complexity scaling has not been analyzed. Here we investigate the cost of QEVE applied to transcorrelated electronic Hamiltonians of second-row atoms and compare it to the cost of applying standard qubitization to non-transcorrelated Hamiltonians. We find that, with the xTC approximation, the T gate count of QEVE in the minimal STO-6G basis is between those of standard qubitization in the cc-pVTZ and cc-pVQZ bases. The accuracy of the transcorrelated energy differs between systems: for Li and Be, it is more accurate than the cc-pVQZ energy, while for larger atoms, the error gradually increases, exceeding the cc-pVDZ level for O, F, and Ne.
The electron--electron Coulomb singularity forces cusp structure in electronic wavefunctions. This cusp structure limits wavefunction regularity and slows the convergence of expansions in smooth bases, requiring large basis sets to reach a target accuracy. The transcorrelated (TC) method incorporates this short-range behavior into the Hamiltonian, thereby improving the regularity of the transformed wavefunction and accelerating convergence toward the complete-basis-set limit. The resulting Hamiltonian is non-Hermitian and non-normal, which limits the direct applicability of standard quantum algorithms. For such non-normal operators, the complexity of several recent quantum algorithms depends explicitly on quantities related to eigenvector conditioning and eigenvalue sensitivity. Understanding these quantities is essential for assessing both algorithmic complexity and quantum resource requirements. In this work, we address these issues in two stages. We first study a solvable model, which provides a controlled setting for isolating how transcorrelation affects basis convergence and eigenvector conditioning. Within this setting, we investigate two distinct one-parameter Jastrow correlators. We derive the asymptotic energy-error scaling: N−1/2N^{-1/2}N−1/2 for the bare projection and N−3/2N^{-3/2}N−3/2 for both TC correlators at generic fixed parameters, with NNNdenoting the basis size. Furthermore, we show that this faster convergence can yield a polynomial improvement in the asymptotic query-complexity upper bound. We then extend the analysis to second-quantized electronic-structure Hamiltonians and use particle-number-sector diagnostics to distinguish target-sector from auxiliary-sector conditioning. Our work provides a framework for understanding the central accuracy--cost trade-off in TC Hamiltonians.
Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.
Abstract. Many problems in linear algebra—such as those arising from non-Hermitian physics, transcorrelated quantum chemistry, and differential equations—can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task because eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate [Formula: see text] Fourier coefficients with [Formula: see text] gates compared to prior approaches with linear cost.
Many problems in linear algebra-such as those arising from non-Hermitian physics, transcorrelated quantum chemistry and differential equations-can be solved on a quan-tum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited to this task, as eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum super-position of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generatennnFourier coefficients with 0 (poly log (nnn)) gates compared to prior approaches with linear cost.
Accurately treating electron correlation in the wavefunction is a key challenge for both classical and quantum computational chemistry. Classical methods have been developed which explicitly account for this correlation by incorporating inter-electronic distances into the wavefunction. The transcorrelated method transfers this explicit correlation from the wavefunction to a transformed, non-Hermitian Hamiltonian, whose right-hand eigenvectors become easier to obtain than those of the original Hamiltonian. In this work, we show that the transcorrelated method can reduce the resources required to obtain accurate energies from electronic structure calculations on quantum computers. We overcome the limitations introduced by the non-Hermitian Hamiltonian by using quantum algorithms for imaginary time evolution.