Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground-state wave function directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B 99, 075119 (2019)] have demonstrated that the use of momentum-space representation, combined with a nonunitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more ``compact'' ground-state wave function, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the Ansatz-based quantum imaginary-time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to four orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength ($U/t=4$), enabling the use of shallower quantum circuits for wave-function Ans\"atzes. In addition, we propose a more efficient implementation of the quantum imaginary-time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers.
Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground-state wave function directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B 99, 075119 (2019)] have demonstrated that the use of momentum-space representation, combined with a nonunitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more ``compact'' ground-state wave function, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the Ansatz-based quantum imaginary-time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to four orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength (U/t=4), enabling the use of shallower quantum circuits for wave-function Ans"atzes. In addition, we propose a more efficient implementation of the quantum imaginary-time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers.
Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground state wavefunction directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B, 99(7), 075119, (2019)] have demonstrated that the use of momentum-space representation, combined with a non-unitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more compact ground state wavefunction, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the ansatz-based quantum imaginary time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to 4 orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength (U/t=4), enabling the use of shallower quantum circuits for wavefunction ansatzes. In addition, we propose a more efficient implementation of the quantum imaginary time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers.
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.
NISQ era devices suffer from a number of challenges like limited qubit connectivity, short coherence times and sizable gate error rates. Thus, quantum algorithms are desired that require shallow circuit depths and low qubit counts to take advantage of these devices. We attempt to realize this with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically 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 transcorelated Hamiltonians are able to describe both ground and excited states of molecular systems in a balanced manner. Using the fermionic-ADAPT-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 much larger cc-pVTZ basis. This leads to a potential reduction in the number of required CNOT gates by more than three orders of magnitude for the chemical species studied in this work. Furthermore, using the qEOM formalism in conjunction with the transcorrelated Hamiltonian, we reduce the errors in excitation energies 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.
Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields, including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and limited connectivity) hamper the straightforward implementation of most quantum algorithms and call for more noise-resilient solutions. In quantum chemistry, the limited number of available qubits and gate operations is particularly restrictive since, for each molecular orbital, one needs, in general, two qubits. In this study, we propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach, which transfers -- without any approximation -- correlation from the wavefunction directly into the Hamiltonian, thus reducing the number of resources needed to achieve accurate results with noisy, near-term quantum devices. In particular, we show that the exact transcorrelated approach not only allows for more shallow circuits but also improves the convergence towards the so-called basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, therefore, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using just 4 and 6 qubits, respectively. Conventional methods require at least ten times more qubits for the same accuracy.
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.
Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and connectivity) hamper the implementation of most quantum algorithms and call for more noise-resilient solutions. We propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach to target these major roadblocks directly. This method transfers, without any approximation, correlations from the wave function directly into the Hamiltonian, thus reducing the resources needed to achieve accurate results with noisy quantum devices. We show that the TC approach allows for shallower circuits and improves the convergence toward the complete basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, thus, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using two and four qubits, respectively. To demonstrate our approach's current and near-term potential, we perform hardware experiments, where our results confirm that the TC method paves the way toward accurate quantum chemistry calculations already on today's quantum hardware.