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arXiv:2008.08694

Momentum-Space Unitary Coupled Cluster and Translational Quantum Subspace Expansion for Periodic Systems on Quantum Computers

David Zsolt Manrique, Irfan T. Khan, Kentaro Yamamoto, Vijja Wichitwechkarn, David Muñoz Ramo

Published Aug 19, 2020• 18 citations
arXivHTMLPDF

Abstract

We demonstrate the use of the Variational Quantum Eigensolver (VQE) to simulate solid state crystalline materials. We adapt the Unitary Coupled Cluster ansatz to periodic boundary conditions in real space and momentum space representations and directly map complex cluster operators to a quantum circuit ansatz to take advantage of the reduced number of excitation operators and Hamiltonian terms due to momentum conservation. To further reduce required quantum resources, such as the number of UCCSD amplitudes, circuit depth, required number of qubits and number of measurement circuits, we investigate a translational Quantum Subspace Expansion method (TransQSE) for the localized representation of the periodic Hamiltonian. Additionally, we also demonstrate an extension of the point group symmetry based qubit tapering method to periodic systems. We compare accuracy and computational costs for a range of geometries for 1D chains of dimerized hydrogen, helium and lithium hydride with increasing number of momentum space grid points and also demonstrate VQE calculations for 2D and 3D hydrogen and helium lattices. Our presented strategies enable the use of near-term quantum hardware to perform solid state simulation with variational quantum algorithms.

DOI
10.48550/arxiv.2008.08694

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We demonstrate the use of the Variational Quantum Eigensolver (VQE) to simulate solid state crystalline materials. We adapt the Unitary Coupled Cluster ansatz to periodic boundary conditions in real space and momentum space representations and directly map complex cluster operators to a quantum circuit ansatz to take advantage of the reduced number of excitation operators and Hamiltonian terms due to momentum conservation. To further reduce required quantum resources, such as the number of UCCSD amplitudes, circuit depth, required number of qubits and number of measurement circuits, we investigate a translational Quantum Subspace Expansion method (TransQSE) for the localized representation of the periodic Hamiltonian. Additionally, we also demonstrate an extension of the point group symmetry based qubit tapering method to periodic systems. We compare accuracy and computational costs for a range of geometries for 1D chains of dimerized hydrogen, helium and lithium hydride with increasing number of momentum space grid points and also demonstrate VQE calculations for 2D and 3D hydrogen and helium lattices. Our presented strategies enable the use of near-term quantum hardware to perform solid state simulation with variational quantum algorithms.
Running quantum algorithms on real hardware is essential for understanding their strengths and limitations, especially in the noisy intermediate scale quantum (NISQ) era. Herein we focus on the practical aspect of quantum computational calculations of solid-state crystalline materials based on theory developed in our group by using real quantum hardware with noise mitigation techniques. We select two periodic systems with different level of complexity for these calculations. One of them is the distorted hydrogen chain as an example of very simple systems, and the other one is iron crystal in the BCC and FCC phases as it is considered to be inaccessible by using classical computational wavefunction methods. The ground state energies are evaluated based on the translational quantum subspace expansion (TransQSE) method for the hydrogen chain, and periodic boundary condition adapted VQE for our iron models. In addition to the usual state preparation and measurement noise mitigation, we apply a novel noise mitigation technique, which performs post-selection of shot counts based on Z2Z_{2}Z2​ and U1U_{1}U1​ symmetry verification. By applying these techniques for the simplest 2 qubit iron model systems, the energies obtained by the hardware calculations agree with those of the state-vector simulations within $\sim$5 kJ/mol. Although the quantum computational resources used for those experiments are still limited, the systematic resource reduction applied to obtain our simplified models will give us a way to scale up by rolling approximations back as quantum hardware matures.
Running quantum algorithms on real hardware is essential for understanding their strengths and limitations, especially in the noisy intermediate scale quantum (NISQ) era. Herein we focus on the practical aspect of quantum computational calculations of solid-state crystalline materials based on theory developed in our group by using real quantum hardware with a noise mitigation technique referred to as partition-measurement symmetry verification, which performs postselection of shot counts based on Z2{Z}_{2}Z2​ and U1{U}_{1}U1​ symmetry verification. We select two periodic systems with different levels of complexity for these calculations. One of them is the distorted hydrogen chain as an example of very simple systems, and the other one is iron crystal in the bcc and fcc phases as it is considered to be inaccessible by using classical computational wave-function methods. The ground-state energies are evaluated based on the translational quantum subspace expansion method for the hydrogen chain, and periodic boundary condition adapted variational quantum eigensolver for our iron models. By applying these techniques for the simplest two-qubit iron model systems, the correlation energies obtained by the hardware calculations agree with those of the state-vector simulations within $\ensuremath{\sim}5$ kJ/mol. Although the quantum computational resources used for those experiments are still limited, the techniques applied to obtain our simplified models will be applicable in essentially the same manner to more complicated cases as quantum hardware matures.
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.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4–8. With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H4-square, H4-chain, and H6-hexagon molecules, while the number of cnot gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient “score” parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using the H4 square and H4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.
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.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing.Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace.In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4-8.With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H 4 -square, H 4 -chain, and H 6 -hexagon molecules, while the number of CNOT gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35.Furthermore, we introduced an efficient "score" parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first.Using the H 4 square and H 4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.