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arXiv:2109.08401 · Physical Review Research

Quantum hardware calculations of periodic systems with partition-measurement symmetry verification: Simplified models of hydrogen chain and iron crystals

Kentaro Yamamoto, David Zsolt Manrique, Irfan T. Khan, Hideaki Sawada, David Muñoz Ramo

Published Aug 9, 2022• 24 citations
arXivHTMLPDF

Abstract

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 ${Z}_{2}$ and ${U}_{1}$ 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.

DOI
10.1103/physrevresearch.4.033110

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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∼5\ensuremath{\sim}5\ensuremath∼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.
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.
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.
Abstract Quantum computers promise to solve electronic structure, lattice dynamics, and strongly correlated problems beyond classical reach. However, near-term noisy intermediate scale quantum (NISQ) processors require systematic benchmarking to establish where variational algorithms with error mitigation can achieve chemical accuracy and cor rectly predict physical phenomena. Here we present 28 quantum material simula tions implemented with Qiskit, spanning molecular hydrogen (H 2 ) through the iron molybdenum cofactor (FeMoco), one-dimensional Hubbard chains, Kitaev honeycomb spin liquids, topological insulators (SSH model), BCS superconductors, and disordered Anderson insulators. Using the variational quantum eigensolver (VQE) with UCCSD ansatz, we achieve chemical accuracy (±1.6 mHa) for H 2 across the full dissociation curve and recover exact ground-state energies for H4 within 9 mHa after zero-noise extrapolation (ZNE) — reducing raw VQE errors from 152 mHa to 9 mHa. We accu rately identify the transverse-field Ising quantum critical point (h c = J) with 4 qubits, resolve Majorana Dirac dispersions in the Kitaev spin liquid, and reproduce the Mott transition in the Fermi-Hubbard model at U c = 6t. Resource estimation for FeMoco reveals O(N 2.5 ) scaling, projecting chemical accuracy with ∼200 logical qubits. Across all benchmarks, we quantify the accuracy-circuit depth tradeoff, demonstrating that depth-6 VQE ans¨atze achieve chemical accuracy for small molecules while depth-10 is required for correlated lattices. These results establish a comprehensive benchmark suite for evaluating quantum hardware, error mitigation protocols, and variational al gorithms for materials science on NISQ processors.
The Variational Quantum Eigensolver (VQE) is a promising quantum algorithm for applications in chemistry within the Noisy Intermediate-Scale Quantum (NISQ) era. The ability for a quantum computer to simulate electronic structures with high accuracy would have a profound impact on material and biochemical science with potential applications e.g., to the development of new drugs. However, considering the variety of quantum hardware architectures, it is still uncertain which hardware concept is most suited to execute the VQE for e.g., the simulation of molecules. Aspects to consider here are the required connectivity of the quantum circuit used, the size and the depth and thus the susceptibility to noise effects. Besides theo-retical considerations, empirical studies using available quantum hardware may help to clarify the question of which hardware technology might be better suited for a certain given application and algorithm. Going one step into this direction, within this work, we present results using the VQE for the simulation of the hydrogen molecule, comparing superconducting and ion trap quantum computers. The experiments are carried out with a standardized setup of ansatz and optimizer, selected to reduce the number of required iterations. The findings are analyzed considering different quantum processor types, calibration data as well as the depth and gate counts of the circuits required for the different hardware concepts after transpilation.
Variational quantum algorithms are among the most promising approaches for simulating interacting quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. However, the practical success of variational quantum eigensolvers (VQE) critically depends on the structure of the chosen variational ansatz. In this work, we investigate the ground-state properties of the one-dimensional antiferromagnetic Heisenberg spin-1/2 chain using both generic hardware-efficient ansatz and physics-informed, symmetry-preserving variational circuits. We benchmark variational results against exact diagonalization and noiseless simulations, and subsequently validate the approach on real IQM Garnet quantum hardware. Our results demonstrate that incorporating physical symmetries into the circuit design leads to significantly improved energy estimates, enhanced robustness against hardware noise, and clearer convergence behavior when compared to hardware-efficient ansatz under identical resource constraints. These findings highlight the importance of problem specific ansatz construction for reliable quantum simulations in the NISQ era.
molecule, emphasizing practical implementation and performance on current quantum hardware. Our research presents a comparative study of HEA and UCCSD ansätze on noiseless and noisy simulations and implements VQE on recent IBM quantum computer noise models and a real quantum computer, IBM Fez, providing a fully functional code employing Qiskit 1.2. Our experiments confirm UCCSD's reliability in ideal conditions, while the HEA demonstrates greater robustness to hardware noise, achieving chemical accuracy on state-vector simulation (SVS). The results reveal that achieving ground-state energy within chemical accuracy is feasible without error mitigation during VQE convergence. We demonstrate that current quantum devices effectively optimize circuit parameters despite misestimating simulated energies. The SVS-evaluated energies provide a more accurate representation of the solution quality compared to QPU-estimated energy values, indicating that VQE converges to the correct ground state despite quantum noise. Our study also applies noise mitigation as a postprocessing technique, using zero-noise extrapolation (ZNE) on a real quantum computer. The detailed methodologies presented in this study, including Hamiltonian construction and Fermionic-to-qubit transformations, facilitate flexible adaptation of the VQE approach for various algorithm variants and across different levels of algorithmic implementation.